
505 F.Supp. 224 (1980)
Joan Rance VUYANICH
v.
REPUBLIC NATIONAL BANK OF DALLAS.
Ellen JOHNSON
v.
REPUBLIC NATIONAL BANK OF DALLAS.
Nos. CA-3-6982-G, CA-3-7949-G.
United States District Court, N. D. Texas, Dallas Division.
October 22, 1980.
*225 *226 *227 *228 *229 JoAnn Peters, Anderson & Peters, Dallas, Tex., for Vuyanich.
Linda N. Coffee, Palmer, Palmer & Coffee, Dallas, Tex., for Johnson.
Richard L. Arnold, Tobolowsky & Schlinger, Dallas, Tex., for intervenor Fenton.
Wayne S. Bishop, Seyfarth, Shaw, Fairweather & Geraldson, Washington, D. C., for defendant.

MEMORANDUM ORDER
PATRICK E. HIGGINBOTHAM, District Judge.


                                 TABLE OF CONTENTS
     I.   History of Case                                               230
    II.   Class Reevaluation                                            233
          A.    "Across-the-Board" Suits                                234
          B.    EEOC Charges by Intervenors                             237
          C.    Class Redefinition                                      238
  III.    The Bank and Its History                                      242
   IV.    Bank Personnel Policy                                         243
          A.    Personnel Division and Hiring                           243
          B.    Affirmative Action                                      247
          C.    Categorization of Employees                             248
    V.    Statistical Evidence                                          251
          A.    Sources of the Data                                     251
          B.    Problems with the Data                                  255
                1.   Plaintiffs' Challenges                             256
                2.   The Bank's Challenges                              256
          C.    Technical and Institutional Competence                  258
          D.    The "Anecdotal Evidence"                                259
   VI.    The Theory Behind the Parties' Mathematical Modeling          261
          A.    Introduction                                            261
          B.    Job Relatedness and Equal Treatment                     262
          C.    Controlling for Productivity                            265
          D.    The Mathematics of Regression Analysis                  267
                1.   Uses of Multiple Regression                        267
                2.   Econometrics and the Ordinary
                     Least Squares Form of Multiple
                     Regression Analysis                                268
                     a.   Estimating Multiple Regressions               269
                     b.   Statistical Inference                         271
                3.   What Can Go Wrong?                                 273
          E.    Econometric Indication of Discrimination                275



*230
  VII.    Compensation                                                  279
          A.    The Legal Standard for Compensation Discrimination      279
          B.    Compensation Data                                       285
                1.   Plaintiffs' Models                                 285
                2.   The Bank's Models                                  299
          C.    Applying the Law to the Data                            304
          D.    Summary                                                 319
 VIII.    Initial Placement and Promotion Analysis                      319
          A.    The Data Presented                                      319
                1.   Plaintiffs' Data                                   320
                2.   The Bank's Data                                    331
          B.    Applying the Law to the Data                            338
                1.   Black and Female Nonexempts                        339
                2.   Female Exempts                                     342
                3.   Black Exempts                                      343
          C.    Summary                                                 344
   IX.    Hiring                                                        344
          A.    Statistical Inference                                   345
          B.    Plaintiffs' Case: The Data                              350
          C.    Plaintiffs' Case: The Legal Standards                   354
          D.    Plaintiffs' Case: Applying the Law to the Data          357
          E.    The Bank's Case: The Data                               362
          F.    The Bank's Case: The Legal Standards                    369
                1.   Occuptional and Educational Weights                369
                2.   Geographical Weights                               375
          G.    The Bank's Case: Applying the Law to the Data           376
          H.    Summary                                                 385
    X.    Terminations                                                  385
   XI.    Terms and Conditions                                          386
          A.    Departmental Segregation                                386
          B.    Maternity Leave                                         389
          C.    Marriage Policy                                         392
          D.    Training                                                392
          E.    Dress                                                   393
          F.    Summary                                                 393
  CONCLUSION                                                            394

This order constitutes the court's findings of fact and conclusions of law entered after a five-week trial to the court on the Phase I liability issues of a class action race and sex discrimination case under Title VII of the Civil Rights Act of 1964, 42 U.S.C. งง 2000e et seq. ("Title VII").[1] For the sake of clarity, and so that this opinion may stand as a self-contained unit, the facts and procedural history of the case will be set forth in full.

I. History of the Case

Plaintiff Joan Rance Vuyanich, a black female, was first employed with defendant Republic National Bank ("the Bank" or "Republic") on April 28, 1969, as an agent contact clerk in the Money Order Department. She was then the only black employee of her department. Soon after her arrival, she began to have problems with two white female co-workers, whom Ms. Vuyanich believed to be shouldering less than a fair share of the workload. Complaints to her supervisor resulted (in her view) in only temporary improvement. On June 29, 1969, Ms. Vuyanich married a white male whom she had met during previous employment. Her supervisors first met her new husband *231 approximately one month later. A few days after this introduction, Ms. Vuyanich was called to her supervisor's office and told that there was a clash of personalities between herself and her co-workers, that the complaints about the workload were her fault, that she was not suitable for the job, and that she should resign. When asked about a transfer, her supervisor replied that Ms. Vuyanich probably did not need a job anymore since her husband was white.[2] Ms. Vuyanich was discharged from the Bank on July 28, 1969.
On August 13, 1969, Ms. Vuyanich filed a charge against the Bank with the Austin Regional Office of the Equal Employment Opportunity Commission ("EEOC"), setting forth the above events and charging the Bank with violation of Title VII. On May 23, 1972, the Dallas District Director of the EEOC issued his findings of fact, in which he found, inter alia, that Vuyanich's supervisor had discharged her instead of taking other steps to resolve the racially motivated personality conflict between her and her co-workers. A determination of reasonable cause to believe that a violation of Title VII had occurred was issued on November 6, 1972. Conciliation efforts were unsuccessful, and on March 6, 1973, the EEOC issued a statutory right-to-sue letter. On March 22, 1973, three years and eight months after her discharge, Ms. Vuyanich filed the first of these two consolidated actions.
Plaintiff Ellen Johnson, a black female, applied for a job at the Bank on September 23, 1971. Ms. Johnson was a 1971 graduate of the University of Texas at Arlington with a major in government. She first applied for a position as a management trainee or in personnel administration, but was told that no such positions were available. She then expressed her willingness to accept any position available; she was not offered a position of any kind.[3]
On October 15, 1971, Ms. Johnson filed a discrimination charge with the Dallas District Office of the EEOC. This charge alleged across-the-board race and sex discrimination by the Bank with respect to hiring, recruiting, job requirements, training, promotion, and personnel rules. On August 14, 1973, the District Director issued a determination of reasonable cause with respect to most of these allegations. On November 1, 1973, the EEOC issued a right-to-sue letter, and on December 3, 1973, the second of the present actions was filed.
The procedural history of these cases is complex, the Vuyanich case having been assigned at one time or another to five different judges of this court. The first significant event in the cases took place on November 7, 1974, when Judge Mahon conditionally certified the Vuyanich case as a class action on behalf of female and black employees and potential employees of the Bank who were adversely affected by the practices alleged by Ms. Vuyanich. Except for sporadic discovery activities, the cases remained dormant until March 12, 1976. On that date, Judge Mahon consolidated the Vuyanich and Johnson cases for discovery purposes. Judge Mahon also granted plaintiffs' motions to strike the Bank's jury demand, basing his decision on Curtis v. Loether, 415 U.S. 189, 94 S.Ct. 1005, 39 L.Ed.2d 260 (1974), and Johnson v. Georgia Highway Express, Inc., 417 F.2d 1122 (5th Cir. 1969). At the same time, he denied the Bank's motion to strike Ms. Vuyanich's allegations of sex discrimination, holding her EEOC charge sufficient under Sanchez v. Standard Brands, Inc., 431 F.2d 455 (5th Cir. 1970), and Gamble v. Birmingham Southern Railroad, 514 F.2d 678 (5th Cir. 1975), to support claims both of race and sex discrimination. Vuyanich, supra n.1, 409 F.Supp. at 1086-90.
On March 15, 1978, the cases having been transferred to the present judge, the class status of the cases was updated and redefined. *232 Based on statistical evidence presented at a two-day hearing, the court certified a class consisting of:
All females of all races and all blacks of either sex; 1) who are or have been employed by the Republic National Bank on or after February 16, 1969, and 2) who applied for employment but were not hired at the Republic National Bank on or after February 16, 1969 to date.
Vuyanich, supra n.1, 78 F.R.D. at 354.[4] At the same time, the court consolidated the cases for all purposes. Trial of the so-called "Phase I liability" issues, see Swint v. Pullman-Standard, Inc., 539 F.2d 77, 94 (5th Cir. 1976); Baxter v. Savannah Sugar Refining Corp., 495 F.2d 437, 443-44 (5th Cir. 1974), cert. denied, 419 U.S. 1033, 95 S.Ct. 515, 42 L.Ed.2d 308 (1975), was severed from trial of individual damage issues.
In early 1979, the court conducted a further hearing for the purpose of refining the class definition. Both sides presented sophisticated statistical analyses in support of their respective positions. On April 25, 1979, the court reaffirmed its earlier class certification order, dividing the original class into five subclasses.[5] The court simultaneously approved requests for designation of three class members as additional class representatives. These three new class representatives have been referred to as intervenors.[6]
*233 The five certified subclasses are as follows:


                                        Subclass                Subclass
          Subclass                      Representative(s)       Attorney(s)
black and female exempt                 Ellen Johnson           Linda Coffee with
employees                               Marjorie Lee Jackson    Joann Peters
female nonexempt employees[7]        Marisu Fenton           Richard Arnold
black nonexempt employees[7]         Joan Vuyanich           Joann Peters with
                                        Dorothy Hooks           Linda Coffee
unsuccessful black and female           Ellen Johnson           Linda Coffee with
applicants for exempt positions                                 Joann Peters
unsuccessful black applicants           Ellen Johnson           Linda Coffee with
for nonexempt positions                                         Joann Peters

The employee subclasses include employees who have worked for the Bank during the period from February 16, 1969, to the date of trial. The applicant subclasses include those who applied for positions at the Bank during the period from February 16, 1969, to the date of trial.[8]
The Phase I trial commenced on October 15, 1979. In the course of 24 days of testimony, the court heard from over three dozen witnesses and received thousands of exhibits into evidence. Ten of these witnesses were experts in the fields of computer science, statistics, business, or economics.[9] The testimony of these experts, together with their written reports, forms the evidentiary base for the statistical analyses at the heart of plaintiffs' case and the Bank's rebuttal of that case.

II. Class Reevaluation

Before turning to substantive issues, we deal with posttrial challenges by the Bank to the status of this case as a class action. Although the issues of class certification have already been considered repeatedly and exhaustively, see n.1, supra, the court has a continuing duty to reevaluate class status on the basis of events which transpire and circumstances which develop as the litigation unfolds. E. g., Guerine v. J & W Investment, Inc., 544 F.2d 863, 864 (5th Cir. 1977); Cooper v. University of *234 Texas, 482 F.Supp. 187, 190 (N.D.Tex.1979), appeal docketed, No. 80-1412 (5th Cir. Apr. 15, 1980). This reevaluation process is complex, and will be discussed in detail in due course. The spokes of all the Bank's challenges to the class as it is now structured run in the final analysis to a central hub. That hub is an attack on the maintainability of so-called "across-the-board" suits. We turn then to their role in the evolving framework of Title VII class action litigation.

A. "Across-the-Board" Suits.

The concept of an "across-the-board" class action, i. e., an action challenging a wide range of employment practices alleged to result from a common discriminatory animus, owes its origin to Johnson v. Georgia Highway Express, Inc., 417 F.2d 1122 (5th Cir. 1969). The Fifth Circuit in Johnson reversed a ruling by the trial court that a discharged black employee could only represent other discharged black employees. The court found such a limitation to be error "as it is clear from the pleadings that the scope of appellant's suit is an `across-the-board' attack on unequal employment practices alleged to have been committed by the appellee pursuant to its policy of racial discrimination." 417 F.2d at 1124. Noting the "Damoclean threat of a racially discriminatory policy," Hall v. Werthan Bag Corp., 251 F.Supp. 184, 186 (N.D.Tenn.1966), hanging over the entire racial group, the court held that the plaintiff could represent all black employees of the defendant in their claims of alleged discrimination in hiring, firing, promotion, and maintenance of facilities. 417 F.2d at 1124.
The Johnson decision received continued support in Fifth Circuit Title VII jurisprudence. Thus in Carr v. Conoco Plastics, Inc., 423 F.2d 57 (5th Cir.), cert. denied, 400 U.S. 951, 91 S.Ct. 241, 27 L.Ed.2d 257 (1970), a group of black employees was permitted to maintain an "across-the-board" suit challenging hiring and internal personnel policies. In Jack v. American Linen Supply Co., 498 F.2d 122 (5th Cir. 1974), the court held that a former employee could properly represent a class of present and future employees. Similarly, in Long v. Sapp, 502 F.2d 34 (5th Cir. 1974), the court, reaffirming Johnson, held that a terminated employee could challenge racially discriminatory policies allegedly "pervad[ing] all aspects of the employment practices of" her former employer:
Having shown herself to be a black and a former employee ... she occupies the position of one she says is suffering from the alleged discrimination. She has demonstrated the necessary nexus with the proposed class for membership therein. As a person aggrieved, she can represent other victims of the same policies, whether or not all have experienced discrimination in the same way.
502 F.2d at 43.
The Fifth Circuit approach, characterized as "pioneering," Worley v. Western Electric Co., 22 Empl.Prac.Dec. ถ 30,600, at 14,224-25 (N.D.Ga.1979), received widespread but not universal support in other circuits. The "across-the-board" approach was recognized and approved by the Third Circuit, Wetzel v. Liberty Mutual Insurance Co., 508 F.2d 239, 247 (3d Cir.), cert. denied, 421 U.S. 1011, 95 S.Ct. 2415, 44 L.Ed.2d 679 (1975), the Fourth Circuit, Barnett v. W. T. Grant Co., 518 F.2d 543, 547-48 (4th Cir. 1975), the Sixth Circuit, Senter v. General Motors Corp., 532 F.2d 511, 523-24 (6th Cir.), cert. denied, 429 U.S. 870, 97 S.Ct. 182, 50 L.Ed.2d 150 (1976); Tipler v. E. I. duPont deNemours & Co., 443 F.2d 125, 130 (6th Cir. 1971), the Eighth Circuit, Donaldson v. Pillsbury Co., 554 F.2d 825, 829-32 (8th Cir.), cert. denied, 434 U.S. 856, 98 S.Ct. 177, 54 L.Ed.2d 128 (1976); Reed v. Arlington Hotel Co., 476 F.2d 721, 722-23 (8th Cir.), cert. denied, 414 U.S. 854, 94 S.Ct. 153, 38 L.Ed.2d 103 (1973); Parham v. Southwestern Bell Telephone Co., 433 F.2d 421, 425 (8th Cir. 1970), and a variety of district courts, e. g., Mack v. General Electric Co., 329 F.Supp. 72, 75-76 (E.D.Pa.1971); Wilson v. Monsanto Co., 315 F.Supp. 977, 979 (E.D.La.1970); see Hall v. Werthan Bag Corp., supra ("across-the-board" action as to injunctive relief but not as to back pay). See generally Developments in the Law-Employment *235 Discrimination and Title VII of the Civil Rights Act of 1964, 84 Harv.L. Rev. 1109, 1219-21 (1971). The "across-the-board" approach was rejected in the Tenth Circuit, Taylor v. Safeway Stores, Inc., 524 F.2d 263, 270-71 (10th Cir. 1975), and by a number of district courts, e. g., Tolbert v. Daniel Construction Co., 332 F.Supp. 772, 775 (D.S.C.1971); White v. Gates Rubber Co., 53 F.R.D. 412, 413 (D.Colo.1971); Hyatt v. United Aircraft Corp., 50 F.R.D. 242, 245-47 (D.Conn.1970); Burney v. North American Rockwell Corp., 302 F.Supp. 86, 90-91 (C.D.Cal.1969); Colbert v. H-K Corp., 295 F.Supp. 1091, 1093 (N.D.Ga.1968), vacated on other grounds, 444 F.2d 1381 (5th Cir. 1971).
The continuing propriety of "across-the-board" class actions under Title VII was called into question in 1977 by East Texas Motor Freight Systems, Inc. v. Rodriguez, 431 U.S. 395, 97 S.Ct. 1891, 52 L.Ed.2d 453 (1977). In that case, the district court had denied certification of an "across-the-board" suit, and had rejected the named plaintiffs' claims on their merits after trial. The Fifth Circuit reversed, certifying a class, and itself found classwide liability on the basis of the trial record. The Supreme Court in turn reversed the court of appeals, holding, based upon the failure of the named plaintiffs' claims, their failure to move for class certification, and their conflicts of interest with members of the class, that the named plaintiffs were not proper class representatives. In this regard, the Court stated:
We are not unaware that suits alleging racial or ethnic discrimination are often by their very nature class suits, involving classwide wrongs. Common questions of law or fact are typically present. But careful attention to the requirements of Fed.Rule Civ.Proc. 23 remain nonetheless indispensable. The mere fact that a complaint alleges racial or ethnic discrimination does not in itself ensure the party who has brought the lawsuit will be an adequate representative of those who may have been the real victims of that discrimination.
431 U.S. at 405-06, 97 S.Ct. at 1897-1898. Citing Schlesinger v. Reservists Committee to Stop the War, 418 U.S. 208, 216, 94 S.Ct. 2925, 2929, 41 L.Ed.2d 706 (1974), the Court noted that "a class representative must be part of the class and `possess the same interests and suffer the same injury' as the class members." 431 U.S. at 403, 97 S.Ct. at 1896.
Mirroring the pre-Rodriguez dissension among the circuits, courts are divided on the question of whether "across-the-board" suits survive Rodriguez. In Payne v. Travenol Laboratories, Inc., 565 F.2d 895, 900 (5th Cir.), cert. denied, 439 U.S. 835, 99 S.Ct. 118, 58 L.Ed.2d 131 (1978), the Fifth Circuit, without discussing Rodriguez, stated:
Plaintiffs' action is an "across-the-board" attack on unequal employment practices alleged to have been committed by Travenol pursuant to a policy of racial discrimination. As parties who have allegedly been aggrieved by some of those discriminatory practices, plaintiffs have demonstrated a sufficient nexus to enable them to represent other class members suffering from different practices motivated by the same policies.
In Satterwhite v. City of Greenville, supra n.7, the court again stated:
Nor is Rodriguez or this opinion contrary to the policy favoring "across-the-board" Title VII class actions. See Johnson v. Ga. Highway Express, supra. It is not necessary that the representative suffer discrimination in the same way as other class members, but it is necessary that she suffer from the discrimination in some respect.
578 F.2d at 993-94 n.8. Finally, in Falcon v. General Telephone Co., 626 F.2d 369 (5th Cir. 1980), the court settled any remaining doubt, holding squarely that "an employee complaining of one employment practice [may] represent another complaining of another practice, if the plaintiff and the members of the class suffer from essentially the same injury." The Fourth Circuit has reached a contrary result, Hill v. Western Electric Co., 596 F.2d 99 (4th Cir.), cert. denied, 444 U.S. 929, 100 S.Ct. 271, 62 *236 L.Ed.2d 186 (1979); see Vuyanich, supra n.1, 82 F.R.D. at 433 n.8, and district courts are split on the issue.[10]
The Payne, Satterwhite and Falcon cases are controlling precedent in this circuit for the proposition that "across-the-board" suits remain appropriate in proper circumstances notwithstanding Rodriguez. Even were the slate clean, however, the factors favoring "across-the-board" suits identified by this court in its earlier order, see Vuyanich, supra n.1, 82 F.R.D. at 432-33, would remain cogent today. The fundamental thesis of "across-the-board" actions remains the existence of a discriminatory animus cutting across a variety of employment practices; where there is substantial evidence that such an animus exists, the class representative "possesses the same interests" in its elimination and "suffers the same injury" from its presence, within the meaning of Rodriguez, as the class members she represents.[11] Whether the direct approach typical of the Fifth Circuit, in which a broad class is certified, or the two-step approach of the Third Circuit, where a narrower class is certified but broader issues examined, see Alexander v. Gino's, Inc., supra n.10, is employed, an "across-the-board" suit serves to vindicate classwide rights on a classwide basis.
Notwithstanding the propriety of "across-the-board" actions in appropriate cases, it must be kept in mind that not every Title VII suit, nor even every Title VII class action, is an "across-the-board" suit. Only where the plaintiff alleges that the particular discrimination she has suffered is due to a racist or sexist animus pervading the defendant's practices, and only where the plaintiff presents sufficient proof of that allegation at the class certification stage, should an action be certified as an "across-the-board" class action. See Falcon v. General Telephone Co., supra, at 376 n.9. The thrust of this court's earlier opinion was that the plaintiffs in this case have met that burden. At the class certification stage, the plaintiffs presented statistical evidence showing discrimination in hiring, pay, promotion, and termination. These statistics constituted substantial evidence that discrimination at the Bank, if indeed such discrimination were provable at trial, was the product of a common discriminatory *237 animus characteristic of "across-the-board" actions. It was on that basis that this action was so certified.[12]
One additional factor, not fully present at the time of the court's 1979 certification order, counsels in favor of "across-the-board" treatment for this case. The court noted at that time that challenged practices such as promotion, pay, training, testing, transfer, job assignment and classification, job content, and constructive discharge were intertwined. Vuyanich, supra n.1, 82 F.R.D. at 433. What was not then apparent, but which now reinforces the decision to treat this as an "across-the-board" suit, was the fact that much of the statistical proof presented at trial would mirror this intertwined relationship. The introduction of multiple regression econometric studies, discussed more fully in sections VI through VIII, infra, which can in certain circumstances measure discrimination in a wide variety of forms through related statistical tests,[13] suggests that "across-the-board" treatment, far from fragmenting the proof adduced at trial, may in fact be a preferred method for testing for discrimination by an employer with a large diverse work force.

B. EEOC Charges by Intervenors.

The Bank next challenges this court's earlier holding, see Vuyanich, supra n.1, 82 F.R.D. at 437-38, that Marisu Fenton, Dorothy Hooks, and Marjorie Jackson could intervene as the representatives of certain subclasses despite having failed to file EEOC charges. Citing Hodge v. McLean Trucking Co., 607 F.2d 1118 (5th Cir. 1979), it asserts that the case of Wheeler v. American Home Products Corp., 563 F.2d 1233 (5th Cir. 1977), on which this court's holding was based, has been overruled. This argument must fail for two reasons. First, one panel of the Fifth Circuit has no power to overrule a decision of a previous panel. E. g., Ford v. United States, 618 F.2d 357, 361 (5th Cir. 1980); Gates v. Collier, 616 F.2d 1268, 1272 (5th Cir. 1980). Second, even were a panel possessed of such a power, examination of Hodge reveals it to be fully consistent with, if not dictated by, the Wheeler doctrine.
The Fifth Circuit in Oatis v. Crown Zellerbach Corp., 398 F.2d 496 (5th Cir. 1968), first established the rule that unnamed plaintiffs in a Title VII class action need not have exhausted administrative remedies by filing a discrimination charge with the EEOC. The court in that case reasoned that it would "be wasteful, if not vain, for numerous employees, all with the same grievance, to have to process many identical complaints with the EEOC." 398 F.2d at 498. Thus it held that the EEOC complaint of the named class representative was sufficient to support an action on behalf of the entire class.
This rule was extended to intervenors in Wheeler v. American Home Products Corp., supra, an action in which class status had been denied. See Romasanta v. United Airlines, Inc., 537 F.2d 915, 919 n.7 (7th Cir. 1976), aff'd sub nom. United Airlines, Inc. v. McDonald, 432 U.S. 385, 97 S.Ct. 24, 64, 52 L.Ed.2d 423 (1977). The Oatis and Wheeler cases both rely on the EEOC complaint of the original plaintiff to stand in lieu of EEOC complaints by others. Hence it is not surprising that intervention was denied *238 in Hodge, where the original plaintiff had failed to file an effective EEOC complaint, since in that case the intervenors could not rely on the original plaintiff's charge. This being the case, Hodge cannot be interpreted as altering the long-standing rule in the Fifth Circuit that intervenors need not exhaust EEOC remedies.
Finally, the Bank argues that there must be an independent basis of jurisdiction for each subclass, i. e., that for each subclass there must be a representative who has filed an EEOC charge.[14] The Bank argues that this requirement is a necessary concomitant of an asserted requirement under Fed.R.Civ.P. 23(c)(4)(B) that each subclass meet all the requirements of Rule 23. See Monarch Asphalt Sales Co. v. Wilshire Oil Co., 511 F.2d 1073, 1077 (10th Cir. 1975); Weathers v. Peters Realty Corp., 499 F.2d 1197, 1200 (6th Cir. 1974); cf. Chmieleski v. City Products Corp., 71 F.R.D. 118, 150 (W.D.Mo.1976) (numerosity). Assuming arguendo that subclasses must always satisfy each requirement of Rule 23, it does not follow that an independent jurisdictional base is required for each subclass. The ratio decidendi of Oatis and Wheeler was that the named plaintiff's EEOC charge will be typical of those which would have been filed by those she represents. Subclassing was undertaken in this case for the dual purposes of facilitating presentation of issues, Vuyanich, supra n.1, 82 F.R.D. at 433, and eliminating potential conflicts of interest, id. at 435. Since the court found plaintiffs' claims to be an "across-the-board" attack on discriminatory practices, typicality, while serving as a convenient basis for separation of subclasses, was not a factor compelling subclassing. Otherwise stated, the entire class, notwithstanding subclassing, continues to complain of "across-the-board" discrimination. Hence, just as absent class members in an "across-the-board" suit may rely on the named plaintiff's charge, Oatis, supra, and just as intervenors who assert claims common to those asserted by existing plaintiffs may travel on those plaintiffs' complaints, Wheeler, supra, the subclass representatives may rely on the EEOC charges of plaintiffs Vuyanich and Johnson.[15]See Vuyanich, supra n.1, 82 F.R.D. at 437-38.

C. Class Redefinition.

This court has recognized its "continuing duty to monitor and modify the class according to the facts that develop." Cooper v. University of Texas, supra, at 190. Behind this seemingly simple phrase, however, lies considerable complexity. Throughout the evaluation process, the legal standards remain the same: the case must satisfy the four prerequisites of Fed.R.Civ.P. 23(a), and must fit into one or more of the categories of Fed.R.Civ.P. 23(b). Nonetheless, the extent to which a determination that these requirements have been satisfied may be reexamined during the course of the litigation must vary from requirement to requirement and must depend on the procedural posture of the case. Just as class certification is not a ritual exercise which once done may be laid to one side, class reevaluation cannot depend on a formalistic invocation of the five class action requirements.
The four prerequisites of Fed.R.Civ.P. 23(a) serve to protect the differing and *239 occasionally antagonistic interests of the named parties, the unnamed class members, and the court. The numerosity requirement of Fed.R.Civ.P. 23(a)(1), for example, protects the interests of absent class members, who might otherwise be unnecessarily deprived of the right to control their own litigation, and those of the court, "in assuring a full and fair exposition of views by all affected parties when it is practicable to join them in a single proceeding." Scott v. University of Delaware, 601 F.2d 76, 88 (3d Cir.), cert. denied, 444 U.S. 931, 100 S.Ct. 275, 62 L.Ed.2d 189 (1979). The commonality requirement of Fed.R.Civ.P. 23(a)(2) serves to focus the issues, protecting both the court and the defendant by ensuring that resources are not wasted through inquiry into a multitude of diverse controversies. The adequacy of representation test of Fed.R.Civ.P. 23(a)(4) is for the primary benefit of absent class members, whose rights might otherwise be adjudicated in a binding fashion without a complete presentation of the facts and legal arguments supporting their contentions. See generally C. Wright & A. Miller, Federal Practice and Procedure ง 1765 (1972). The typicality requirement of Fed.R.Civ.P. 23(a)(3) has been characterized as a "double schizophrenic." 3B Moore's Federal Practice ถ 23.06-2, at 23-192 (2d ed. 1980). It serves both the class and the court, complementing the commonality requirement as well as providing an intuitive check on the court's determination that the class representative will adequately protect the rights of absent class members. The differing nature of these purposes suggests that differing levels of inquiry apply to each of the requirements as the litigation evolves.
At the outset of the case, all four requirements must be satisfied.[16] Fed.R. Civ.P. 23(c)(1) requires the court, "[a]s soon as practicable after the commencement of" the action, to determine whether it may be maintained as a class action. The standard of proof for such a determination is, however, ill-defined. As the court stated in its earlier order with regard to commonality:
Of course a plaintiff must do more to demonstrate the existence of the question than simply assert its existence. Bare bones conclusions are insufficient. At the same time, a plaintiff need not make out a prima facie case of liability. The higher courts have not yet articulated where between these marks a plaintiff must place his proof.
Vuyanich, supra n.1, 82 F.R.D. at 431. Similar uncertainty exists as to how far a plaintiff must go in proving numerosity, adequacy, and typicality. We do know that a plaintiff need not show at the class certification stage that she has a winning individual claim as a sine qua non to the typicality of her claim or the adequacy of her representation. Huff v. N. D. Cass Co., 485 F.2d 710 (5th Cir. 1973) (en banc). At this early stage, the typicality requirement will commonly be met by a showing that the issues which will likely be raised at trial by the named plaintiff are typical of those of the class, and the adequacy requirement by a demonstration that plaintiff will likely be an adequate representative at trial due to her incentive and ability to represent the class and the absence of conflicts of interest between herself and the class.
Fed.R.Civ.P. 23(c)(1) also authorizes the court to alter or amend a class certification order as the litigation progresses. Hence if pretrial activity demonstrates that a case is not appropriate for class treatment, the district court should decertify the action. Lamphere v. Brown University, 553 F.2d 714, 720 (1st Cir. 1977). Considerations such as this led this court to decertify the portion of the class composed of unsuccessful female applicants for non-exempt positions. Vuyanich, supra n.1, 82 F.R.D. at 438. As discovery progresses, it is reasonable to hold the plaintiff to a higher *240 standard of satisfaction of such requirements as numerosity and commonality, as to which more precise proof will become available. At the same time, reliance on incentive, ability, lack of conflict, and typicality as measures of adequate representation, while still present, will give way to an evaluation of the extent to which the plaintiff's litigation performance demonstrates adequacy. Thus if it becomes apparent that adequate representation is not being provided, the court must withdraw class status from the suit. Guerine v. J & W Investments, supra, at 864-65.
The duty to continually reevaluate class status does not end with the commencement of trial. Because, however, by the end of trial the court's interests in efficiency and manageability have for better or for worse been realized or frustrated, the focus of reevaluation at this stage must be on the factors protecting the parties, named and unnamed. Principal among these is adequacy of representation, and the court must not hesitate to decertify a class in whole or in part if the plaintiff has failed to present at least minimal evidence, or has otherwise demonstrated that her representation of all or part of the class is less than adequate. Cooper v. University of Texas, supra, at 198. Likewise, if the court finds that the numerosity requirement, which is based in part on protection of the absent class members' right of autonomy, is not satisfied, the court must decertify. Scott v. University of Delaware, supra, at 88-89. By this stage of the proceedings, the emphasis is on actual rather than predictive measures of adequacy and numerosity: the proof has presumably been fully developed, and the court may evaluate plaintiff's actual adequacy at trial.
The role of typicality in the post-trial class reevaluation scenario is uncertain. To the extent that typicality mirrors the commonality requirement, whose role is diminished once the trial is concluded, its role will also be diminished. To the extent that typicality serves as a predictor of adequacy of representation, its importance is overshadowed by the more objective measures of adequacy which become available after prolonged observation by the court of the conduct of plaintiff and her counsel. While the typicality requirement does serve a residual role of providing a check on the accuracy of that evaluation, that role is limited: unless the named plaintiff's claim appears at trial to be so atypical of the those of the class that the adequacy of her representation is drawn into question, the court should not decertify the class merely because the individual plaintiff's proof differs from that presented on behalf of the class. This is especially the case where, as here, consideration of the named plaintiff's individual claim has been severed from trial of class issues.[17]
The Bank argues that there must be a "continuing nexus" at all stages of the litigation between the named class representation and the class members she represents. While this assertion is undoubtedly correct in the abstract, the "continuing nexus" test cannot be mechanically applied without regard to the procedural posture of the case. In each of the cases cited by the Bank, an appellate court held that a class representative whose individual claim has failed cannot continue to represent the class, either on appeal or on remand following appeal. East Texas Motor Freight Systems, Inc. v. Rodriguez, supra; Armour v. City of Anniston, 597 F.2d 46 (5th Cir. 1979), vacated, 445 U.S. 940, 100 S.Ct. 1334, 63 L.Ed.2d 774, remanded, 622 F.2d 1226 (5th Cir. 1980); Davis v. Roadway Express, Inc., 590 F.2d 140 (5th Cir. 1979), on rehearing, 621 F.2d 775 (5th Cir. 1980) (reaffirmed, but on other grounds than in earlier opinion); Camper v. *241 Calumet Petrochemicals, Inc., 584 F.2d 70 (5th Cir. 1978); Satterwhite v. City of Greenville, supra n.8.[18] It is a far different matter, however, to hold that the failure of a named plaintiff's claim at trial requires the retroactive decertification of the class and consequent failure of the class claims.[19]
This distinction is recognized by the "continuing nexus" cases themselves. In Rodriguez, the progenitor of this line of cases, the Court stated:
Obviously, a different case would be presented if the District Court had certified a class and only later had it appeared that the named plaintiffs were not class members or were otherwise inappropriate class representatives. In such a case, the class claims would have already been tried, and, provided the initial certification was proper and decertification not appropriate, the claims of the class members would not need to be mooted or destroyed because subsequent events or the proof at trial had undermined the named plaintiffs' individual claims. [Citations]. Where no class has been certified, however, and the class claims remain to be tried, the decision whether the named plaintiff should represent a class is appropriately made on the full record, including the facts developed at the trial of the plaintiffs' individual claims.
431 U.S. at 406 n.12, 97 S.Ct. at 1898 n.12.[20] The Fifth Circuit in Satterwhite outlined the reasons underlying this distinction:
Where a class is certified, and class claims tried, before the lack of merit or mootness of the representative's claim is discovered, the class representative has already assiduously asserted the claims of the constituents. The conservation of both litigants' and judicial resources makes it desirable not only to avoid abortion of the litigation but also to prevent prejudice to the members of a certified class who, in the midst of a law suit, suddenly discover that their representative's claim is no longer viable.
578 F.2d at 994. Accord, Drayton v. City of St. Petersburg, 477 F.Supp. 846, 857 n.19 (M.D.Fla.1979). Indeed, in addition to wasting the resources of the parties and the court and frustrating the reasonable reliance of absent class members, decertification on the basis of failure of the individual plaintiff's claim would in many cases disserve the defendant, by depriving it of an adjudication of nonliability binding on the class.
It remains only to apply these standards of class reevaluation to the present case. The Bank makes no argument that the numerosity and commonality requirements are no longer satisfied, and indeed no such argument could be made given the state of the record. The course of the trial has revealed no continuing pattern of inadequacy of representation warranting decertification. Decertification of particular subgroups within the class and subclasses involves an evaluation of whether the plaintiffs have produced at least minimal evidence on the issues applicable to those subgroups, and is best deferred until consideration and evaluation of those parts of the *242 plaintiffs' case. As will be seen, no decertification is warranted.
Finally, the testimony of the named class representatives reveals their claims to be sufficiently typical that no substantial doubts are raised as to the adequacy of their representation. The testimony of plaintiffs Vuyanich and Johnson has already been described in section I, supra, and will not be repeated here. The testimony of intervenors Jackson, Fenton, and Hooks at trial mirrored that given by them during the class certification hearing and summarized in n.6, supra. The wide variety of discriminatory practices testified to by these individuals reinforces the court's earlier determination that the essence of their claims was the presence of a racially and sexually discriminatory animus pervading the Bank's personnel practices. Whether this animus in fact existed is a question which must await more detailed evaluation of the statistical and other evidence presented at trial. The court is convinced at this juncture, however, that the claims of the named plaintiffs and intervenors, whether meritorious or not, are typical of those asserted on behalf of the class.

III. The Bank and Its History[21]
With 2,400 full-time employees and assets exceeding $8 billion, Republic National Bank is among the 25 largest banks in the United States and is the largest bank in the South. Through its sister companies, its corporate influence is further extended, as it is the "flagship" bank of the Republic of Texas Corporation, a bank holding company incorporated under the laws of Texas with some 30 wholly-owned subsidiaries. The Bank's principal business is commercial lending. Republic lends to consumers as well as businesses, but consumer lending accounts for not more than 5% of its loan portfolio. Indeed, Republic is the third largest nonretail unit bank in the United States.
The "line" function of commercial lending is carried out by the Bank's five commercial banking departments. The Banking Department, consisting of the Cash Management, Commodity, Metropolitan, Southwestern, and National Divisions, provides business loans and financial consultation to the customers of those divisions' respective geographic regions. Loans to individuals are provided by the Consumer Lending Group of the Metropolitan Division. Loans to other banks are generated by the Correspondent Banking Department, while the Real Estate Department offers single mortgage lending services and interim financing for the building and construction industry. The Petroleum and Minerals Department makes loans to companies in the petroleum and related minerals industry, and the International Department provides financial assistance to United States based companies who wish to deal in foreign markets, in addition to financial service to those markets.
Other departments in the Bank include the Trust and Investment Department, the Finance and Credit Administration Department, the Operations Department, and the Funds Management Department. The Trust and Investment Department is responsible for managing money and other assets for individuals and organizations, according to specified conditions set out in trust agreements. The department consists of the following divisions: Securities Management Services, Personal Services, Operations Services, Corporate Services, Legal Counsel, Taxes, and Business in Trust.
The Funds Management Department is responsible for making investment judgments affecting the flow of funds in and out of the Bank. The objective of this department is to ensure that surplus funds are properly invested to obtain maximum return, and that funds needed to meet the Bank's money commitments are secured at the lowest cost. This department also houses the Bank's municipal bonds and sales and *243 trading operation, as well as bond portfolio management.
The Finance and Credit Administration Department is responsible for the day-to-day business of the Bank. This department includes the following divisions: Controller, Personnel, Credit and Corporate Finance, and Loan Review and Special Loans. The Operations Department is responsible for all of the Bank's banking customer services functions and data processing, programming, production, and development. Other staff divisions at the Bank include Corporate Planning, Legal Counsel, Marketing and Public Affairs, and Economic Research, all of which report directly to the President, and the Audit Division, which reports to the Chairman of the Board.
The Bank's Personnel Division is responsible for coordinating and administering the Bank's equal employment and personnel policies, including hiring, promotion, compensation and benefits, counseling, discipline, and termination. These responsibilities are carried out by three group managers. The Personnel Group, headed by Vice President Thomas E. Barksdale, has responsibility for the salary administration section and the payroll and benefits section. The Personal Development Group, led by Vice President Dan White, coordinates Bank employee participation in external employee development programs, and the development and presentation of internal programs leading to personal development. The Employment Group is the responsibility of Jerry M. Watson, Vice President and Manager of Interview and Selection. This group carries out the Bank's staffing and recruiting functions. Watson also has responsibilities relating to affirmative action, internal employee transfers, terminations, counseling, and record keeping. Each group manager reports to Thomas Croft, Senior Vice President and Director of Personnel, who in turn reports to the Executive Vice President and Manager of the Bank's Finance and Administration Department.
The Bank's structure thus reflects independent staff divisions serving the core lending function, including economic research, legal counsel, marketing, and public affairs. Over the decade spanned by this lawsuit, that structure has been staffed with a work force that has ranged in size from a low of 1,450 employees to a high of approximately 2,400. Over this time the internal lines of the structure have occasionally shifted, with resulting changes in departmental or division categories. Today the work force falls into nine separate organizational departments, with the staff divisions earlier mentioned.
Roughly speaking, the Bank's work force may be sliced horizontally according to the Fair Labor Standards Act categorization of exempt and nonexempt employees. During the period from 1969 to 1972, 71% of the work force was nonexempt and 29% was exempt (Plaintiffs' Exhibit 641). The same approximate percentage existed throughout.
Before 1965 there was a virtual absence of black employees from the Bank's work force. From 1968 through 1972, there were no blacks in the exempt category. The first black officer of Republic Bank did not arrive until 1973. As late as 1972, females were significantly better represented than blacks in their employment in the exempt category. Like blacks, however, and throughout the 10-year period at issue, there was a concentration of females in the nonexempt categories, and in turn in the lower grades of the nonexempt range. The numbers of females hired into exempt positions, and the movement of females upward within the nonexempt category, have risen gradually over the 10-year period. Today there are approximately 15 black employees in the exempt category, all of whom are below the vice president level. Of 570 Bank officers, 132 are female, 13 of whom are at the level of vice president.

IV. Bank Personnel Policy


A. Personnel Division and Hiring.

The Personnel Division of the Bank is headed by a Vice President. The Vice President is responsible for the Bank's personnel activities, including day-to-day staffing, campus recruiting, EEO activities, the infirmary, counseling, and career advancement. *244 Eight exempt employees work under his direct supervision. Hiring is done in many ways. Nonexempt employees are recruited largely through word of mouth and applicants for nonexempt positions are usually "walk-ins." The experience of the Personnel Division has been that applicants for exempt positions are attracted by advertisements, referred by other companies to the Bank, or recruited at college campuses. The use of employment agencies and search firms is principally confined to the filling of secretarial positions or hiring of accountants and data processing specialists. Recruiting is also done through governmental and civic agencies, including the Texas Employment Commission, National Association of Bankers, the Dallas Interracial Council, Minority Women's Employment Council, the Black Chamber of Commerce, and the Mexican-American Chamber of Commerce. In exempt job offerings, the Personnel Division has throughout the period covered by this suit placed great emphasis on the interview process. The Bank has used from time to time both black and female interviewers. The ultimate hiring decisions for exempt employment have, however, been made by white males.
Throughout the 10 years covered by this suit, the Bank has had a large number of job titles and positions. The number of such titles has ranged to upwards of 3,500, with as many as 700 to 950 titles in use at a given time. In mid-1979, as part of the defense of this suit and, according to the Bank, for other business purposes as well, the entire job structure was regrouped by occupational codes and families. The purpose of this effort was said to be to achieve horizontal symmetry in job functions by looking past job titles to job functions. Employees with similar job functions albeit with different descriptive titles, were grouped together in the same job family. This division of the work force, whose results are the benchmarks for labor availability in the statistical hiring studies and other studies, is reviewed in depth where those issues are discussed. The point here is that in an overview of the Bank structure, we find a shifting organization as the Bank grew in size followed by a dramatic adjustment accomplished in one sweep. The validity of that precipitate change must be examined. See section IX(G), infra.
With recruiting for entry positions leading to "line" jobs, the effort begins on college campuses, and has historically been concentrated in the Southwest. Recruiting at college campuses is confined with little exception to recruitment of exempt employees entering the credit analyst program.[22] The credit analyst program is the main entry channel for future bank executives. As stated by the Bank, "the Credit Training Program is the means by which [the Bank] seeks out and develops persons with the potential to assume positions of managerial responsibility with the bank" (Defendant's Post Trial Brief at 286). In hiring into this program, the Bank places great emphasis upon a business-related degree with emphasis in accounting and finance, a preference that will be examined later in the context of hiring and availability of potential employees. See section IX(G), infra. In the time period 1970-1978, 181 persons were hired into the credit training program. Of this group, 34 were hired for their experience. Of the remaining 147, only two did not have a business-related degree. These two held degrees in mathematics and law, respectively. This is consistent with the overall hiring pattern for the years 1970-1978, when there were a total of 997 exempt hires, of which 860 (86%) had at least a bachelor's degree. Of those who did not, virtually all had experience related to the jobs for which they were hired (Defendant's Exhibits 508, 566).
Credit analysts assist loan officers while they are being "trained" and provide a pool from which the loan "floor" draws. The credit training program is said by the Bank to be the breeding ground for its loan officers. And except for occasional lateral hiring of an individual with equivalent training, *245 it is the only means for staffing loan officer positions. An examination of the pattern of Bank hires and track of progression followed by incumbent management of the Bank reveals that as a practical matter, it is the main track for higher level executives within the Bank. This is not surprising in view of the Bank's core function of commercial lending and its insistence upon college degrees in business fields for persons in commercial lending. Indeed, throughout, the majority of all Bank officers have been in lending and marketing jobs.
The credit training program at the Bank has existed throughout the time period. In fact, the current president of the Bank is a product of that program. The program's changes have been primarily in increased intensity in curriculum. In 1969, the Bank employed a professor at the Harvard Business School to design a more intensive program and the Bank has implemented that program. In a nutshell, there is no other internal training or internship program of a formal nature at the Bank for exempt employees.
The Credit Department has approximately 36 desks. A new employee historically spends approximately nine months in the department before entering one of the respective loan divisions. The new employee is expected to progress through levels of credit analyst, unit manager, and supervisor. In this nine month period, the analyst receives increasing responsibility for the credit analysis work of prospective borrowers or existing bank customers. Part of the program consists of a "live-in" system. Under this program, analysts, after a few months at the Bank, are assigned to a particular division and work for that division alone for approximately a month. This rotating assignment of live-ins allows the analyst to become familiar with the somewhat differing emphases of the respective areas of the Bank while allowing the lending officers in those areas to observe the work of the credit analyst. Credit analysts are not officers of the Bank. It is anticipated, however, that upon completion of the credit program and acceptance of a position in one of the respective lending divisions the new employee will be elected to officer status.
When we turn to the filling of vacancies for nonexempt positions, the emphasis shifts from the campus to the Bank's internal staff and to more local labor pools. Moreover, there is far more lateral hiring for nonexempt jobs than for exempt jobs. There is also some movement from nonexempt into the lower exempt jobs as well as some overlap of exempt and nonexempt personnel hiring responsibility. With nonexempt employees, the Bank's promotional and training policies have changed over the years. Before 1975, the movement of employees from job to job within the Bank lacked formal structure. There were no predetermined paths, and no specific prohibitions or established channels. Movement was usually accomplished through one or a combination of three methods: supervisory referral; employee request to appropriate managers; and employee-initiated contact with interviewers in the Personnel Division. Not surprisingly, this loosely structured process caused personnel decisionmaking to be more subjective than the system which came into effect in 1975.
In February of 1975, the Bank created a "career advancement program." The career advancement program consists of formalized job posting and bidding, enabling employees to learn of the existence of other positions within the Bank. The program contemplates that job openings will be posted outside the employee cafeteria, at the drive-in motor bank, and at the Spring Valley Commuter center, with listings updated daily. In April of 1977, the program was expanded to include exempt job openings except those with the title of officer, manager, or administrator. Under the program, an employee interested in a transfer completes a career advancement interview request form and forwards it to the Personnel *246 Division. As applications for transfers are received in the Personnel Division, the personnel files of the applicants are reviewed for eligibility for transfer by a personnel representative. An employee is not eligible for transfer consideration until he has been in his current job for six months. Eligible applicants are then scheduled for screening interviews in the Personnel Division. Interviews are conducted by the supervisor in the section to which the transfer is sought. Bank policy is that effort is to be made to fill vacancies by internal transfer or promotion before outside hiring is attempted, although this policy does not appear to be uniformly followed.
When the Bank turns to outside employees, authorization must be granted by the Personnel Division upon receipt of a personnel requisition by a line manager or a request for addition to staff.[23] The Personnel Division's employment group interviewers begin efforts to fill the job. Applicants are screened by the Personnel Division and those applicants considered to possess the qualifications for the requested job are referred to management in the target division.
As earlier mentioned, hiring for most nonexempt positions at the Bank begins with an applicant's visit to the Personnel Division. Walk-ins are required to sign an applicant log at one end of the Personnel Division offices. The log is maintained by the interview coordinator, but no information has been maintained regarding qualifications of particular applicants and information regarding race and sex was not kept before 1974. Sources of Bank hires also include unsolicited resumes mailed to the Bank and college recruiting programs. While exempt and nonexempt hiring have some common sources, these resumes and the college recruiting program have been the richest source for exempt applicants. Despite this small overlap in sources, the hiring approaches were different. Before 1974, the Personnel Division used a "clerical" application form and a "professional" application form. These forms were the same except that the professional form contained places for professional and technical references while the clerical form contained a space for office machine operation, typing speed, and shorthand speed. The Bank claims that the professional form was given to those who expressed an interest in exempt positions or those who had a college degree. The form, utilized by the Bank since 1974, instructs walk-in applicants to indicate specific areas of banking in which they are interested and for which they are qualified. Upon receipt of the completed application form, the interview coordinator arranges an interview with one of the Personnel Division interviewers if the applicant has the "appropriate" qualifications. That is, there is some pre-interview screening: while the Personnel Division interviews many persons, not all applicants are interviewed. But the failure to interview was not the subject of separate evidentiary focus, and we do not know, apart from inferences from mere general hiring and placement models, whether the failure to interview itself cut along racial or sexual lines. In the interviewing process there is a channeling effect, because applicants have historically been directed to interviewers who specialize in either exempt or clerical or nonexempt positions, and the route to the exempt positions not surprisingly results in a more rigorous interview process. If the *247 interviewer determines that the applicant is qualified for an open position in which she has expressed interest, the applicant is referred to the appropriate supervisor or manager. If the manager determines that the applicant is the one desired, attempts are then made to verify the applicant's history. The supervisor/manager makes the final decision to hire or reject the applicant upon receipt of verified work histories. With exempt employees, more than one manager may participate in the decision. The salary level of the incoming employee is chosen by the manager by application of salary guidelines issued by the Salary Administration Section of the Personnel Division. An offer of employment is then extended by the interviewer.

B. Affirmative Action.

Beginning in the late 1960's, and continuing with increasing emphasis in 1974 and later years, the Bank has engaged in various affirmative action efforts. In 1974, the Bank came under new and younger upperlevel management in the person of President Charles Pistor. At approximately the same time, the Bank began a more intensified effort to engage in affirmative action efforts. At least since Pistor assumed the presidency, there is little question that it has been the announced policy of the Bank's senior management not to discriminate.[24]
In 1964, the first written EEO policy was adopted and circulated within the Bank. The first Affirmative Action Plan was drawn in 1970 and was distributed to managers in 1971. The plan was submitted to and approved by the Treasury Department in early 1971. Through 1975, the Treasury Department monitored the Affirmative Action Plan. Under the Bank's Affirmative Action Plan, the Vice President for Personnel receives monthly status reports revealing the percentage of hires that are minorities. The Bank's newspaper advertisements, at least since 1971, have contained the usual addendum that the Bank is an equal opportunity employer. In the 1968-1973 period, five charges of discrimination were filed with the EEOC by employees, as part of a total of 43 charges for the 10-year period of this suit (Defendant's Exhibit 177). The EEOC in July, 1974, charged the Bank with discrimination, and a conciliation agreement was reached in December, 1976 (Plaintiffs' Exhibit 733). The agreement settled four employee charges and set various hiring and distributive goals. In 1975, the Treasury Department presented a list of "deficiencies" which resulted in a conciliation agreement in March, 1977.
Beginning in the latter part of 1968, of 19 colleges visited by the Bank, eight were predominately black and one was predominately female. As Jerry Watson advised his supervisors in April, 1975:
The selection of colleges and universities ... was strongly influenced by: (1) the Treasury Department's compliance review (May, 1974); (2) the class action discrimination charge filed by Mr. John Powell, Commissioner of the Equal Employment Opportunity Commission (July, 1974); and (3) Republic National Bank's commitment, as expressed in the Affirmative Action Plan, to increase minority and female representation in officer and exempt positions.
Defendant's Exhibit 68. Watson also commented at the same time about the Bank's hiring success, observing that:
... a very large number of minority students must be screened in order to find a disproportionately few acceptable candidates ..... *248 Id. In December, 1976, Watson commented to his supervisors about the small number of invitations the Bank was extending as a result of colleges visited, 61% of which had predominate black enrollments: "... the small number of invitations ... is an indication of our continued `selective' approach to invitations and hiring." Defendant's Exhibit 174. He further observed as of the end of the third quarter of 1976, 25.8% of hires in 1976 were minority: "... the majority of minority hiring is accounted for in nonexempt positions." Id.


C. Categorization of Employees.

As earlier mentioned, before 1973, the Bank's structure viewed vertically consisted of exempt and nonexempt employees. In turn, the nonexempt range was divided into 17 grades. There were five practical levels in the exempt categories. The first four ran through the vice president level with senior vice president and above categorized as number five. In the 1969-70 period, the Personnel Division decided that the top management of the Bank should occupy a separate structure in that their pay values were disproportionate with the lower ranks. Except for that separation of the few officers at the level of top management, the levels of bank officers were not changed. In 1973, as part of a reevaluation of pay structures and personnel assignments, the Personnel Division reexamined the 17 nonexempt grade divisions and decided that a 10-grade structure would be more effective. Among other things, it found that although there were 17 grades, less than 14% of the total of nonexempt employees actually fell in grades 11 through 17. In an effort to avoid this bunching and to correlate the nonexempt jobs with a sound pay structure, the reduction to 10 grades was made.
Beginning around 1970 and continuing until 1973, the Bank undertook a transition to a system of job evaluation known as the "Hay System." It is not necessary to set out in detail this relatively complex technique, because the court finds that nothing intrinsic to the Hay System itself creates Title VII concerns. Of course if a wage or hiring system is otherwise skewed, the Hay System is no guarantee against liability. A general appreciation of its operation, however, is important to a comprehension of the Bank's personnel policy, its employee structure and the system's usefulness in the multiple regressions in section VII(A), infra. We will return to the system's utility in the statistical measurement efforts.
The reduction of the number of nonexempt grades from 17 to 10 was part of the implementation of the Hay System. This system, in use presently at the Bank, is intended to be a technique for comparing a variety of jobs within a firm in an orderly way. The job evaluation phase of the system is claimed to be premised on the know-how needed, the problem solving involved, and the accountability of the job being analyzed. More specifically:
As Harriette Weiss, a Senior Principal with Hay and Associates, testified, the "Hay System" is basically a euphemism for the Hay Guide Chart-Profile Method of job evaluation, a proprietary method utilized by the management consulting firm for evaluating and ranking jobs by job content within an organization. With the aid of job evaluation, a firm is able to establish a rational and consistent method of calculating the relative worth of jobs, which in turn can be used as a basis for payment of compensation for the performance of those jobs.

Defendant's Post-Trial Brief at 353 (emphasis supplied). The system is summarized by Hay Associates in Figure 1.
The Bank utilized the Hay method from January 1, 1970, forward with its exempt *249 employees, the method being set up in consultation with Hay Associates. See Defendant's Exhibit 202. The first phase of the Hay study involved an effort to establish equitable internal relationships among all positions. This was attempted through a job evaluation process conducted by an in-house committee of Republic executives, usually one or two levels above the job being evaluated. The result, after review, was the assignment of "points," based on job content (using know-how, problem-solving, and accountability as factors to be taken into account) to each position being evaluated. These "points" have been referred to as Hay points throughout this litigation, though perhaps they are more accurately referred to as "client points"; comparisons with other firms are made possible by conversion of client points to "Hay points." See Defendant's Post-Trial Brief at 354 n.100.
The second phase, an effort to develop a competitive salary structure within the Bank, depends on the point values assigned to the exempt jobs in the first phase, and what other employers pay for jobs of equivalent point value:
Compensation practices are compared among [Hay Compensation Comparison] survey participants utilizing "Hay points" as the standard of reference for comparing jobs of equivalent value. A conversion formula enables each Hay client to convert Hay points to a dollar figure representing the average salary paid by survey participants or the midpoint of a salary range for a job at any given Hay point level.
Defendant's Post-Trial Brief at 359. *250
The salary range contains a minimum figure that is set as a hiring level rate of pay for applicants meeting position specifications. The maximum figure is said to be one that is only exceeded rarely by an employee with a highest rating, that of "path *251 finder." The salary range has a 50% spread. That is, the maximum is 150% of its minimum. For example, a salary range with a $8,000 minimum would have a maximum of $12,000. Under the plan the midpoint of a salary range (for example, $10,000) would be the area within which the Bank would pay for "consistently satisfactory performance on a sustained basis of the functions assigned to the incumbent." Defendant's Exhibit 202.
Under the plan, employees sometimes can find themselves below minimum if, for example, they are rapidly promoted. The plan allows the Bank to give inequity increases to reduce any such inequities. The minimum has attracted the attention of the Department of Treasury. Since 1977, as a result of a reconciliation agreement with the Treasury Department, the total males, the total females, and the total minorities below minimum are monitored.
The Bank has throughout required performance appraisals by the immediate supervisors of employees. The Hay system defines certain ranges of performance. In the exempt position category, the ranges included inadequate, marginal, beginning, fair, competent, commendable, distinguished, and pathfinder.
The nonexempt structure, to which the Hay System was applied in 1973, differs somewhat from the exempt structure in that nonexempt salaries are quoted in monthly sums whereas exempt salaries are quoted in annual sums. Of course there is a difference in this salary spread. With a grade one position, the minimum to maximum spread is 25% while with a grade ten it is 60%. There is also a difference when an employee becomes eligible for changes, as well as difference in allowable percentage salary increases. For example, for nonexempts, higher percentage increases keyed to performance, are allowed. In addition, different survey data was used for exempts and nonexempts: at the exempt level the competitive salary inquiry was national, while for nonexempts the metroplex area was the primary data source.

V. Statistical Evidence

All statistical studies in this case share a common goal: to compare the personnel decisions which have actually been made at the Bank since 1969 with the decisions which hypothetically would occur in the absence of discrimination. In the areas of compensation, initial placement, promotion, and termination, this comparison relates one segment of the Bank population to another, determining whether females or blacks at the Bank are treated differently from males or whites. In the hiring area, the Bank's hiring decisions are compared with data external to the Bank-the availability of blacks and females as measured by Census Bureau data. Each of these comparisons depends upon the existence of an accurate and complete data base reflecting the multitude of individual personnel decisions made by the Bank during the relevant time period. While the statistical models employed by the parties possess varying degrees of "robustness," i. e., ability to withstand the effect of errors in the data or violation of the assumptions on which the models rely, it is generally true that the utility of a statistical comparison is directly dependent on the accuracy and completeness of the data being compared. For this reason, we must, before turning to the statistical models and the results generated by them, carefully examine the sources of data on which those models rely. We must then evaluate the parties' challenges to those data. Having examined the input to the statistical models, we next must determine the extent to which the court is confined to the output of the studies in its presented form, and whether it may or must perform statistical calculations beyond those provided by the parties. Finally, we discuss the role of nonstatistical evidence and its relationship to the statistical studies.

A. Sources of the Data.

With some exceptions, all data used by the parties in this case were ultimately derived from employment records maintained by the Bank as its permanent business records. For each present and past employee, the Bank maintains a traditional "personnel *252 jacket," containing such items as the employee's vital statistics and salary, promotion, and performance data. The Bank also maintains a variety of personnel summaries and payroll and other accounting records. In addition, the Bank has since 1973 maintained computerized personnel records. Each of these types of personnel record has played a role in the development of the data bases used by the parties.
The data bases used by the parties, either for statistical analysis or for the generation of other data bases, are summarized in the margin.[25] The data bases used for analysis *253 were derived in principal part from a series of tapes maintained by the Bank and designated by it as the "A020 tapes." These tapes represent semi-monthly "snapshots" of the Bank's work force, and were used by the Bank to generate employee paychecks and day-to-day personnel records. The tapes contain a variety of information on each employee, including the items of information listed in n.25, supra, and pension, insurance, and actuarial information.
The Bank also maintains a personnel history tape which it designated as the "A030 tape." This tape, which has not been used on a day-to-day basis, was produced in an attempt to avoid the use of individual personnel jackets, and contains personal data on each employee, together with work histories and information about preemployment education. The A030 tape is the Bank's only computerized record of information about education.
In August of 1978, the Bank developed a tape designated as "A035," which it used for the bulk of its statistical studies. The A035 tape represents a continuous record of the employment history of all Bank employees from January 1, 1973, forward. The *254 tape was constructed by extracting information from the A020 tapes and merging that information with employee names and educational data from the A030 tape. Unlike the A020 and A030 tapes, the A035 tape contained a variable length record for each employee, with one to 95 fields per record.[26] A total of 6,395 present or past employees were represented. Eighteen of these employees, who had accidently been omitted from the A030 tape, were manually added to the A035 tape. Based on a belief that many Equal Employment Opportunity codes on the A020 tapes were in error,[27] EEO codes were changed on the A035 tape to reflect "appropriate" values.
After assembling the A035 tape, the Bank audited the tape for accuracy. A random sample of 375 employee records was chosen, and computerized information for each record was compared with the same information as recorded in the employee's personnel jacket, which the Bank believed to constitute the most reliable source of information on its employees. The audit demonstrated, to the Bank's satisfaction, that most fields contained accurate information.[28] Some fields, however, were found to be, in the Bank's opinion, too unreliable for use. These included salary ratings (24% missing and 6% erroneous), annual performance appraisal ratings (26% missing and 4% erroneous), and, most importantly, information about education.[29]
Before 1973, the Bank did not maintain any computerized personnel records, and for the pre-1973 period it was necessary to manually assemble personnel data from paper records. From information contained in personnel jackets, salary change sheets, and payroll earnings journals, the Bank assembled a computer tape, designated the "A500 tape" (sometimes designated as the "A040 tape"). This tape contained information for 1969-72 similar to that contained on the A035 tape for 1973-78.
For certain studies performed by Drs. Stoikov, Stolzenberg, and Snyder, it was necessary to go beyond the Bank's personnel records. To this end, a questionnaire was prepared and submitted to a randomly selected sample of 286 current Bank employees on June 28, 1979. Employees were asked to verify information obtained from personnel jackets and to provide responses to a variety of questions about their work habits, working conditions, motivations, and expectations. Ninety-two percent of the employees surveyed fully completed the questionnaire.
For their regression study on involuntary terminations, Drs. Stolzenberg and Snyder prepared their own ad hoc data base. This data base consisted of information from the A035 and A500 tapes, together with a variety of data on education, skills, and experience.
*255 For the purpose of developing weighted average availability figures for use in hiring studies, see section IX(F), infra, Dr. Stoikov used a variety of data on the education, experience, and geographical origin of those hired by the Bank between 1970 and 1978. These data were obtained from personnel jackets.
While plaintiffs were furnished with most of the computerized data bases developed by the Bank, including the A030, A035, and A500 tapes and selected A020 tapes, they, unlike the Bank, chose not to place primary reliance on the A035 tape. Instead, plaintiffs' computer expert, Dr. David Morgan, prepared a tape for 1973-78 by merging data from the A020 tapes with date of birth and highest grade completed from the A035 tape. For 1969-72, Dr. Morgan merged data from the A500 tape with date of birth, highest grade completed, and salary taken from excerpts from personnel jackets supplied by the Bank. Dr. Morgan's primary purpose in developing these separate tapes was to facilitate use of the Statistical Package for the Social Sciences, a group of standardized statistical computer programs used in a variety of social science applications. See generally N. Nie, C. Hull, J. Jenkins, K. Steinbrenner, & D. Bent, Statistical Package for the Social Sciences (2d ed. 1975).

B. Problems with the Data.

The parties mount a variety of challenges to the accuracy and reliability of each other's data. Each such challenge is made on two levels: First, the challenging party argues that the data are so infected with inaccuracy and omission that they are unfit for use, and that statistical studies relying on the data should be disregarded. As a fallback position, the parties argue that flaws in the data should be kept in mind in evaluating the strengths and weaknesses of the respective statistical presentations. The purpose of this section is to identify the challenges, and to evaluate whether any challenges have sufficient merit to warrant the total disregard of studies relying on the challenged data. Any inaccuracies not rising to this level will be considered in the evaluation of the statistical presentations which rely on the challenged data.
We begin by noting that the Bank's personnel records from an unusually rich source of data for statistical analysis of employment discrimination. Cf. Testimony of Dr. Francine Blau at 207 (Bank's data pool a "really large impressive sample"). Only in the case of a large employer with substantial resources is a comprehensive computerized personnel record spanning many years typically available. To this must be added the fact that a certain measure of inaccuracy in data is a daily fact of life for the social scientist. As Dr. Blau put it,
I think if social scientists in general waited for perfect data that was completely error free, no social science research would be done so the real question for the researcher is determining when any of these bias[es] or errors are serious enough to precluded [sic] analysis....
Id. at 37-38. Nor are courts immune to this pervasive imprecision: the latitude allowed a district court in computing Title VII back pay awards, see, e. g., United States v. Allegheny-Ludlum Industries, Inc., 517 F.2d 826, 852 n.29 (5th Cir. 1975), cert. denied, 425 U.S. 944, 96 S.Ct. 1684, 48 L.Ed.2d 187 (1976), is but one example of the judicial response to limitations of employer personnel data.
In light of these factors, together with the fact that the parties' approaches to the data assembly problem are facially reasonable and the fact that the parties' respective data bases are at their core quite similar, a heavy burden must be met before a party can justify the rejection in toto of any statistical analyses on grounds of errors or omissions in the data. Identification of flaws is helpful for later use in evaluating the results, but identification is not alone sufficient to warrant rejection. Instead, the challenging party bears the burden of showing that errors or omissions bias the data, i. e., that erroneous or omitted items are not distributed in the same way as items which are present and correct. That *256 party must then show, at the least, that this bias alters the result of the statistical analyses in a systematic way, i. e., that errors or omissions make the models more (in the case of challenges) or less (in the case of the plaintiffs' challenges) likely to show discrimination than would correct and complete data. See section VII(C) (introduction). Having established these ground rules, we turn to the specific challenges made by the parties.

1. Plaintiffs' Challenges.

The plaintiffs eschewed use of the A035 tape because, in their words, it was "prepared for litigation." That the tape was indeed prepared with this litigation in mind is not seriously questioned by the Bank. Nevertheless, Bank officials have testified that it is their intention to use the A035 tape as a master personnel history tape in day-to-day operations. In particular, they expect the tape to play a role in the preparation and monitoring of future Affirmative Action Plans. Moreover, to say that the tape was "prepared for litigation" says nothing about its accuracy absent some indication that that fact has resulted in erroneous, omitted, or biased data.
Plaintiffs' expert Dr. Janice Madden questioned the validity of the hiring qualifications data used for availability calculations, arguing that "recall" ex post facto of the qualifications for which employees were hired renders those data suspect. While it is true that the primary reason for which an employee was hired was determined after the fact for the purpose of the availability analyses, this classification was not accomplished without objective guidelines. Messrs. Stotts and Barksdale testified that in preparing the breakdown (Defendant's Exhibit 549) they used as a rule of thumb the presence or absence of a degree conferred within the preceding twelve months: those with such a degree were presumed to have been hired for their educational qualifications, and those without such a degree for their experience. Apart from the allegedly subjective nature of the classification process, plaintiffs offer no reason to believe that the classifications were systematically biased in any way.
The plaintiffs finally attack the data obtained through the questionnaire which was submitted to a sample of current Republic employees. They argue that employees who did complete the questionnaire "may not have understood what they were considering," and that those who did not complete it may not have wanted to cooperate in the Bank's defense of this lawsuit. The high participation rates in the questionnaire survey discredit these assertions. Of 313 current employees selected for the questionnaire, 286 (91%) were actually given the questionnaire, the remainder being on vacation, maternity leave, or foreign assignment. Of these 286, 285 (99.7%) completed the questionnaire at least partially, and 268 (93.7%) completed the questionnaire fully. There is no indication that responses from absent or uncooperative employees would have been differently distributed from those of other employees. While it is impossible to know whether the answering employees understood the questionnaire, the questionnaire is not difficult to understand and those who answered must be presumed to have known what they were doing.[30]

2. The Bank's Challenges.

The Bank first challenges the plaintiffs' use of year-end A020 tapes as the source for most of their data. It argues that errors in the year-end tapes will be perpetuated until the next year's tape in the plaintiffs' analyses, while such errors will be corrected in the Bank's analyses by information from the A020 tape for the next payroll period. While this is indeed a flaw in the plaintiffs' data, the court is convinced that it is a minor one: the errors (presumably few in number to begin with) will be propagated only if they happen to appear in *257 the year-end tape, and then only until the next year-end tape. Moreover, no systematic bias has been shown.
The Bank also attacks the plaintiffs' inclusion of hourly employees in the data for some but not all years. The short median tenure of such employees (about one year; Defendant's Exhibit 481) and the small proportion of Bank employees they represent (averaging 160 hourly employees; Defendant's Exhibit 480) suggest that any error caused by their inclusion is small. Again, no systematic bias has been shown.
The Bank's most substantial challenge to the plaintiffs' data focuses on the use of the "highest grade completed" field. For 1973-78, this challenge relies on the results of the A035 audit, see n.28, supra, which shows an omission rate of 31% and an error rate of 10% for the highest grade completed field. The plaintiffs performed an analysis (Plaintiffs' Exhibit 1308, Tables 11 & 12) which purports to show that errors and omissions in this field display no systematic pattern and hence do not affect the regression results. This attempt must fail, however, since there is no mathematical way to determine the extent or effect of measurement error. D. Baldus & J. Cole, Statistical Proof of Discrimination ง 9.11, at 298 (1980). In common sense terms, if correct data were present with which to compare the data which were used, those correct data could have been used in the first place and the problem would never have arisen. The problem with plaintiffs' error/omission analysis is that there is error and omission in both groups under comparison: the total Bank population, as to which there is a 31% omission rate and 10% error rate in the highest grade completed field, is compared with the regression sample (total Bank population less those missing one or more critical data items). Not surprisingly, the comparison shows little difference.
For 1969-72, the Bank performed an audit of the manually added fields from the plaintiffs' tape. Results of this audit are shown in the margin.[31] For these years, there is a 12.8% omission rate and a 24.5% error rate in the highest grade completed field. The Bank attributes these flaws to the use of excerpts from personnel jackets rather than the entire jackets, and to imprecision in the translation of high school, college, and graduate degrees into years of education by Dr. Morgan and his coders.
As with other challenges, the Bank has failed to show that these flaws in the data produce any systematic bias. This court does not accept the Bank's assertion that an error rate of, e. g., 10% is alone sufficient to warrant rejection of the data. See Defendant's Post-Trial Brief at 65. There is no indiction that any errors or omissions are not randomly distributed among sexes and races[32], or that the errors and omissions *258 caused an overestimate or underestimate of education for any particular racial or sexual group. No guidance is given as to what was treated as an "error": a discrepancy of one year between education as recorded in a personnel jacket and education as recorded on the plaintiffs' tapes is likely recorded as an "error," yet there is no indication of the proportion of such small errors.
The flaws in the plaintiffs' educational data are indeed serious enough to warrant care in evaluating the studies which rely on those data. Those flaws do not, however, warrant the extreme sanction of disregarding the studies as irrelevant. The high proportions of the data which are both present and correct (59% for 1973-78 and 63% for 1969-72) and the even higher proportions of correct data among all included data (86% for 1973-78 and 72% for 1969-72) render the data at least marginally probative of the existence vel non of discrimination. Cf. Fed.R.Evid. 401.

C. Technical and Institutional Competence.

An important question which must be resolved before turning to the analysis of the parties' statistical presentations relates to the proper role of the court in evaluating those studies. Of course, the record on which the court must base its decision consists of the exhibits and testimony, principally the reports and other exhibits prepared by the experts and their live testimony concerning those reports and exhibits. But a nagging question remains: To what extent may (and should) the court, using data contained in the record and mathematical and statistical techniques explained by the experts in their reports and testimony, perform its own calculations? Two limitations on such activity must be examined: the technical competence of the court to perform such calculations; and the extent to which voluminous calculations can be required by a party which did not perform such calculations as part of its trial presentation.
The role of the court as factfinder requires that it bring to its task its own fund of common knowledge and general reasoning ability. To go beyond these sources of decisionmaking power, however, presents three related dangers. First and foremost among these is the risk of error. Many judges (including this court) have little or no formal training in statistics, econometrics, or higher-level mathematics. True to the aphorism that a little knowledge is a dangerous thing, such a court may be led into the temptation to apply its limited fund of such knowledge to the solution of mathematical problems in the case before it, without a full appreciation of the risks and limitations of the techniques it is employing and without assurance of their suitability for the task to which they are put. Thus the explanation by an expert of a statistical technique or formula and its applicability to a particular problem does not warrant the application of that technique or formula to an unrelated problem, no matter how closely related the two may seem to the uninitiated. Only where the expert has justified the use of an approach for a particular problem, evaluating the risks and limitations of the approach in the context of the problem, ought the court to apply the technique in performing calculations not performed in the evidentiary presentation.
An overwillingness to undertake computational efforts also creates the risk of decisional disparity. Courts with varying degrees of inclination and ability in the application *259 of complex technical approaches to the solution of social problems will reach varying results, and the outcome of cases requiring such application will turn more on the plaintiffs' choice of forum and the luck of the draw in the District Clerk's office than on the merits of the cases. Indeed, absent some neutral and consistently applied approach to computation by the court, a court must risk the accusation that it has picked and chosen among the data, performing or not performing additional computations to suit a preconceived result.
Going too far beyond the calculations performed for and explained at trial is also unfair to the parties. In a typical complex Title VII class action, the parties have by the commencement of trial deposed each other's experts, discovered those experts' reports, counseled with their own experts, explored possible cross-examination, developed rebuttal evidence, and otherwise prepared for the evidentiary onslaught they expect to meet. For the court to appoint itself as an additional statistical expert, ex parte, after trial, and without opportunity for challenge by the parties, frustrates the expectations of the parties and diminishes the reliability which the adversary system was designed to insure. Similarly, complex calculations performed for trial but not explained will not receive the benefit of sustained expert scrutiny of their validity and implications, and so cannot be relied upon.
Beyond problems of technical competence are those of what may be referred to as institutional competence. Obviously, a court cannot be called upon to perform multiple regression analyses requiring a computer merely because the raw data which would support such analyses have been introduced into evidence. More generally, the introduction of raw data and the explanation of a statistical technique ought not, at least where the data are voluminous or the technique complex, place the burden on the court to perform the calculations. While the court cannot shirk its duty to review the evidence and decide the case, to require a myriad of calculations based on file cabinets full of statistical exhibits would exact not diligence but fanatical devotion to detail. See n.117, infra.
To say that the court should be reluctant to go far afield from the parties' own calculations does not mean, however, that in all cases the court is hamstrung by a party's failure to perform calculations necessary or helpful to the decision of the case. Where a simple statistical technique has been explained through expert testimony in terms understandable by one with no advanced statistical background, and where the use of that technique has been justified for the problem in question, the performance of simple calculations using the technique is not the creation of new evidence but only a more detailed examination of existing evidence. Cf. L.C.L. Theatres, Inc. v. Columbia Pictures Industries, 421 F.Supp. 1090, 1103 & n.9 (N.D.Tex.1976), rev'd in part on other grounds, 566 F.2d 494 (5th Cir. 1978) (post-trial summaries permissible where underlying data are in evidence); 75 Am. Jur.2d Trial ง 991 (1974) (jury experimentation permissible if effect is not to introduce extraneous evidence); Annot., 95 A.L.R.2d 351 (1964) (same). This type of calculation aids rather than harms the parties, enabling the court to fill gaps created when minor flaws not recognized by the parties are discovered in the data. Limited calculations may alternatively enable the court to transform the parties' statistical data into a form which gives greater insight into the existence of discrimination, a result which likewise ought to be welcomed.

D. The "Anecdotal Evidence."

The Second Circuit recently stated that "statistics showing a significantly disparate racial impact have consistently been held to create a presumption of Title VII discrimination." Guardians Association of the New York City Police Department, Inc. v. Civil Service Commission, 630 F.2d 79, at 88, 23 Empl.Prac.Dec. ถ 31,153, at 16,973-74 (2d Cir. 1980). In Hazelwood School District v. United States, 433 U.S. 299, 307-08, 97 S.Ct. 2736, 2741, 53 L.Ed.2d 768 (1977), the Court held that "[w]hen gross statistical disparities can be shown, they alone may in a proper case constitute prima facie proof of *260 a pattern or practice of discrimination." See also Johnson v. Uncle Ben's, Inc., 628 F.2d 419, (5th Cir. 1980).
Like those courts, we make inferences from statistical patterns in this litigation. Indeed, the court must rely on such statistical evidence: neither the testimony of individual applicants and employees, nor that of Bank executives,[33] provides in this case a sufficient independent basis for a finding of the presence or absence of class-wide discrimination of the sorts here alleged. See section VIII(A), infra. Anecdotal evidence does, however, serve a useful function in this case, in light of the principle that tests of statistical correlation cannot by themselves identify the causative factors which produce the observed results. See D. Baldus & J. Cole, supra, ง 9.42, at 320; R. Wonnacott & T. Wonnacott, Econometrics 173 (2d ed. 1979). See generally F. Mosteller & J. Tukey, Data Analysis and Regression-A Second Course in Statistics 260-62 (1977). It serves a useful function as one of the three ways in which the plaintiffs have bolstered the thesis that it is real-world behavior rather than random statistical patterns which causes there to be correlations and modelling results damaging to the Bank. We leave discussion of two of the ways-statistical significance and the use of theory in designing the models-to later sections. See section VI(D) nn.58 & 54, infra; sections VI(C) and IX(A), infra.
The anecdotal evidence-evidence of individual experiences, specific employment practices, and testimony of bank officials-which in itself is not weighty enough to alter the results obtained from the statistical evidence, lends support to the idea that any discrimination found in the statistical analyses is due to discriminatory behavior rather than to chance. Cf. Waintroob, The Developing Law of Equal Employment Opportunity at the White Collar and Professional Level, 21 Wm. & Mary L.Rev. 45, 96-98 & 103-105 (1979).
While we do not here detail such anecdotal evidence, we have considered all such evidence carefully. That there is such evidence can hardly be challenged. For example, the following exchange took place at the deposition of Thomas G. Croft, Senior Vice President and Director of Personnel:
Q When do you believe that the bank began to be in compliance or began to offer equal employment opportunity to females?
A When? I don't understand your question.
Q Well, my previous questions concerning being in compliance and offering equal employment opportunity to blacks and females, you said something about historically, and then we narrowed it to presently, and you said yes, you believed that you were or that the bank was. Therefore, I presume that you intended that historically perhaps it was not, and I'm saying when did the bank begin, as far as you're concerned, to offer equal employment opportunity to women and to blacks.
A I think we probably began in the early '70's.
Q When you began in the early '70's-excuse me just a minute.
(Off-the-record discussion.)
MS. PETERS: (To the reporter.) What was our last question and answer?

(Thereupon, the last question and answer were read by the reporter.)
A I would, if I may, I'd like to add to that. At least there's been a spiritual change, if you will, in the thinking of the management of the bank which started in the early '70's, and historically these laws and the executive orders are relatively new. So it's difficult to say when it came about or when it started or where we are now, but I must say, to the best of my knowledge, backed up by what the government agencies have to say, *261 that we are in compliance with the law of the land. Again, we've had people like Sharon Jobe, who is vice president and general counsel for the holding company. Sharon has done a lot of things to change people's minds about the capability of a female executive. She's doing some work with the-or she's preparing to do some work with the affiliates of the holding company, again working with Bob Stoller. So I guess maybe spirtually [sic] is not the best word, but it's the one that comes to mind, that change did begin to take place in the early '70's.
Deposition of Thomas G. Croft at 126-27 (December 7 and 8, 1977).[34]See also Deposition of Tom Croft at 106-07 (December 9, 1977) and his trial testimony concerning the December 9 answers. Croft's candid remarks form a backdrop to the statistical evidence, to which at long last we now turn.

VI. The Theory Behind the Parties' Mathematical Modeling


A. Introduction.

Both the plaintiffs and the Bank rely on econometric models rooted in principles of labor economics in order to perform one task: to represent in mathematical terms the Bank's employment practices. Neither side posits that every time a Bank administrator is faced with a personnel decision he "evaluates each applicant according to a predetermined rule for accounting and weighting key characteristics [using a system where] [t]he relevant characteristics are specified in advance, and so is the rule for combining them to produce a score for each applicant."[35] Both parties instead offer models claimed to represent the Bank's behavior on the whole.
The mathematical models draw heavily on quantitative techniques long used by social scientists and more recently used in judicial resolution of antitrust, securities, and employment discrimination disputes. The use of quantitative techniques in this case is unique only in that the techniques used here, especially those involving multiple regressions, are more sophisticated and are more heavily relied on by counsel for both sides than in most litigation. By pushing the techniques to the center of the fray, the parties have compelled the court to examine the techniques' theoretical under-pinning-the "human capital" theory of labor economics and general productivity theory-as well as the operational mechanics of multiple regression itself. Only by doing so will we understand the strengths and limitations of the models, and thus determine whether any particular model offered is sufficient to establish or rebut a prima facie case. Once it is determined that a particular model generates probative evidence, the court must determine whether a model represents a behavioral pattern violative of Title VII. For this second step, an understanding of both the mathematical techniques and Title VII are necessary.
Despite the quantitative social science nature of its analysis, the court fashions no new policy based on its own concept of the public good. In attempting to evaluate the models against the background of an understanding of both the mathematics and the law, we seek to ensure that any hidden value choices are not introduced by the quantitative techniques. We confess to a determination not to reject the unfamiliar for its unfamiliarity, and to a hope that the translation of Title VII equal opportunity doctrine to technocratic terms will make possible more sensitive and accurate detection of the presence or absence of discrimination, as defined not by the court's or the economist's view of public policy but by the court's view of policy set by Congress in the form of Title VII.
*262 Stripped of jargon, the parties are agreed as to the fundamental theory behind the Bank's employment decisions as to compensation, initial placement, and promotion: the Bank treats people differently according to their productivity.[36] The plaintiffs argue that the Bank considers, in addition, the racial and sexual group status of the individual. The Bank counters that it does not. In section VI(B), infra we show that Title VII does not proscribe differentiation according to productivity.
In Section VI(C), infra, we describe the human capital theory, which here provides the intellectual justification for inclusion of productivity-related factors in the mathematical constructs. In section VI(D), infra, we describe the mechanics of modeling, with focus on "single equation, ordinary least squares regression" models. Finally, in section VI(E), infra, we discuss how particular econometric results can determine whether employees are being treated according to their group-status as well as their productivity.

B. Job Relatedness and Equal Treatment.

There are two concepts of discrimination employed in interpreting fair employment laws. The first is "equal treatment," a doctrine wherein all persons are to be treated by employers without regard to their racial or sexual group status. It seeks to assure that in competing for jobs a person is not handicapped by his group status. The focus of this concept is upon individuals and acknowledges their varying skills. The second concept is that of "equal achievement." Its focus is not upon individuals with varying skill levels, but is upon the actual distribution of employment among groups. That is, it looks to the results of the contest, not to whether the rules are the same for everyone. Equal achievement subscribers contend that equality of treatment of minorities is spurious rhetoric because minorities remain hobbled by the effects of past discrimination.[37] The choice between these two interpretations is not an academic exercise. Equal achievement subscribers would place on employers the burden of correcting for the disabling effect of societal discrimination such as inadequate education and training. This court is persuaded that in doing so, they would frustrate the stated purpose of Congress.
The construction of "job relatedness," to be discussed more fully herein, turns directly upon the choice between equal treatment and equal achievement. An employer does not discriminate by paying a white more for a higher productivity level in a job than a black otherwise similarly situated. Judicial decisions, although not always true to an equal treatment subscription in their analysis of Title VII cases, generally do not hold to the contrary. The difficulties stem from failure to give to courts accurate measures of productivity of individuals. Left at sea, we simultaneously have recognized that productivity is a valid differentiating factor but have indulged in a tendency to hedge asserted indicators of such productivity with high standards of proof because we recognize that absent accurate measure they contain great potential for masking differentials actually based on race or sex. Excessively high standards of proof have sometimes resulted in adoption of the equal achievement approach by evidentiary default.[38]
*263 The choice between the equal treatment and equal achievement approaches is not for the courts because Congress made that choice when it passed Title VII; and it left an unequivocal record that its choice was critical to the very enactment of the legislation. Congress had equal treatment in mind and the courts cannot, if they accept institutional limitations of a tripartite government, do otherwise โ either directly or by unspoken decisions that erode the premise, whether done intentionally or unwittingly.
Title VII was directed solely at unequal treatment of individuals;[39] it was not intended to infringe on independent decision-making based on reasons of business efficiency as long as the employer did not take the group status of the individual into consideration.[40] Senators Clark and Case, floor managers of Title VII, stated in an interpretative memorandum:
It has been suggested that the concept of discrimination is vague. In fact it is clear and simple and has no hidden meanings. To discriminate is to make a distinction, to make a difference in treatment or favor, and those distinctions or differences in treatment or favor, which are prohibited by section 704 are those which are based on any of the five forbidden criteria: race, color, religion, sex and national origin ....
110 Cong.Rec. 7213 (1964), quoted in No-Alternative Approach, supra n.37, at 103 n.29. Responding to opponents of Title VII, who were concerned that the bill would lead to quotas and other equal achievement oriented reactions, the supporters stated:
The language of ... [Title VII] simply states that race is not a qualification for employment. Every man must be judged according to his ability. In that respect, all men are to have an equal opportunity to be considered for a particular job ....
.   .   .   .   .
... It is possible that although a ... particular business will contain no Negroes, no charge of discrimination will be made. But businesses ... may not systematically exclude Negroes, when the only ground for exclusion is the color of a man's skin.
110 Cong.Rec. 8921 (1964), quoted in Disparate-Impact Liability, supra n.37, at 927. Senator Case also explained:
Whatever its merit as a socially desirable objective, title VII would not require, and *264 no court should read title VII as requiring, an employer to lower or change the occupational qualifications he sets for his employees simply because proportionately fewer Negroes than whites are able to meet them ... [nor would it require lower standards] because prior cultural or educational deprivation of Negroes prevented them from qualifying ....
[T]he very purpose of title VII is to promote hiring on the basis of job qualifications, rather than on the basis of race or color.
Id. at 928.
The concept of job relatedness, born in Griggs v. Duke Power Co., supra n.38, is an application of the equal treatment philosophy. In Griggs, the Supreme Court held that an employment practice nondiscriminatory on its face, but which has a disparate impact on a protected class, is illegal under Title VII unless the employer can show the practice is a "business necessity." Griggs struck down the requirement of a high school diploma and satisfactory aptitude test scores for hiring into supervisory jobs because the employer showed "no demonstrable relationship to successful performance" of the requirements in the jobs for which they were used. 401 U.S. at 431, 91 S.Ct. at 853. As the Court stated, Title VII prohibits "practices that are fair in form, but discriminatory in operation. The touchstone is business necessity. If an employment practice which operates to exclude Negroes cannot be shown to be related to job performance, the practice is prohibited." Id. at 431, 91 S.Ct. at 853. Duke Power Company's qualifications were unrelated to efficiency, being surrogates for a direct racial criterion. See Disparate-Impact Liability, supra n.37, at 929. The Court's chief concern was not equal achievement, but equality of treatment. Id. Later Supreme Court cases confirm the equal treatment interpretation of Title VII. Id. at 930-33.[41]
The bare words "business necessity" have been interpreted as involving more than "business purpose," under which a practice can be justified by showing that any benefit accrues to the employer through the use of the practice. No-Alternative Approach, supra n.37, at 100. The typical formulation of the rule is that the practice must be one that is essential to the safe and efficient operation of the business. Disparate-Impact Liability, supra n.37, at 916, 918-19.
While some commentators assert that the requirement of more than business purpose to justify the use of a race-correlated practice is more consistent with equal achievement than with the equal treatment interpretation of Title VII,[42] this is not always true for two reasons. First, while a particular race- or sex-correlated predictor[43] may indeed be indicative of productivity (and thus one that would have business purpose), the racist or sexist employer may be placing a heavier emphasis on that predictor than would a non-racist or non-sexist employer.[44] Thus, if the possessor of a particular degree is marginally more productive-if, for example, he is worth twenty cents an hour more-but if the racist or sexist employer pays not twenty cents an hour more, but fifty cents an hour more or even refuses to hire those without the degree, then the inference that the employer is using the predictor as an excuse for paying one group more is both rational and fair. Second, if there is an alternative predictor available which would serve equally well with lesser disparate impact, then to use the initial predictor would have undue *265 impact on minorities not justified by productivity considerations. Cf. No-Alternative Approach, supra n.37, at 115-16; Parson v. Kaiser Aluminum & Chemical Corp., 575 F.2d 1374, 1389 (5th Cir. 1978), cert. denied, 441 U.S. 968, 99 S.Ct. 2417, 60 L.Ed.2d 1073 (1979).
On the other hand, a "balancing" approach, wherein a court balances the impact on an employer of disallowing a practice and the impact of blacks or females of allowing the practice, is not consistent with the equal treatment construction of Title VII. Cf. No-Alternative Approach, supra n.37, at 101 & 119. For instance, in Bing v. Roadway Express, Inc., 444 F.2d 687, 690 (5th Cir. 1971), the court appears to have compared the benefits accruing to the employer from the use of a practice and the extent of the disparate impact-a mixed equal treatment-equal achievement approach. See Swint v. Pullman-Standard, 624 F.2d 525 (5th Cir. 1980); Note, 12 Ga.L. Rev. 104, 106 n.12 (1977). Often, these courts are looking at equality of results not because they fail to adhere to the equal treatment interpretation of Title VII, but because they were not presented with sophisticated quantitative indicators of equality of treatment itself, and so had to make do with looking at results. This court, however, having direct quantitative evidence of whether there has been equality of treatment, need not deviate from such an interpretation. This court will, and must, adhere to the equal treatment philosophy intended by Congress.
We defer discussion to section VI(E), infra, of the specifics of how econometric modeling may indicate whether differential treatment is due entirely to productivity differences or to group status as well. We can, however, describe intuitively three ways in which such non-productivity-based behavior may manifest itself. First, the mathematical relationship found may indicate that a non-job-related or otherwise improper predictor is being used as a basis for differentiation. Second, the mathematical relationship found may indicate that even though a black and white are identical in all productivity-characteristics, they are being treated differently ("unequal treatment of twins"). Third, though such identical blacks and whites may be treated equally, the productivity characteristics disproportionately possessed by the favored group may be rewarded more than they would be rewarded by a nonracist (or nonsexist) employer (an aspect of "improper treatment across twins"). See section VI(E), infra.

C. Controlling for Productivity.

As earlier stated, the fundamental behavioral assumption of both the plaintiffs and the Bank is that the Bank treats people with different productivities (or true indicators of productivity) differently. Thus the mathematical models are designed to determine if there is any differential treatment not entirely attributable to such productivity differences. The productivity of an individual is gauged, in part, by those observable characteristics, such as schooling, thought to affect productivity. Cf. Gwartney, Asher, Haworth, & Haworth, supra n.40, at 636-37.
The plaintiffs' experts explicitly rely on what is called "human capital theory" to determine what observable characteristics out to be controlled for in their multiple regression models. See Plaintiffs' Exhibit 504, at 1; Testimony of Dr. Janice Madden at 11. See generally Sahota, Theories of Personal Income Distribution: A Survey, 16 J.Econ.Lit. 1 (1978); Blaug, The Empirical Status of Human Capital Theory: A Slightly Jaundiced Survey, 14 J.Econ.Lit. 827 (1976); P. Samuelson, Economics 751-52 (11th ed. 1980). Dr. Blau explained the theory behind her modeling as follows:
Economists define wage (or salary) discrimination as occurring when pay differences between groups exist that cannot fully be explained by productivity differences between those groups. (Becker, 1957). This definition matches our intuitive or common sense view that if one individual is more productive than another, he or she is entitled to higher pay. However, pay differences that cannot be explained by productivity differences are *266 suspect, especially if they are associated with race and/or sex differences.
One problem with applying this definition of salary discrimination is that firms generally do not keep records of employees' productivity (e. g., output per work hour). Indeed in some industries, like banking, employee output would be very hard to measure. For this reason, economists generally focus upon the characteristics of individuals that make one more or less productive than another, rather than upon productivity itself. The human capital theory (Becker, 1964; Mincer, 1973) is a widely accepted analysis of the determinants of earnings differences among individuals. The theory focuses upon the investment that individual workers and their employers make that increase the workers' knowledge and skills and thus make them more productive. The following factors are particularly important:
(1) Increases in knowledge and skills through formal education (generally measured by years of schooling).
(2) Increases in knowledge and skills through informal training acquired on-the-job. This is of two types:
(a) General training which is useful in a variety of firms (generally measured by years of labor market experience).
(b) Specific training which is useful only in the firm in which the training is acquired (generally measured by years of experience with the firm).
Thus, human capital theory also leads us to some common sense conclusions. If one individual has more education and experience than another, he or she is probably entitled to higher pay (with the size of the pay difference depending on the size of the difference in education and experience). However, pay differences in qualifications are suspect, especially if they are associated with race and/or sex differences. Economists generally attribute such unexplained pay differences to discrimination.

How does such salary discrimination arise? Two major mechanisms have been identified:
(1) UNEQUAL PAY FOR EQUAL WORK. Pay differentials among individuals with similar qualifications that are due to pay differences within occupational categories.
(2) UNEQUAL WORK. Pay differentials among individuals with similar qualifications that are due to differences in occupational distributions (i. e., differences in access to occupations on the basis of race and/or sex). Such discrimination in access to occupations may be due to discriminatory hiring and/or promotion policies of the firm.
Plaintiffs' Exhibit 504, at 1-2.
The Bank's principal expert, Dr. Stoikov, similarly offers multiple regressions which control for characteristics a productivity-minded employer might wish to use as predictors. For instance, as to her compensation analysis, Dr. Stoikov stated:
Sex- or race-related pay disparities are estimated using a model which relates an employee's pay to those characteristics which are thought to influence the quantity and quality of his/her work. These characteristics include the skills an individual brings to his/her employment at the time of hire that have been acquired through education and previous employment experience, the skills an individual has acquired since the time of hire, and an individual's motivation to be productive at his/her present job assignment and to assume future assignments which entail larger amounts of responsibility, skill and effort.
Individuals bring different quantities and types of skills to their employment, as well as different degrees of willingness to acquire new skills or to assume additional responsibilities. Moreover, when these factors are averaged separately for men and women (nonblacks and blacks) differences usually are observed. Therefore, we can anticipate that there will be a difference in the average pay of men and women (nonblacks and blacks) that reflects the differences in the quantity and quality of work-related characteristics *267 they possess. This difference is not attributable to their employer's discriminatory behavior, but rather is an expected pay disparity attributable to differences in employee-determined, work-related characteristics. Remaining sex- or race-related pay disparities are pay differences that cannot be related to male/female (nonblack/black) differences in these employee-determined, work-related characteristics. They may or may not stem from an employer's discriminatory behavior.
Defendant's Exhibit 32, at 16-18 (footnote omitted).

D. The Mathematics of Regression Analysis.

In today's society, among those who claim special insight denied the common run of men, the "esoteric language is mathematics; the special means of inspiration, the computer, the forbidden path of truth, science." B. Ackerman, et al., The Uncertain Search for Environmental Quality 1 (1974). Econometricians-who with multiple regression analysis can provide an important addition to the judicial toolkit necessary for reconstructing from bits and pieces of data the framework of past events-are no exception.
The practical use of multiple regression has grown markedly over the past 25 years due to the development of statistical methodology itself, increasing availability of statistical data, and most importantly, the development of the computer. Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. 702, 702 (1980). Regression analysis is increasingly being used in legal proceedings and commentary. Id.; Statistical Evidence on the Deterrent Effect of Capital Punishment: Editors' Introduction, 85 Yale L.J. 164, 167 n.15 (1975) (Editor's Introduction). To record this court's understanding, correct or not, of the regression analyses presented by the experts in this case, and the limitations of those analyses-an understanding made necessary by technically complex trial challenges to their validity-we here describe how they are used, how they work, and when they do not. Cf. id. at 167 n.15 & 169; Fisher, supra, at 702.[45]

1. Uses of Multiple Regression.

The two primary uses of multiple regression analysis can be illustrated through the following examples where such analyses have actually been used:
(i) For years after the disappearance of coal-burning locomotives, there was dispute on the preservation of the jobs of railroad firemen. One issue was whether the presence of a fireman on trains contributed to railroad safety.
(ii) Cable television systems (CATVs) have been involved in administrative proceedings where one issue is the magnitude of the effect of the entry and activity of CATVs upon the profits and growth of broadcast television stations. This issue presents such questions as the influence of CATVs on the viewing audience of particular stations and the effect of changes in a station's audience on its revenues. Of course some claim that such effects are small while others insist they are large.
Fisher, supra, at 703.[46]
In the first case, multiple regression is being used to "test hypotheses"-does a particular variable[47] (presence or absence of *268 firemen) have any effect on some other variable (railroad safety). In the second case, multiple regression is being used for "parameter[48] estimation"-there being little doubt that audience size affects television revenue, and the real question being how much. Fisher, supra, at 704.
Both the firemen and CATV cases above involve "conditional forecasting"-a prediction of what will happen to the "dependent variable"[49] (such as railroad safety) if an "independent variable"[50] (such as the number of firemen) is changed or, looking retrospectively, what would have happened to the dependent variable had the value of an independent variable been different. Fisher, supra.
Determining whether firemen do affect railroad safety faces two difficulties in the absence of multiple regression analyses. First, the factor whose influence one wishes to test or measure is usually not the only major factor affecting the dependent variable. Thus, for instance, the amount of traffic on the railroads affects accidents as well. If we could make controlled experiments, it would be easy to quantify the relationship. A controlled experiment here would involve varying number of firemen, traffic on railroads, and the other variables expected to affect the number of accidents one at a time, holding everything else constant and observing the resulting number of accidents. This would be difficult and costly. We are left then with analyzing nature's experiments. See id. at 705; cf. R. Wonnacott & T. Wonnacott, Econometrics 7 (2d ed. 1979). Second, even if the effects of other systematic factors can be accounted for, there typically remain elements of chance. Id. Falling objects follow a physical law, but the behavior of individuals does not. Cf. R. Wonnacott & T. Wonnacott, supra at 6.

2. Econometrics and the Ordinary Least Squares Form of Multiple Regression Analysis.

Put in more formal terms, econometrics may be viewed as the science of model building, using quantitative tools used to construct and test mathematical representations of parts of the real world. R. Pindyck & D. Rubinfeld, Econometric Models and Economic Forecasts xi (1976). The fundamental underpinning of econometrics is the basic idea of relationships among economic variables. Relationships are grouped to form a model, the number of relationships included in an economic model depending on the objectives for which the model is constructed and the degree of explanation that is being sought.[51]
The models presented in this case involve, for the most part, one type of econometric modeling. They are "single-equation regression models." R. Pindyck & D. Rubinfeld, supra, at 1. Most of the single equation regression models are of a common variety: the behavior of the "endogenous" variable (a variable determined within the economic system under study) is assumed to be a linear function[52] of a set of "exogenous" *269 variables (those determined outside the system),[53] and the variables are assumed to possess certain other properties such that the convenient "ordinary least squares" method of estimating the relationships among those variables can be used. R. Pindyck & D. Rubinfeld, supra, at 1, 225; Editors' Introduction, supra; G. Maddala, supra n.49, at 5; R. Wonnacott & T. Wonnacott, supra, at 334-35.
Multiple regression begins by specifying the major variables believed to affect the dependent variable. Fisher, supra, at 705. For instance, in our railroad example, we may wish to include as explanatory variables the number of firemen and the amount of railroad traffic, using as the dependent variable the number of railroad accidents. This involves using independent variables which reflect the important or systematic influences that may affect railroad safety. The "minor influences" are placed in a "random disturbance term," treating their effects as due to chance. Id. at 705-06. The relationship between the dependent variable and the independent variable of interest-for example, the relationship between the number of accidents and the number of firemen-is then estimated by culling the effects of the other major variables. Multiple regression is thus a substitute for controlled experimentation. Id. at 706. The results of multiple regressions-such as what we will call "coefficients" in the ordinary least square methodology-can be read as showing the effect of each independent variable on the dependent variable, holding the other independent variables constant. Moreover, relying on statistical inference, one can make statements about the probability that the effect described is due only to a chance fluctuation. Cf. id.
Central to the validity of any multiple regression model and resulting statistical inferences is the use of a proper procedure for determining what explanatory variables should be included and what mathematical form the equation should follow. The model devised must be based on theory, prior to looking at the data and running the model on the data. If one does the reverse, the usual tests of statistical inference do not apply. And proceeding in the direction of data to model is perceived as illegitimate.[54] Indeed it is important in reviewing the final numerical product of the regression studies that we recall the model's dependence upon this relatively intuitive step.

a. Estimating Multiple Regressions.

If the relationship of interest is to include only one independent variable ("x1") to explain the behavior of a dependent variable ("Y"), and it is believed that the relationship of Y to x1 is a straight line, then the relationship is expressed mathematically as:
(1) Y = a + b1x1
where a and b1 are constants, Y is the dependent variable, and x1 is the independent variable. Diagrammatically, the relationship is illustrated by the straight line in Figure 1. The econometrician thus seeks to determine the value of "a" (called the "intercept" or "constant") and the value of "b1" (the "coefficient" or "slope" of x1). Once he obtains these two numbers, for any value of x1, he would know the exact value of Y. See G. Maddala, supra n.49, at 74; J. Johnston, supra n.51, at 122. Because there are random influences in life, it is unlikely that the relationship between Y and x1 will be so *270 exact. Instead, plotting values of Y against values of x1 will likely produce a scatter of points as in Figure 2. Thus the correct relationship is not described by equation (1) but instead by:
(2) Y = a + b1x1 + u

where u represents random influences and is called the "residual" or "error." See G. Maddala, supra n.49, at 74. The econometrician attempts to cut through the noise generated by these random disturbances and extract the "signal"-that is, the line around which the points are scattered. He passes the line through the points so that it is as close as possible to the scatter of points, in the sense that the sum of the squared deviations between the predicted and actual Y values is minimized. Not inappropriately, this is called "least squares regression." G. Maddala, supra n.49, at 75. See Figure 3.
In general, numerical estimates of a and b1 are obtained by entering the scatter of points in a computer programmed to perform "least squares" calculations. These empirical estimates, based on the relatively crude "least squares" method of processing empirical observations, will be "good"-in that they are likely to be quite close to the true values-if and only if certain assumptions hold as to the nature of and relationship among the dependent variable, the independent variable(s), and the error term. See, e. g., R. Pindyck & D. Rubinfeld, supra, at 20-24, 55; R. Wonnacott & T. Wonnacott, supra, at 55-69; J. Johnston, supra n.51, at 126; J. Kmenta, Elements of Econometrics 9-14, 161 (1971).[55]
Regressions involving more than one explanatory variable (e. g. number of firemen as well as railroad traffic) are more frequently used and are known as "multiple regressions." Id.; R. Wonnacott & T. Wonnacott, supra, at 71. To illustrate, we may assume that there is a suspected relationship between Y, the dependent variable, and the explanatory variables, x1, x2, x3, ... xk, such that:
(3) Y = a + b1x1 + b2x2 + b3x3 + ... + bkxk + u
where a, b1, b2, b3, ... bk are constants. As with a model with only one explanatory variable, a is called the intercept or constant; b1, the coefficient of x1; b2, the coefficient of x2; and so forth; and u, the "error term" or "residual."[56] When empirical observations are placed in the computer programmed to do least squares manipulations (just as we input the "points" in the model with one explanatory variable), it will generate numerical estimates of a, b1, b2, ... bk based on how the dependent variable changes when the independent variables move in a variety of ways. Cf. Fisher, supra, at 712. These numerical estimates of b1, b2, b3 ... bk, are analogous to the b1 in Figure 3. When only one explanatory variable is used, b1 is the slope of the line: that is, it tells us how much Y will change for a unit change in the value of x1. With a multiple regression, b1 tells how much Y will change for a unit change in the value *271 of x1, holding the other explanatory variables constant; b2 is the change in Y corresponding to a unit change in the value of x2, holding the other explanatory variables constant; and so forth. Thus if the proper model for railroad safety were described by:
(4) Y = a + b1x1 + b2x2 + u
where Y is the number of railroad accidents, x1 is the number of firemen, and x2 the miles of railroad traffic, we can "run" this model on the appropriate data, and obtain, for example, Y = 50 + ฝ x1 + พ x2.
This equation tells us, among other things, that with each additional fireman, the number of railroad accidents is reduced by ฝ of an accident, while with each additional mile of railroad traffic, the number of railroad accidents is increased by พ of an accident.[57]

b. Statistical Inference.

With certain assumptions about the particular probability distribution of the error term, one can go beyond estimating effects of independent variables on the dependent variable to gauging the certainty or accuracy of those effects. Fisher, supra, at 716; G. Maddala, supra n.49, at 79. For instance, one can perform calculations that support statements about the range of values likely to contain the true coefficient of any of the explanatory variables.
In particular, one can determine the size of the interval on either side of the estimated coefficient which has a given probability (e. g., .95) of containing the true parameter. This range of variables is referred to as a "confidence interval." R. Pindyck & D. Rubinfeld, supra, at 31. For example, after performing the appropriate calculations, we might find that there is a 95% chance that the true coefficient of firemen in our railroad example is larger than ผ and smaller than พ. In other words, the 95% confidence interval for the firemen coefficient is ฝ ฑ ผ.
One can also perform calculations to test the hypothesis, at a given level of statistical significance,[58] that the true coefficient is actually zero; that is, that the independent variable to which it corresponds has no effect on the dependent variable. Fisher, supra, at 717. If what is known as the "t-statistic" for the particular coefficient is large enough, then we can reject the hypothesis that the true coefficient of the variable in *272 question is equal to zero. See R. Pindyck & D. Rubinfeld, supra, at 31; R. Wonnacott & T. Wonnacott, supra, at 86. For example, if a 5% level of significance is used, a sufficiently large t-statistic for the coefficient indicates that the chances are less than one in 20 that the true coefficient is actually zero. Fisher, supra, at 717. The magnitude of the t-statistic necessary to reject such hypothesis varies with the desired level of significance. Thus the t-statistic would need to be larger with a 5% level of significance than with a 10% level of significance.
The magnitude of the t-statistic is also dependent on the particular type of hypothesis to be tested. When an analyst uses a "two-tailed hypothesis"-that is, where he wishes to determine whether there is any relationship, positive or negative, between the independent and the dependent variable-a t-statistic of approximately two means (in the case of large samples) that the chances are less than one in 20 that the true coefficient is actually zero. Cf. id. An explanatory variable is, however, usually included in the equation because of a prior theoretical reason for expecting it to affect the dependent variable in a specific direction. R. Wonnacott & T. Wonnacott, supra, at 86. For instance, we expect additional firemen to decrease, not to increase, the number of accidents. In such situations, we test a more specific hypothesis-a "one-tailed hypothesis"-not whether or not a particular coefficient is positive or negative as with the two-tailed test, but whether or not, in our railroad example, it is negative or zero. In this circumstance, the "one-tailed test" is one in which 5% would be the probability of observing some negative coefficient if the true value were zero. Cf. Fisher, supra, at 717 n.26. See also n.148, infra. The t-statistic required for significance at the 5% level on a one-tailed test is only approximately 1.6. Fisher, supra, at 717 n.26.
Speaking purely from a statistical point of view, this does not mean that only results significant at the 5% level should be considered; less significant results may be suggestive. Id. at 718-19. Thus even if the t-statistic is small for a particular dependent variable, this does not mean there is no relationship between that dependent variable and the independent variable. R. Wonnacott & T. Wonnacott, supra, at 88. This is so because the size of the t-statistic corresponding to statistical significance is ultimately dependent on the level at which statistical significance is arbitrarily set. "The most commonly used significance level is five percent, but this is purely arbitrary." G. Maddala, supra n.49, at 45. It follows that in many instances, reporting only whether a particular coefficient is significant or not should be avoided; it is more informative to report confidence intervals, test statistics, or other quantitative measures of significance. Id. at 45-46.
Moreover, significance tests and confidence intervals are controversial when the test data includes an entire universe of decisions. For example, in a promotion case, data on every promotion decision may be available, rather than a mere random sample of such decisions. When all such data are available, some statisticians argue that statistical tests are either meaningless or misleading, while others say that they can be useful. D. Baldus & J. Cole, supra n.55, ง 9.32, at 316. See also Testimony of Dr. John Spaulding, Jr., at 1094-95; Johnson Post-Trial Brief at 4 n.4.
Considering the arbitrary nature of the adoption of the 5% level of significance, it is not surprising that courts show flexibility in determining what level of significance to be required in a legal context. Cf., e. g., United States v. Georgia Power Co., 474 F.2d 906, 915 n.11 (5th Cir. 1973); D. Baldus & J. Cole, supra n.55, ง 9.221, at 308 n.36 & ง 9.41, at 318. See generally section IX(A), infra. Indeed, the Fifth Circuit has specifically stated that a 10% level of significance rather than the statisticians' more conventional and more stringent 5% level "might be acceptable" in the context of the validity of job tests under Title VII. Watkins v. Scott Paper Co., 530 F.2d 1159, 1187 n.40 (5th Cir.), cert. denied, 429 U.S. 861, 97 S.Ct. *273 163, 50 L.Ed.2d 139 (1976) (emphasis in original).[59]
Statistical tests more elaborate than t-statistics and confidence intervals for individual coefficients are also available. For instance, "R2" is often calculated; this is a statistic which provides an overall index of how well Y can be explained by all the independent variables, that is, now well a multiple regression fits the data. R. Wonnacott & T. Wonnacott, supra, at 180-81. The higher the R2, the greater the association between movements in the dependent and independent variables. Fisher, supra, at 720. There are problems, however, associated with the use of R2. A high R2 does not necessarily indicate model quality. See R. Pindyck & D. Rubinfeld, supra, at 58; G. Maddala, supra n.49, at 122-24. Cf. Fisher, supra; D. Baldus & J. Cole, supra n.55, ง 8.22, at 266-67. For instance, the addition of more explanatory variables to the regression equation can never lower R2 and is likely to raise it. Thus one could increase R2 by simply adding more variables, even though, because of "over-inclusion" and "multicollinearity" (terms we will later describe) it may be improper econometrically to do so. Cf. R. Pindyck & D. Rubinfeld, supra.

3. What Can Go Wrong?

A "perfect" model would explain completely the process under study. While such models are found in the physical sciences, they are rare in the social sciences. D. Baldus & J. Cole, supra n.55, ง 8.21, at 264. Indeed, it has been argued that no model in the social sciences ever meets the requirements for a perfect regression analysis. Id., ง 8.22, at 266. But this does not mean that because a model is subject to challenge, its results are valueless. Cf. Editors' Introduction, supra, at 169. Small departures from assumptions necessary for a perfect regression may have small deleterious effects. Cf. Fisher, supra, at 711.
The value of a regression would obviously be affected by problems in the underlying data or by mismeasurements of the explanatory variables. Cf., e. g., R. Pindyck & D. Rubinfeld, supra, at 194-202; Kmenta, supra at 336-41; G. Maddala, supra n.49 at 201-07; J. Johnston, supra n.51, at 281-91; Beyond the Prima Facie Case, supra, n.57, at 417-21. See generally section V, supra, and section VII(C), infra (introduction). And, as discussed earlier, the model must be one justified by theory. We turn here to a few of the less intuitively obvious ways in which ordinary least squares can provide unreliable results important to this case.
In a perfect regression model using the ordinary least squares technique,[60] three major assumptions would hold:
(a) that the effects of the random disturbance term are independent of the effects of the independent variable; (b) that the values of the random term for different observations are not systematically related and that the average squared size of the random effect has no systematic tendency to change over observations; and (c) that the sum of random effects embodied in the disturbance term is distributed normally, in the "bell curve" generally characteristic of the distribution *274 of the sum of independent random effects.
Fisher, supra, at 708.
One way the first assumption may be violated is if some relevant explanatory variable has been left out of the analysis. This is one type of "misspecification" or "specification error." For instance, if yield of a crop is dependent on both amount of fertilizer and rainfall, and if in our regression we include only fertilizer, there will be improper omission of an explanatory variable. The problem caused by omission of variables is that the regression coefficient(s) of the included explanatory variable(s) (e. g., of fertilizer) would be "biased" (that is, not likely to be correct "on the average"),[61]and the usual tests of significance concerning the included regression coefficient(s) (such as calculation of a confidence interval) will be invalid. Kmenta, supra, at 392-95. See J. Johnston, supra n.51, at 169. Certain statistical tests are available to suggest whether this sin of omission has occurred. See, e. g., Kmenta, supra, at 405.
However, in at least one circumstance, this problem may not be a serious one in cases where the issue is whether discrimination exists or does not exist. Where it is possible to use as proxy for the presence (or absence) of discrimination against a particular group a "dummy" or "group status" explanatory variable,[62] such an omission will not threaten the validity of the group status coefficient (and hence, the validity of the model's suggestion of the existence or nonexistence of discrimination) unless the omitted variable is related to the group status variable. D. Baldus & J. Cole, supra n.55, ง 8A.1, at 273. Thus, here, where the plaintiffs' model had compensation as its dependent variable, and various explanatory variables (including dummy variables for race and sex), Dr. Madden stated:
Q. Do you agree that your report would have been more valid if you measured all potential productivity?
A. Well, since my purpose was to analyze sex and race my report would have been more valid had there-to the extent that there are any omitted productivity variables that are correlated with race or sex. To the extent they're not correlated with race or sex [it] makes no difference whatsoever.
Testimony of Dr. Janice Madden at 104. Cf. Testimony of Dr. Judith Stoikov at 91.
Another type of specification error occurs when one or more irrelevant variables are included in the model. This overinclusion by itself causes fewer problems than under-inclusion. See R. Wonnacott & T. Wonnacott, supra, at 413; R. Pindyck & D. Rubinfeld, supra, at 190; Kmenta, supra, at 399; J. Johnston, supra n.55, at 169. Overinclusion of variables, however, increases the risk of "multicollinearity."
Multicollinearity refers to a situation where due to the high (but not perfect) correlation of two or more variables (or combination of variables), it becomes difficult to disentangle their separate effects on the dependent variable. R. Pindyck & D. Rubinfeld, supra, at 67; G. Maddala, supra n.49, at 183. Multicollinearity creates broad confidence intervals, and estimates of coefficients become sensitive to particular sets of sample data: a multicollinear model makes it difficult to establish that an individual explanatory variable influences the dependent variable. R. Wonnacott & T. Wonnacott, supra, at 353; J. Johnston, supra n.51, at 160. Thus, even if two explanatory variables should be included in the regression, if multicollinearity is serious it may be necessary to drop one of them. This, in turn, may cause problems associated *275 with omission of variables, but those problems might in certain circumstances be acceptable in the face of more serious problems of multicollinearity. See R. Pindyck & D. Rubinfeld, supra, at 68; G. Maddala, supra n.49, at 190. Cf., e. g., Kmenta, supra, at 390-91 (an alternate solution to multicollinearity is acquisition of more data). There are some rough rules of thumb to judge whether multicollinearity is serious or not. G. Maddala, supra n.49, at 186; Kmenta, supra, at 389-91. Thus, in a discrimination model, when too many qualification variables are included, the patterns of correlation among the explanatory variables will be such that the confidence interval of the group status coefficient is inflated. Cf. D. Baldus & J. Cole, supra n.55, ง 8A.1, at 274. Hence, if multicollinearity exists, the probability will be increased that the net impact of group status will be judged statistically nonsignificant, even in cases where there are actual differences in the treatment. Id. at 275.
Another form of specification error, occurs in the case where the analyst chooses to use a regression equation that is linear in the explanatory variables when the true regression model is nonlinear in the explanatory variables. For instance, the analyst may think the relationship is best described by, and thus runs the regression on:
(5) Y = a + b1x1 + u
while the true relationship is:
(6) Y = a + b1x1 + b2(x1)2 + b3(x1)3 + u
Specification of a linear model when the model is nonlinear-an error in the "form" of the specification-can lead to biased estimates. R. Pindyck & D. Rubinfeld, supra, at 190-91.
We have discussed the first major assumption underlying ordinary least squares with reference to specification errors. When the second assumption is violated, i.e., when the scatter or variance of the error terms about zero (the point of perfect prediction) is not approximately the same for all values of each independent variable, "heteroscedasticity" is said to exist. R. Pindyck & D. Rubinfeld, supra, at 17; R. Wonnacott & T. Wonnacott, supra, at 194-95; D. Baldus & J. Cole, supra n.55, ง 8A.42, at 284. Heteroscedasticity can produce errors such as errors in confidence intervals. Id. at 285.
The third major assumption underlying ordinary least squares is that the error term follows the "normal distribution."[63] With respect to this assumption, basic least squares regression models are "quite `robust' in that they will tolerate substantial deviations without affecting the validity of the results." D. Baldus & J. Cole, supra n.55, ง 8A.41, at 284. Nonnormality of errors can be detected, through the use of such techniques as the Kolmorgorov-Smirnov test. G. Maddala, supra n.49, at 306.

E. Econometric Indication of Discrimination.

In econometric terms, one way used (here and in the economic literature) to study differences among individuals' wages is to estimate a regression such as:
(7) Y = a + b1x + b2x2 + ... + bkxk + u
where Y is the level (or natural logarithm)[64] of earnings, income or wage rate, and x1,x2 ..., xk are observable productivity characteristics. Cf. Blinder, Wage Discrimination: Reduced Form and Structural Estimates, 8 J. Human Res. 436, 437 (1973). To illustrate, we could assume that the employee's wages are a function of his years of *276 schooling and years of relevant experience, both factors being postulated by human capital theory to be productivity-enhancing. In such a model, Y would be earnings, x1 would be the number of years of schooling, and x2 would be the number of years of relevant experience. If the assumptions necessary for ordinary least squares hold, and if this model is a valid one, then by running the model on actual data for the firm, we would obtain actual numbers for a, b1, and b2. To determine the predicted earnings for any employee, we would plug his particular values of x1 and x2 into equation 7, using the estimates of a, b1, and b2.
Recalling from our discussion in section VI(B), supra, there are three ways the econometric models, if true mathematical representations of real-world behavior, may indicate that the employer is discriminatory: (1) use of non-job-related criteria with disparate impact; (2) unequal treatment of twins; and (3) improper treatment across twins.
The first type of indication of discrimination may occur, for instance, where the employer states that he behaves according to a certain model, and that model includes explanatory variables which do not meet the legal standard of business necessity. The process of mathematically representing an employer's behavior should at this stage be external to the legal framework, such as those concerned with the legality of the employer's use of various predictors. In determining whether the employer is violating Title VII through the use of non-job-related criteria with disparate impact, one must follow a three-step procedure:[65]
(i) On purely econometric grounds, derive a mathematical model which best represents how the employer behaves. At this stage, it is not necessary for there to be more than business purpose to justify inclusion of a variable. Indeed, to require such a higher standard would be flawed methodologically. This is not to say that all relevant productivity-related variables should always be included: the problems which may be introduced by overinclusion and multicollinearity, described in section VI(D), supra, must always be considered.
(ii) Examine the variables with disparate impact and see if they are adequately justified under the "business necessity" standard.
(iii) For those variables justified under (ii), see if those variables have been subject to group status-based manipulation by the employer.
Step (ii) above is in part a restatement of the job-relatedness requirement:
Once the racially adverse impact of an examination is demonstrated by statistical or other evidence, the burden of proof shifts to the employer to prove that the exam is job-related. Albemarle Paper Co. v. Moody, 1975, 422 U.S. 405, 426, 95 S.Ct. 2362, 2375, 45 L.Ed.2d 280, 301. See Griggs v. Duke Power Co., supra, 401 U.S. at 432, 91 S.Ct. at 854, 28 L.Ed.2d at 164. Employment practices having a racially adverse effect are subject to the same scrutiny: if a prima facie case of discriminatory practice is shown, it becomes the employer's burden to demonstrate the job performance validity of its practices. Washington v. Davis, [426 U.S. 229, 96 S.Ct. 2040, 48 L.Ed.2d 597], supra. The employer's burden is not satisfied by establishing merely a rational basis for a test; the test must be validated. Washington v. Davis, supra, 426 U.S. at 247, 96 S.Ct. at 2051, 48 L.Ed.2d at 611. Validation, in general, requires a demonstration that "the qualifying tests are appropriate for the selection of qualified applicants for the job in question." Id. Validation by any one of a number of methods of making such a showing, see id. at 247 n.13, 96 S.Ct. at 2051, 48 L.Ed.2d at 611, suffices to refute statistical evidence that a test has had a disproportionate racial impact. Id. See also Albemarle Paper *277 Co. v. Moody, supra, 422 U.S. at 425-31, 95 S.Ct. at 2375-78, 45 L.Ed.2d at 300-304. The same sort of validation will rebut the inferences drawn from statistical evidence of racial discrimination based on work practices.
Scott v. City of Anniston, 597 F.2d 897, 901-02 (5th Cir. 1979), cert. denied, 446 U.S. 917, 100 S.Ct. 1850, 64 L.Ed.2d 271 (1980).[66]See also Smith v. Olin Chemical Corp., supra (re those criteria which are "manifestly job-related"); Garcia v. Gloor, supra n.36, at 269 n.6, 270 & 272 ("there is no disparate impact if the [employment] rule is one the affected employee can readily observe and nonobservance is a matter of individual preference"); Dothard v. Rawlinson, 433 U.S. 321, 331, 97 S.Ct. 2720, 2727, 53 L.Ed.2d 786 (1977) (re directness of predictors).
What is to be proven in job-relatedness is separate from the methods used in such proof. We leave description of the methods to section IX(F)(1), infra. The employer must show that he is using the predictor to the same extent as would a race-(or sex-) blind employer.[67]
Finally, predictors must be such that if "the legitimate ends of safety and efficiency could be served by a reasonably available alternative system of classification with less discriminatory effect, the classification cannot be continued." Note, 12 Ga.L.Rev. 104, 106 n.12 (1977). See United States v. Jacksonville Terminal Co., 451 F.2d 418, 451 (5th Cir. 1971), cert. denied, 406 U.S. 906, 92 S.Ct. 1607, 31 L.Ed.2d 815 (1972). More specifically, the Fifth Circuit has held that:
[F]or a practice, which is not intentionally discriminatory or neutral but perpetuates consequences of past discrimination, to be justified by business necessity, the practice must "not only foster safety and efficiency, but must be essential to that goal ... and there must not be a acceptable alternative that will accomplish that goal equally well with a lesser differential racial impact." Parson v. Kaiser Aluminum & Chemical Corp., 575 F.2d 1374 (5th Cir.), cert. denied, 441 U.S. 968, 99 S.Ct. 2417, 60 L.Ed.2d 1073 (1979).
Swint v. Pullman-Standard, 624 F.2d 525, 536 (5th Cir. 1980). But cf. Garcia v. Gloor, supra n.36, at 271.
Step (iii) of our three-step analysis relates to use of predictors which are susceptible to manipulation by the employer. If a predictor which, while not valueless as a predictor of productivity, is subject to manipulation by a racist employer, the econometric results generated may well be biased in favor of a finding of nondiscrimination. Cf., e. g., Greenspan v. Automobile Club of Michigan, 495 F.Supp. 1021, 22 Empl.Prac. Dec. ถ 30,812, at 15,193-94 (E.D.Mich.1980); Finkelstein, The Judicial Reception of Multiple Regression Studies in Race and Sex Discrimination Cases, 80 Colum.L.Rev. 737, 738-41 (1980). Thus, for instance, in James v. Stockham Valves & Fittings Co., 559 F.2d 310 (5th Cir. 1977), cert. denied, 434 U.S. 1034, 98 S.Ct. 767, 54 L.Ed.2d 781 (1978), the defendant presented an earnings regression study which included a merit rating as one of its explanatory variables. The court observed, "If there is racial bias in the subjective evaluations of white supervisors, then that bias will be injected into ... [the] earnings analysis." Id. at 332. Use of manipulable predictors in multiple regressions may cloak the employer's "personal biases in the mantle of a scientific judgment." Cf. Underwood, supra n.35, at 1443.
*278 Since to include manipulable predictors as explanatory variables may be to bias the econometric results in favor of the employer, if despite such inclusion the plaintiffs are able to show non-productivity-related pay differentials, the results are all the more impressive. But in the defendant's hands, while models which include manipulable predictors among the explanatory variables are not valueless, especially if those predictors are shown not to have been manipulated, they may be suspect.
The second way in which econometrics may indicate discrimination occurs when the coefficients indicate that a white and a black are being rewarded differently for the possession of the same productivity characteristic(s): there is not equality of treatment of twins. See section VII(C), infra. Referring to the econometric model described in the beginning of this section (wherein x1 is the number of years of schooling and x2 is the number of years of relevant experience), we can see that there are at least two possible econometric indications of this sort of inequality:[68]
1. First, one can run the same equation on data twice, the initial time running it on data solely for one group, and second time on data solely for the other group. After doing this, one compares the coefficients obtained for the two groups. Cf. Testimony of Dr. Francine Blau at 66.
So if we obtain for males Y = 10000 + 500x1 + 350x2, and for females Y = 9000 + 400x1 + 300x2 at the same job, it is clear that females are being discriminated against. That is, any female regard less of her particular combination of schooling and experience is receivingless than a corresponding male: since the regression coefficients (a, b1, and b2) are each higher for males than for females, males are being rewarded more for each productivity-related characteristic. This is the simple situation, where it is clear that females are being discriminated against-where a (female) is less than a (male), and b1 (female) is less than b1 (male), b2 (female) is less than b2 (male), and so on.
A more difficult case would be that where males are rewarded more for one characteristic, while females are rewarded more for another: a "mixed" case. Cf., e. g., Defendant's Exhibit 32, at Table 5. Here some men would be paid more than women with identical portfolios of productivity characteristics, while some women would be paid more than men with identical portfolios of productivity characteristics. For instance, suppose that Y = 10000 + 500x1 + 350x2 for males and Y = 10000 + 400x1 + 1000x2 for females.
(a) A woman with no schooling and two years of experience will be paid 10000 + 0 + 2000 or $12,000; a similar man will be paid 10000 + 0 + 700 or $10,700. Here a woman earns more than a man with the identical portfolio of productivity characteristics.
(b) A woman with three years of schooling and no experience will be paid 10000 + 1200 or $11,200; a similar man will be paid 10000 + 1500 + 0 or $11,500. Here a woman earns less than a man with the identical portfolio of productivity characteristics.
In such a "mixed" case, one might, for example, offer statistical evidence to show that as a result of different regression coefficients for the two groups, because of the empirical distribution patterns of productivity characteristics in the two groups one group is favored at the expense of the others. Cf., e. g., Oaxaca, Male-Female Wage Differentials in Urban Labor Markets, 14 Int'l Econ.Rev. 693, 694-97 (1973); Defendant's Exhibit 32, at 24-25.[69]

*279 One manner of comparing the regression coefficients for the two groups without actually running two separate regressions as described above is to use what are called "interaction" terms. By adding such terms to the regression equation to be estimated, one can produce estimates of the difference in the weights placed on each qualification. Thus, if we added an interaction term for race and experience, and one for race and schooling, the coefficient of the race-experience interaction term indicates the difference between the coefficient for experience for whites and blacks, and the coefficient of the race-schooling interaction term indicates the difference between the coefficients for schooling for whites and blacks. See D. Baldus & J. Cole, supra n.55, ง 8.123[2], at 262-64.
2. The other major way of detecting discrimination between two groups is to use "dummy variables." This method, in contrast to the more general method outlined above, can only be used if it is assumed that all the coefficients for the explanatory variables are exactly the same for the two groups being compared; that the only difference with being female (or black) is represented by one overall effect (that is, the coefficient of the "dummy variable"). Cf. Testimony of Dr. Francine Blau at 66.
Thus, suppose that x1 is the "dummy variable" for race ("1" is input for being black, and "0" for being white) and x2 and x3 are the two productivity predictors. If the multiple regression is run and we obtain Y = $5000 - 700x1 + 300x2 + 400x3, then this means that a black will earn $700 a year less than an equivalent white. Cf. D. Baldus & J. Cole, supra n.55, ง 8.02, at 242; R. Wonnacott & T. Wonnacott, supra, at 100-103.
The third way an employer may not be treating people solely according to productivity differences is what we have termed improper treatment across twins: (i) econometrically, this may mean that a particular race- (or sex-) correlative personal productivity characteristic may be rewarded by a racist (or sexist) employer more than it would be by a nonracist (or nonsexist) employer or (ii) the relative pay of various jobs may not correspond to their "worth". As for (i), suppose that a nonsexist employer would pay employees of both sexes according to the equation Y = 5000 + 300x1 + 400x2, and that men tend to possess higher amounts of x1 and lower amounts of x2. If a sexist employer pays instead according to the equation Y = 5000 + 600x1 + 400x2, then men are being paid more than is actually justified by productivity considerations. The weighting is improper.[70] As for (ii), see section VII(A), infra (re "comparable work").
In this section we have been discussing econometric detection of an employer's consideration of group status in addition to his consideration of productivity differences among individuals. In certain rare circumstances, such consideration of group status might not be violative of Title VII. See the discussion of possible Equal Pay Act restrictions on allegations of wage discrimination on account of sex in section VII(A), infra. Finally, we must emphasize that while we do examine certain indicators of discrimination-such as coefficients of dummy variables for group status-we do not mean to canonize particular indices of discrimination. To do so may encourage employers to focus unduly on improving the indices, despite all their shortcomings, rather than on ensuring real-world equality of opportunity. Moreover, if employers were to do so, the value of the indices as accurate indicators of discrimination would be depreciated. Cf. B. Ackerman, et al., supra, at 28-30 (dangers of using a simple number as proxy for an environmental problem).

VII. Compensation


A. The Legal Standard for Compensation Discrimination.

In section VI, supra, the court reviewed the "equality of treatment" objective *280 of Title VII, and the role of mathematical modeling in detecting deviation from such equality. That discussion is applicable to the area of compensation. An employer is permitted to pay people commensurate with their productivity.[71] Focusing primarily on "equality of treatment of twins", discussed in section VI(E), supra, the plaintiffs, relying on coefficients of dummy variables, seek to show that a hypothetical black (or female) would be paid less than a hypothetical white (or male) possessing the same set of productivity characteristics in all the relevant years. The Bank offers analyses designed to show that this is not so.
Not surprisingly, where an employer has employees in differing occupations and of different backgrounds, a simple comparison of the average wage of all white employees and all black employees (or all male employees and all female employees) will not be enough to prove salary discrimination. See, e. g., Pouncy v. Prudential Insurance Co. of America, supra at 449, 23 Emp.Prac.Dec. n.66, at 16,751 ("The Court believes that the proper inquiry in an analysis of salaries by race should focus on whether black and white employees with the same tenure at the same job level are paid the same salaries"); Sweeney v. Board of Trustees of Keene State College, 569 F.2d 169, 179-80 (1st Cir.), vacated on other grounds, 439 U.S. 24, 99 S.Ct. 295, 58 L.Ed.2d 216 (1978); Patterson v. Western Development Laboratories Division of Aeronutronic Ford Corp., No. 74-2177 (13 Fair Empl.Prac.Cas. 772 N.D.Cal. Sept. 14, 1976) (at Section II); Smith v. Union Oil Co., 17 Empl.Prac.Dec. ถ 8411 (N.D.Cal.1977) (at section F); Booth v. Board of Directors of National American Bank, 475 F.Supp. 638, 654 (E.D.La.1979); cf. EEOC v. Akron National Bank & Trust Co., 497 F.Supp. 733, 23 Empl.Prac.Dec. ถ 31,102, at 16,662 (N.D. Ohio 1980). But cf. Kyriazi v. Western Electric Co., 461 F.Supp. 894, 913-14 (D.N.J. 1978), vacated in part on other grounds, 473 F.Supp. 786 (D.N.J.1979). For reasons analogous to the rejection of "raw" work force figures as the basis for establishment of a prima facie hiring case in certain circumstances, described in section IX(C), infra, we reject use of "raw" average pay differentials as the basis for a prima facie compensation case where the pay is averaged across a wide range of jobs and backgrounds. See Pouncy v. Prudential Insurance Co. of America, supra n.66; cf. Patterson v. Western Development Laboratories Division of Aeronutronic Ford Corp., supra. But cf. Booth v. Board of Directors of National American Bank, supra (average pay at hire of all clerical employees (only) hired by defendant in 1976). Such modeling is not sufficiently meaningful to suggest that discrimination has indeed occurred.
One method for controlling for personal characteristics affecting productivity and occupational factors[72] affecting productivity in analysis of compensation discrimination is through multiple regressions. See, e.g., Greenspan v. Automobile Club of Michigan, 495 F.Supp. 1021, 22 Empl.Prac.Dec. ถ 30,812, at 15,179, 15,183, 15,192 n.49, and 15,195 n.51 (E.D.Mich.1980); Pouncy v. Prudential Insurance Co. of America, supra n.66, 499 F.Supp. 427, 23 Empl.Proc.Dec. at 16,751-53; Presseisen v. Swarthmore College, 442 F.Supp. 593, 615-20 (E.D.Pa.1977), aff'd mem., 582 F.2d 1275 (3d Cir. 1978); Mecklenburg v. Montana Board of Regents of Higher Education, 13 Empl.Prac.Dec. ถ 11,438, at 6496-97 (D.Mont.1976). While the selection of the personal productivity *281 predictors to be included in the multiple regression should be based on the general Title VII and econometric principles outlined in section VI, supra, the control for occupation (and hence any productivity differences rooted in occupational differences) is dependent on provisions directed specifically toward compensation discrimination. We turn now to this developing area of the law, and draw out its implications for the sort of controls for occupation an econometric analysis may have to have in order to establish a prima facie case of compensation discrimination.
Title VII provides in part:
It shall be an unlawful employment practice for an employer-(I) to fail or refuse to hire or to discharge any individual, or otherwise to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual's race, color, religion, sex, or national origin....
42 U.S.C. ง 2000e-2(a) (emphasis supplied). "Shortly before the enactment of Title VII in 1964, Senator Bennett proposed an amendment providing that a compensation differential based on sex would not be unlawful if it was authorized by the Equal Pay Act, which had been passed a year earlier." Los Angeles Department of Water & Power v. Manhart, 435 U.S. 702, 711, 98 S.Ct. 1370, 1377, 55 L.Ed.2d 657 (1978) (emphasis supplied). The Bennett Amendment became part of ง 703(h) of Title VII.[73]
In contrast to Title VII, there is no language in the Equal Pay Act which proscribes discrimination in compensation generally; instead, the Equal Pay Act provides in part:
No employer having employees subject to any provisions of this section shall discriminate, within any establishment in which such employees are employed, between employees on the basis of sex by paying wages to employees in such establishment at a rate less than the rate at which he pays wages to employees of the opposite sex in such establishment for equal work or jobs the performance of which requires equal skill, effort, and responsibility, and which are performed under similar working conditions.

29 U.S.C. ง 206(d)(1) (emphasis supplied). The only wage discrimination based on sex proscribed by the Equal Pay Act is that of unequal compensation for equal work. Orr v. Frank R. MacNeil & Son, Inc., 511 F.2d 166, 171 (5th Cir. 1973). The Equal Pay Act is limited in scope. See EEOC v. Aetna Insurance Co., 616 F.2d 719, 724 n.5 (4th Cir. 1980).
As the Fifth Circuit stated in Pearce v. Wichita County, 590 F.2d 128, 133 (5th Cir. 1979) (emphasis supplied):
A prima facie Equal Pay Act case requires a showing that the "employee pays different wages to employees of opposite sexes `for equal work on jobs the performance of which requires equal skill, effort and responsibility, and which are performed under similar working conditions'." Corning Glass Works v. Brennan, 417 U.S. 188, 195, 94 S.Ct. 2223, 2228, 41 L.Ed.2d 1 (1974). To establish "equal work," the employee need not prove that the duties performed are identical, but merely that the "skill, effort and responsibility" required in the performance of the jobs is "substantially equal". Brennan v. City Stores, Inc., 479 F.2d 235, 238-39 (5th Cir. 1973). See 29 CFR ง 800.122 (1977).
See also Stastny v. Southern Bell Tel. & Tel. Co., 628 F.2d 267, at 281, 23 Empl.Prac. Dec. ถ 31,155, at 17,006-07 (4th Cir. 1980). Congress substituted the word "equal" for "comparable" in the Equal Pay Act to show that "the jobs involved should be virtually identical, that is, they would be very much alike or closely related to each other." *282 Brennan v. City Stores, Inc., 479 F.2d 235, 238 (5th Cir. 1973). In looking for substantial equality, courts must look separately at each of skill level, effort level, and responsibility level. See, e. g., Kohne v. Imco Container Co., 480 F.Supp. 1015, 1038 (W.D.Va. 1979). Cf. Blumrosen, Wage Discrimination, Job Segregation, and Title VII of the Civil Rights Act of 1964, 12 U.Mich.J.L.Ref. 399, 477 (1979). Hence, two jobs which have different levels of skill, effort, or responsibility, or different working conditions, even if considered of comparable "worth" for compensation purposes by a firm because the total of the "points" assigned to each of the four factors for the two jobs are equal, will not suffice for Equal Pay Act purposes. Cf. id. at 476-77. Otherwise stated, the "Act's requirements of equal skill, effort, responsibility, and similar working conditions cannot be aggregated to establish job equality." Kohne v. Imco Container Co., supra, at 1038. Indeed, implementing regulations provide that:
Application of the equal pay standard is not dependent on job classifications or titles but depends rather on actual job requirements and performance. For example, the fact that jobs performed by male and female employees may have the same total point value under an evaluation system in use by the employer does not in itself mean that the jobs concerned are equal according to the terms of the statute.
29 C.F.R. ง 800.121 (1979). Cf. Nelson, Opton, & Wilson, Wage Discrimination and the "Comparable Worth" Theory in Perspective, 13 U.Mich.J.L.Ref. 233, 266-67 (1980). See also Murphy, Female Wage Discrimination: A Study of the Equal Pay Act 1963-1970, 39 U.Cinc.L.Rev. 615, 637 (1970).
The Bennett Amendment created an inescapable analytical linkage between Title VII and the Equal Pay Act. The Third and Ninth Circuits have both held that in sex-based wage discrimination cases, recovery is not limited to the Equal Pay Act "unequal pay for equal work" claims. See International Union of Electrical, Radio and Machine Workers v. Westinghouse Electric Corp., 631 F.2d 1094, 23 Empl.Prac.Dec. ถ 31,106A (3rd Cir. 1980); Gunther v. County of Washington, 602 F.2d 882, 623 F.2d 1303 (9th Cir. 1979), rehearing denied, 623 F.2d 1303, 1317 (9th Cir. 1980). On the other hand, the Tenth Circuit has rejected this approach. See Lemons v. City and County of Denver, 620 F.2d 228, 229 (10th Cir. 1980). In a sex discrimination case, the Fifth Circuit recently left open the issue of whether Title VII is broader than the Equal Pay Act. See Burdine v. Texas Dept. of Community Affairs, 608 F.2d 563, 569 n.14 (5th Cir. 1979), cert. granted, 447 U.S. 920, 100 S.Ct. 3009, 65 L.Ed.2d 1112 (1980).
We do not decide this complex issue here. Compare International Union of Electrical, Radio and Machine Workers v. Westinghouse Electric Corp., supra, (A. Leon Higginbotham, J.) with id. (Van Dusen, J., dissenting). Compare also Blumrosen, supra, with Nelson, Opton, & Wilson, supra. We do not decide the issue here because it is not dispositive: for reasons independent of the relationship issue, the plaintiffs fail to establish a prima facie case of wage discrimination as to females.
Whatever the correct interpretation of the interrelationship between the two statutes for sex-based wage discrimination, the Equal Pay Act requirement of substantially equal work does not apply to race-based wage discrimination. But see Chapman v. Pacific Telephone & Telegraph Co., 456 F.Supp. 65, 68 n.5 (N.D.Cal.1978); Patterson v. Western Development Laboratories Division of Aeronutronic Ford Corp., supra. Equal work in order to render unequal pay illegal under Title VII is a product of the Bennett Amendment. Gitt & Gelb, Beyond the Equal Pay Act: Expanding Wage Differential Protections Under Title VII, Loyola U.L.J. 723, 734 (1977). This rationale does not reach race-based claims of wage discrimination; indeed, the Bennett Amendment literally applies only to sex-based compensation claims. And though Congress could have done so, it did not legislate that plaintiffs alleging race or other non-sex-based group-status discrimination *283 were similarly subject to the Equal Pay Act.
Indeed, the Third Circuit has argued as a reason for abandonment of the equal work requirement for sex-based wage discrimination claims that there is no such requirement for wage discrimination based on other group-status. The majority opinion in International Union of Electrical, Radio and Machine Workers v. Westinghouse Electric Corp., supra, at 1100, 23 Empl.Prac.Dec. at 16,682, stated:
[I]t is clear that Title VII prohibits an employer from paying more per hour to welders than plumbers if the reason for the employer paying higher wages to the welder is that the majority of the welders are Protestants and that the majority of the plumbers are Catholics. In such case an employer be "classify[ing] his employees ... in [a] way which would deprive any individual of employment opportunities [high wages] ... because of such individual's ... religion." 42 U.S.C. ง 2000e-2(a). While Westinghouse presumably would not challenge the illegality of the scheme outlined above, it asserts that the scheme would be permissible if the reason for the wage disparity is that the majority of welders are men and the majority of plumbers are women. The Supreme Court has never ruled on the statutory issue raised in this case, but in dicta in Title VII cases, the Court tends to refer to discrimination on the basis of race, religion, sex, or national origin as they are equally nefarious and equally prohibited.
3 .   .   .   .   .
In the absence of explicit statutory language or Supreme Court holdings to the contrary, we are hesitant to conclude that Title VII would allow discriminatory behavior on the basis of sex, when the same behavior would be prohibited if made on the basis of race, religion, or national origin.
Moreover, control for "substantially equal" jobs in multiple regressions offered to show race-based wage discrimination has not been required in at least one Fifth Circuit, albeit non-Title VII, case. In Wade v. Mississippi Cooperative Extension Service, 528 F.2d 508, 517 (5th Cir. 1976), while the opinion is not clear on this point, the court appears to have considered a multiple regression with salary as a dependent variable, performed on black and white professional workers at the Mississippi Cooperative Extension Service, using experience and qualifications as explanatory variables. Applying the Equal Pay Act's "substantially equal work" standard would have required regressions on employees in "substantially equal" jobs (or on a representative sample of "substantially equal" jobs) at the firm[74] or the multiple regression analysis would otherwise have had to be designed to test whether equivalent blacks and whites doing "substantially equal" jobs would be paid the same. Whether because the court was uncomfortable with controlling only for professional status, or for some other reason, the court did not hinge its opinion solely upon the statistical evidence.[75]
*284 Concluding that at least for race-based wage discrimination cases we are not restricted to claims of unequal pay for equal work does not mean that we adopt a policy of independently setting out entire scales of relative wages for dissimilar jobs according to the court's perception of each job's "worth", a hopelessly involved task inappropriate of judicial resolution. Such a failure to recognize institutional competence would risk possible deleterious effects not only on employers but on groups intended to be protected. Cf., Nelson, Opton, & Wilson, supra, at 277 & 288-97; International Union of Electrical, Radio and Machine Workers v. Westinghouse Electric Corp., supra (Van Dusen, J., dissenting), at 1108, 23 Empl.Prac.Dec. at 16,690. As stated in another context, "Judges ... have neither business experience nor the problem of meeting the employees' payroll ...." Garcia v. Gloor, supra n.36, at 271.
Courts need not engage in this sort of independent wage-setting even in the absence of Equal Pay Act restrictions because of the basic principle that it is the plaintiffs' burden to prove discrimination in compensation. It would be difficult for a plaintiff to prove that jobs dissimilar in duties and pay are of the same "worth" to the employer. Cf. Nelson, Opton, & Wilson, supra, at 288 (measurability of wage discrimination); Gunther v. County of Washington, supra, 623 F.2d at 1321; Sullivan, The Equal Pay Act of 1963: Making and Breaking a Prima Facie Case, 31 Ark.L.Rev. 545, 580-81 (1978). Moreover, even were the first job "worth" the same as the second to a race-blind employer, the market may price those two jobs differently.[76] "[N]othing in the text and history of Title VII ... [suggests] that Congress intended to abrogate the laws of supply and demand or other economic principles that determine wage rates for various kinds of work." Christensen v. Iowa, 563 F.2d 353, 356 (8th Cir. 1977).
Here the plaintiffs can show that there has been discrimination in compensation across dissimilar jobs without involving the court in independent wage-setting. The "worth" of particular jobs has already, in a sense, been determined by the Bank itself through the use of the "Hay point system". By controlling for Hay points, such as by including Hay points among the explanatory variables, the regression is comparing salaries of people in jobs "worth" the same amount to the Bank; the other explanatory variables control for the other ways in which an employee may vary in worth to the company (such as by his education or experience). Using the Bank's own valuation system should not give us pause. Indeed, if a plaintiff can show wage discrimination using the Bank's own valuation system, it is all the more impressive because of the possibility of inappropriate valuation by the Bank of jobs. See Blumrosen, supra, at 434-37 (how internal phases of evaluation systems may be discriminatory).
We will be more specific. Where a job evaluation system used by the employer himself assigns values of relative worth (points) to positions, they can be used to control for work. While it may well be true that in certain instances one job worth the same internal points as another job might have different values in the open market, a court can presume for the purposes of the prima facie case that overall, across the many positions of a large employer such as Republic, there is no systematic bias that would occur from using such a control.[77]See section IV(C), supra (re Hay system). Cf. Pouncy v. Prudential Insurance Co. of America, supra n.66, 499 F.Supp. 427, 23 Empl.Prac.Dec. at 16,751.
It would then be up to the Bank on rebuttal to show how this control for work is not adequate in the face of marketplace *285 realities.[78] That is, we do not require that the employer ignore the marketplace value of skills and jobs. See Christensen v. Iowa, supra; Horner v. Mary Institute, 613 F.2d 706, 714 (8th Cir. 1980). Cf. Presseisen v. Swarthmore College, supra, at 615-16 (Physics versus English teachers example); P. Samuelson, Economics 84 (11th ed. 1980) (professors of physics and engineering average higher incomes than do professors of Greek and botany); Defendant's Exhibit 201 (first paragraph). For such a market-place defense to succeed, however, the employer must do more than show that there are isolated instances of deviations from the standard relationship of salary to Hay points.[79]See section VII(C), infra (at introductory part of subsection) (description of the conditions necessary for modeling imperfections to cause the modeling results to lose probative effect).

B. Compensation Data[80]

1. Plaintiffs' Models.

The plaintiffs' statistical proof as to their compensation claims consists almost exclusively of the multiple regressions designed by Drs. Francine Blau and Janice Madden. However, the plaintiffs also present statistical analyses performed by Drs. John Spaulding and David Morgan. These other studies use techniques less powerful than the plaintiffs' multiple regressions, and are best considered superseded by the multiple regressions. Cf. n.183, infra (re Dr. Odell's hiring analysis in relation to Dr. Stoikov's). Nevertheless, we discuss these two experts' analyses briefly, concluding that they do not provide independent probative evidence on the compensation issue.
Dr. Spaulding's[81] major report focused on the change from 17 nonexempt grades to 10 grades and what he refers to as "an apparent change in salary ranges of minimum, midpoint, maximum" on September 1, 1973. Plaintiffs' Exhibit 500, at 1. See also section IV(C), supra. He performed analyses comparing various segments of the Bank (i. e., comparing males and females) as to the relative frequency of "down", "same", and "up" changes in salary and in Hay points experienced by those segments as a result of the switchover. These measures are far too crude to serve as the basis of any inference of salary discrimination. A de minimis change in salary or Hay points would be counted in the same manner as a large change. Cf. section VIII(A)(2)(i), infra (crudeness of proxy for promotion).
Certain of Dr. Spaulding's analyses show the distribution by race and sex of employees in various categories of job performance evaluation ("distinguished", "commendable", "competent", "fair", and "marginal"), and associated magnitudes of salary increases from being placed in these performance categories. The analysis focuses on whether there is a difference in raises across races and sexes after controlling for performance. See Testimony of Dr. John Spaulding, Jr., at 929-30, 944 (Plaintiffs' *286 Exhibits 675-80, 687, 688, 689 admitted), and 1036-46. There are two reasons why this analysis is not of independent probative value on the issue of salary discrimination. First, it does not separate the effects of any salary discrimination from promotion discrimination. That is, an observation that males get higher salary increases might be due to greater promotional opportunities available to them, and not due to any advantage in salary. Second, this sort of study is concerned only with changes in salary, not the underlying salaries themselves. At best, this sort of study is more probative on how salary discrimination may be occurring than on whether salary discrimination is taking place. Third, and foremost, the court is not convinced of the probative value of what is in essence a model for salary increases with only one true explanatory variable, at least in the absence of expert testimony that this is indeed a credible model. Professor Spaulding is an impressive expert on statistics, but not on compensation or other personnel practices. See Testimony of Dr John Spaulding, Jr., at 973-74. In the absence of endorsement of the model by an expert familiar with compensation theory, the court cannot accept it because, drawing on its general knowledge, it can conceive of many non-included factors which might affect the magnitudes of salary increase. Cf. section VI(D)(2), supra (re necessity of theory underlying model).
Dr. David Morgan[82] performed analyses which compared salaries of various segments of the Bank's population (e. g., males and females) controlling for one or two factors at a time. See Plaintiffs' Exhibits 922 and 923. For example, he controlled for exempt status in one analysis, coming up with the following table:


                  Exempt           Exempt      Female Average/Male
                  Females          Males            Average
1973[*]     10,145 (124)     17,079 (683)          .594
1974[*]     10,822 (166)     18,170 (752)          .596
1975[*]     11,754 (188)     20,050 (752)          .586
1976[*]     12,774 (199)     21,012 (696)          .608
1977[*]     13,875 (239)     22,797 (691)          .609
1978[*]     14,595 (233)     24,213 (636)          .603
Note: Numbers in parentheses are counts of persons
      in each category.

In another, he broke down comparisons between exempt females and exempt males by officer grade level (level zero, "entry level", through level nine, "chairman of the board"). See Plaintiffs' Exhibit 922, at 3 & 14-15.
Dr. Morgan's analyses are not probative of salary discrimination. Without reaching other objections, we find that in none of his analyses does he adequately control simultaneously for personal productivity-related characteristics and job characteristics. For *287 instance, comparing the average level 3 (vice president) exempt female salary with the average level 3 (vice president) exempt male salary will not tell us much; greater productivity due to the personal characteristics of the males could be accounting for the differential. (Similarly, such raw comparisons as those presented in Plaintiffs' Exhibit 253(a) are non-probative.)
Drs. Blau[83] and Madden[84] provided econometric analyses for pay differences among various groups at the Bank. We have already described their approach in section VI(C), supra, and the use of dummy variables as indices of discrimination in section VI(E), supra. For these reasons, and because we will be examining the Bank's challenges to the models one by one in sections VII(C) and VIII, infra, we need do no more here than to summarize in tabular form the more important regressions offered by the plaintiffs at trial.[85]
The dependent variable in each regression is the natural logarithm of salary. Different sets of controls for personal characteristics (sets "A", "B1", "B2", "C1", "C2", and "D") and for job characteristics (sets "a" and "b") are used as explanatory variables. All regressions were of the ordinary least squares type. The coefficient of the dummy variable for group status, multiplied by 100 will give the approximate percentage differential in salary, relative to white males, assertedly due to group status.[86],[87]


*288
                                                                                1969                                      Coefficient
                                                  Run on both types of                                             of dummy variables (such
                                                      employees or                                                    that comparison is                          Differential
                          Source                one of (exempt/nonexempt)                                         white female/white male or                 due to differences in
   Regression      # of employees[88]               employees              Explanatory variables                 black male/white male)             occupational distributions[89]
                                                                          Personal characteristic  Job
                                                                              variables         variables      Females        Blacks         Other            Females        Blacks
I.
Regression (1)    Plaintiffs' Exhibit 501                both                        none                    -.628[**]  -.342[**]
                  (at Table 1)
Regression (2)              "                            both            set "A"                   none      -.389[**]  -.201[**]
                                                                                                                                                              -.233          -.05
Regression (3)              "                            both            set "A"                  set "a"    -.156[**]  -.151[**]
II.
Regression (4)    Plaintiffs' Exhibit 501             nonexempts                     none                    -.184[**]  -.201[**]
                  (at Table 2)
Regression (5)              "                         nonexempts         set "A"                   none      -.161[**]  -.166[**]
III.
Regression (6)    Plaintiffs' Exhibit 501              exempts                       none                    -.423[**]
                  (at Table 3)
Regression (7)              "                          exempts           set "A"                   none      -.350[**]
IV.
Regression (8)    Plaintiffs' Exhibit 1308  (1307)       both                        none                   -.6283[**] -.3415[**] -.2223[**]
Regression (9)              "               (1307)       both            set "B1"                  none     -.3892[**] -.1969[**] -.0613[**]
Regression (10)             "               (1307)       both            set "C1"                  none     -.1453[**] -.2327[**] -.0879[**]
V.
Regression (11)   Plaintiffs' Exhibit 1309               both            set "B1"                  none      -.389[**]  -.197[**]
                 (Table 1) [See also Testimony
                 of Dr. Janice Madden at 67-68
                 & 70; compare with regression
                 above; section VIII(A) (1) (iii),
                 infra]                                                                                                                                       -.221          -.045
Regression (12)             "                            both            set "B1"                 set "a"    -.168[**]  -.152[**]


*289
                                                                                1970                                Coefficient
                                             Ran on both types of                                             of dummy variables (such
                                                 employees or                                                    that comparison is                          Differential
                       Source              one of (exempt/nonexempt)                                         white female/white male or                 due to differences in
   Regression      # of employees                 employees                Explanatory variables               black male/white male)                 occupational distributions
                                                                        Personal characteristic  Job
                                                                           variables         variables        Females    Blacks   Other                  Females        Blacks
I.
Regression (1)  Plaintiffs' Exhibit 501             both                             none                     -.603[**]   -.442[**]
                (at Table 1)
Regression (2)            "                         both                set "A"                 none          -.375[**]   -.309[**]
                                                                                                                                                         -.239          -.058
Regression (3)            "                         both                set "A"                set "a"        -.136[**]   -.251[**]
II.
Regression (4)  Plaintiffs' Exhibit 501          nonexempts                          none                     -.164[**]   -.295[**]
                (at Table 2)
Regression (5)            "                      nonexempts             set "A"                 none          -.137[**]   -.266[**]
III.
Regression (6)  Plaintiffs' Exhibit 501           exempts                            none                     -.441[**]
                (at Table 3)
Regression (7)             "                      exempts               set "A"                 none          -.299[**]
IV.
Regression (8)  Plaintiffs' Exhibit 1308            both                             none                    -.5966[**]  -.3608[**]  -.2194[**]
                (at Table 2) (1481)
Regression (9)            "  (1481)                 both                set "B1" plus dummy     none         -.3601[**]  -.2259[**]  -.0735[**]
                                                                        variable for being
                                                                        hired in 1969
Regression (10)           "  (1481)                 both                set "C1" plus dummy     none         -.1432[**]  -.2503[**]  -.0979[**]
                                                                        variable for being
                                                                        hired in 1969
V.
Regression (11)  Plaintiffs' Exhibit 1309           both                set "B1" plus dummy     none          -.360[**]   -.226[**]
                 (Table 1) [See also Testimony                          variable for being
                 of Dr. Janice Madden at 67-68                          hired in 1969
                 & 70; compare with regression
                 (9); section VIII (A) (1) (iii),                                                                                                      -.226          -.054
                 infra.]
Regression (12)           "                         both                       "              set "a"         -.134[**]   -.172[**]


*290
                                                                                            1971                            Coefficient
                                                Run on both types of                                                  of dummy variables (such
                                                    employees or                                                         that comparison is                                 Differential
                       Source                 one of (exempt/nonexempt)                                              white female/white male or                         due to differences in
Regression         # of employees                     employees                Explanatory variables                    black male/white male)                        occupational distributions
                                                                          Personal characteristic   Job
                                                                             variables           variables          Females    Blacks     Other                      Females          Blacks
I.
Regression (1)    Plaintiffs' Exhibit 501              both                            none                          -.620[**]    -.370[**]
                  (at Table 1)
Regression (2)              "                          both                set "A"                  none             -.397[**]    -.220[**]
                                                                                                                                                                     -.262            -.056
Regression (3)              "                          both                set "A"                 set "a"           -.135[**]    -.164[**]
II.
Regression (4)    Plaintiffs' Exhibit 501           nonexempts                         none                          -.141[**]    -.214[**]
                  (at Table 2)
Regression (5)              "                       nonexempts             set "A"                  none             -.131[**]    -.181[**]
III.
Regression (6)    Plaintiffs' Exhibit 501            exempts                           none                          -.461[**]
                  (at Table 3)
Regression (7)              "                        exempts               set "A"     none                          -.333[**]
IV.
Regression (8)    Plaintiffs' Exhibit 1308             both                            none                         -.6198[**]   -.3698[**]   +.2153[**]
                  (at Table 3) (1445)
Regression (9)              "  (1445)                  both                set "B1" plus dummy      none            -.3773[**]   -.2072[**]   -.0736[**]
                                                                           variables (i) for
                                                                           being hired in 1969,
                                                                           and (ii) for being
                                                                           hired in 1970
Regression (10)             "  (1445)                  both                set "C1" plus dummy      none            -.1677[**]   -.2296[**]   -.0918[**]
                                                                           variables (i) for
                                                                           being hired in 1969,
                                                                           and (ii) for being
                                                                           hired in 1970
V.
Regression (11)   Plaintiffs' Exhibit 1309             both                set "B1" plus dummy      none             -.377[**]    -.207[**]
                  (Table 1) [See also Testimony                            variables (i) for
                  of Dr. Janice Madden at 67-68                            being hired in 1969;
                  & 70; compare with regression                            and (ii) for being
                  (9); section VIII(A) (1) (iii),                          hired in 1970                                                                             -.234            -.048
                  infra]
Regression (12)             "                          both                        "               set "a"           -.143[**]    -.159[**]


*291
                                                                                         1972                              Coefficient
                                                     Run on both types of                                            of dummy variables (such
                                                         employees or                                                   that comparison is                    Differential
                            Source                 one of (exempt/nonexempt)                                        white female/white male or           due to differences in
   Regression           # of employees                   employees                Explanatory variables               black male/white male)          occupational distributions
                                                                              Personal characteristic     Job
I.                                                                              variables              variables    Females    Blacks    Other           Females        Blacks  
Regression (1)       Plaintiffs' Exhibit 501               both                            none                     -.626[**]    -.397[**]
                     (at Table 1)
Regression (2)                 "                           both               set "A"                  none         -.391[**]    -.239[**]
                                                                                                                                                         -.262          -.065
Regression (3)                 "                           both               set "A"                  set "a"      -.129[**]    -.174[**]
II.
Regression (4)       Plaintiffs' Exhibit 501             nonexempts                        none                     -.105[**]    -.228[**]
                     (at Table 2)
Regression (5)                 "                         nonexempts           set "A"                   none        -.105[**]    -.193[**]
III.
Regression (6)       Plaintiffs' Exhibit 501              exempts                          none                     -.443[**]
                     (at Table 3)
Regression (7)                 "                          exempts             set "A"                   none        -.333[**]
IV.
Regression (8)       Plaintiffs' Exhibit 1308              both                            none                     -.6279[**]   -.3973[**]   -.2287[**]
                     (at Table 4) (1410)
Regression (9)                 "  (1410)                   both              set "B1" plus dummy       none         -.3739[**]   -.2241[**]   -.0856[**]
                                                                             variables (i) for
                                                                             being hired in 1969, (ii)
                                                                             for being hired in 1970,
                                                                             and (iii) for being hired
                                                                             in 1971
Regression (10)                "  (1410)                   both              set "C1" plus dummy       none         -.1521[**]   -.2492[**]   -.1077[**]
                                                                             variables (i) for
                                                                             being hired in 1969, (ii)
                                                                             for being hired in 1970,
                                                                             and (iii) for being hired
                                                                             in 1971
V.
Regression (11)      Plaintiffs' Exhibit 1309              both              same as Regression        none         -.375[**]    -.224[**]
                     (Table 1) [See also Testimony                           (9) above
                     of Dr. Janice Madden at 67-68
                     & 70; compare with Regression
                     (9) above; section VIII(A) (1) (iii),                                                                                               -.238          -.056
                     infra]
Regression (12)                "                           both                     "                  set "a"      -.136[**]    -.168[**]


*292
                                                                                  1973                            Coefficient
                                             Run on both types of                                           of dummy variables (such
                                                 employees or                                                   that comparison is                  Differential
                         Source            one of (exempt/nonexempt)                                        white female/white male or         due to differences in
   Regression        # of employees               employees                Explanatory variables              black male/white male)        occupational distributions
                                                                      Personal characteristic    Job
                                                                            variables        variables      Females    Blacks    Other         Females       Blacks   
I.
Regression (1)     Plaintiffs' Exhibit               both                          none                     -.6204[**]  -.3379[**]  -.2083[**]
                   504-A and 504-E (1821)
Regression (2)              " (1821)                 both             set "D"                   none        -.3750[**]  -.1677[**]  -.0692[*]
                                                                                                                                               -.2935        -.0680
Regression (3)              " (1614)                 both             set "D"                   set "b"     -.0815[**]  -.0997[**]  -.0489[**]
II.
Regression (4)     Plaintiffs' Exhibit             nonexempt                       none                     -.0525[**]  -.1637[**]  -.1141[**]
                   504 and "Appendix I" (1141)
Regression (5)              " (1141)               nonexempt          set "D"                   none        -.0672[**]  -.1189[**]  -.0702[**]
                                                                                                                                               -.0610        -.0751
Regression (6)              " (1026)               nonexempt          set "D"       set "b" but without     -.0062    -.0438[**]  -.0242[**]
                                                                                    dummy variables for
                                                                                    "bank officer" or
                                                                                    "other exempt"
III.
Regression (7)              " (684)                 exempt                         none                     -.4422[**]  -.4501[*]   -.1835
Regression (8)              " (684)                 exempt            set "D"                   none        -.3114[**]  -.3452[*]   -.1660
                                                                                                                                               -.1687        -.1573
Regression (9)              " (588)                 exempt            set "D"       set "b" but without     -.1427[**]  -.1879[*]   -.0871
                                                                                    dummy variable for
                                                                                    "other exempt"
IV.
Regression (10)    Plaintiffs' Exhibit 1308
                                 (1601)              both                          none                     -.6452[**]  -.3134[**]  -.2009[**]
Regression (11)             "    (1601)              both             set "B2"                  none        -.3755[**]  -.1557[**]  -.0768[**]
Regression (12)             "    (1601)              both             set "C2"                  none        -.2981[**]  -.1618[**]  -.0839[**]
V.
Regression (13)    Plaintiffs' Exhibit 1309          both             set "B2"                  none        -.375[**]   -.156[**]
                   (Table 1) [See also Testimony
                   of Dr. Janice Madden at 67-68
                   & 70; compare with Regression
                   (11) above; section VIII (A) (1)                                                                                            -.296         -.054
                   (iii), infra]
Regression (14)             "                        both             set "B2"                  set "b"     -.079[**]   -.102[**]


*293
                                                                  1974                              Coefficient
                                        Run on both types of                                  of dummy variables (such
                                           employees or                                          that comparison is              Differential
                       Source        one of (exempt/nonexempt)                               white female/white male or      due to differences in
   Regression      # of employees           employees           Explanatory variables          black male/white male)      occupational distributions
                                                           Personal characteristic   Job
                                                                 variables         variables  Females    Blacks    Other      Females       Blacks  
I.
Regression (1)  Plaintiffs' Exhibit 504-E      both                     none                  -.6151[**]  -.3388[**]  -.2069[**]
                and 504-K  (2,023)
Regression (2)           " (2,023)             both          set "D"                 none     -.3755[**]  -.1873[**]  -.0917
                                                                                                                              -.2986        -.0952
Regression (3)           " (2,023)             both          set "D"                 set "b"  -.0769[**]  -.0921[**]  -.0528[**]
II.
Regression (4)  Plaintiffs' Exhibit           nonexempt                     none              -.0327[**]  -.1482[**]  -.0915[**]
                504 and "Appendix I" (1,254)
Regression (5)           " (1,254)            nonexempt      set "D"                 none     -.0514[**]  -.1191[**]  -.0684[**]
                                                                                                                              -.0509        -.0800
Regression (6)           " (1,165)            nonexempt      set "D"     set "b" but without  -.0005     -.0391[**] -.0288[**]
                                                                         dummy variables for
                                                                         "bank officer" or
                                                                         "other exempt"
III.
Regression (7)           "  (769)               exempt                  none                  -.4437[**]  -.4268[**]  -.2527[**]
Regression (8)           "  (769)               exempt       set "D"                 none     -.3237[**]  -.1528    -.1702[**]
                                                                                                                              -.1753        -.1051
Regression (9)           "  (646)               exempt       set "D"     set "b" but without  -.1484[**]  -.0477    -.1005[*]
                                                                         dummy variable for
                                                                         "other exempt"
IV.
Regression (10) Plaintiffs' Exhibit 1308          both                  none                  -.6113[**]  -.3313[**]  -.2031[**]
                           (1,795)
Regression (11)          " (1,795)                both       set "B2"                none     -.3633[**]  -.1875[**]  -.1048[**]
Regression (12)          " (1,795)                both       set "C2"                none     -.2935[**]  -.1928[**]  -.1096[**]
V.
Regression (13) Plaintiffs' Exhibit 1309          both       set "B2"                none     -.363[**]   -.188[**]
                (Table 1) [See also Testimony
                of Dr. Janice Madden at 67-68
                & 70; compare with Regression
                (11) above; section VIII (A) (1) (iii),                                                                       -.289         -.090
                infra]
Regression (14)          "                        both       set "B2"                set "b"  -.074[**]   -.098[**]


*294
                                                                  1975                              Coefficient
                                        Run on both types of                                  of dummy variables (such
                                           employees or                                          that comparison is              Differential
                       Source        one of (exempt/nonexempt)                               white female/white male or      due to differences in
   Regression           # of employees      employees           Explanatory variables         black male/white male)       occupational distributions
                                                           Personal characteristic   Job
                                                                 variables         variables  Females    Blacks    Other      Females       Blacks  
I.
Regression (1)  Plaintiffs' Exhibit 504-L      both                     none                  -.6072[**]  -.3620[**]  -.2087[**]
                504-E (2,064)
Regression (2)           " (2,064)             both          set "D"                 none     -.3621[**]  -.2025[**]  -.0857[**]
                                                                                                                              -.2750        -.0987
Regression (3)           " (1,831)             both          set "D"                 set "b"  -.0871[**]  -.1038[**]  -.0547[**]
II.
Regression (4)  Plaintiffs' Exhibit           nonexempt                 none                  -.0180[**]  -.1462[**]  -.0908[**]
                504 and "Appendix I" (1,263)
Regression (5)           " (1,263)            nonexempt      set "D"    none                  -.0425[**]  -.1270[**]  -.0684[**]
                                                                                                                              -.0444        -.0850
Regression (6)           " (1,170)            nonexempt      set "D"     set "b" but without  -.0019    -.0420[**]  -.0268[**]
                                                                         dummy variables for
                                                                         "bank officer" or
                                                                         "other exempt"
III.
Regression (7)           " (801)               exempt                   none                  -.4619[**]  -.4636[**]  -.1729[*]
Regression (8)           " (801)               exempt        set "D"                 none     -.3375[**]  -.1492[*]   -.0925
                                                                                                                              -.1639        -.0507
Regression (9)           " (-61)               exempt        set "D"     set "b" but without  -.1736[**]  -.0985[*]   -.0650
                                                                         dummy variable for
                                                                         "other exempt"
IV.
Regression (10)  Plaintiffs' Exhibit 1308      both                     none                  -.6100[**]  -.3524[**]  -.2046[**]
                            (1,818)
Regression (11)          "  (1,818)            both          set "B2"          none           -.3397[**]  -.1972[**]  -.0914[**]
Regression (12)          "  (1,818)            both          set "C2"          none           -.2518[**]  -.2013[**]  -.0940[**]
V.
Regression (13)  Plaintiffs' Exhibit 1309      both          set "B2"          none           -.340[**]   -.197[**]
                 (Table 1) [See also Testimony
                 of Dr. Janice Madden at 67-68
                 & 70; compare with Regression
                 (11) above; section VIII (A) (1) (iii),                                                                      -.261         -.092
                 infra]
Regression (14)           "                    both            set "B2"          set "b"      -.079[**]   -.105[**]


*295
                                                                  1976                              Coefficient
                                        Run on both types of                                  of dummy variables (such
                                           employees or                                          that comparison is              Differential
                       Source        one of (exempt/nonexempt)                               white female/white male or      due to differences in
   Regression      # of employees           employees           Explanatory variables          black male/white male)      occupational distributions
                                                           Personal characteristic   Job
                                                                 variables         variables  Females    Blacks    Other      Females       Blacks  
I.
Regression (1)  Plaintiffs' Exhibit 504-M       both                    none                  -.5855[**]  -.3578[**]  -.2000[**]
                504-E (1,937)
Regression (2)           " (1,937)              both         set "D"                 none     -.3349[**]  -.2049[**]  -.0885[**]
                                                                                                                              -.2588        -.1063
Regression (3)           " (1,745)              both         set "D"                 set "b"  -.0761[**]  -.0986[**]  -.0473[**]
II.
Regression (4)  Plaintiffs' Exhibit           nonexempt                 none                  -.0228    -.1346[**]  -.0840[**]
                504 and "Appendix I" (1,198)
Regression (5)           " (1,198)            nonexempt        set "D"               none     -.0478[**]  -.1216[**]  -.0667[**]
                                                                                                                              -.0584        -.0817
Regression (6)           " (1,133)            nonexempt        set "D"   set "b" but without  -.0106    -.0399[**]  -.0179[*]
                                                                         dummy variables for
                                                                         "bank officer" or
                                                                         "other exempt"
III.
Regression (7)           " (739)               exempt                   none                  -.4385[**]  -.4282[**]  -.1534
Regression (8)           " (739)               exempt        set "D"                 none     -.2945[**]  -.1193    -.1221
                                                                                                                              -.1526        -.0355
Regression (9)           " (612)               exempt        set "D"     set "b" but without  -.1419[**]  -.0838    -.0636
                                                                         dummy variable for
                                                                         "other exempt"
IV.
Regression (10)  Plaintiffs' Exhibit 1308       both                    none                  -.5896[**]  -.3416[**]  -.2072[**]
                            (1,738)
Regression (11)          "  (1,738)             both         set "B2"                none     -.3009[**]  -.1897[**]  -.0879[**]
Regression (12)          "  (1,738)             both         set "C2"                none     -.2264[**]  -.1945[**]  -.0909[**]
V.
Regression (13)  Plaintiffs' Exhibit 1309       both         set "B2"                none      -.301[**]  -.190[**]
                 (Table 1) [See also Testimony
                 of Dr. Janice Madden at 67-68
                 & 70; compare with Regression
                 (11) above; section VIII (A) (1) (iii),                                                                      -.234         -.091
                 infra]
Regression (14)           "                     both         set "B2"                set "b"    -.067[**]   -.099[**]


*296
                                                                  1977                              Coefficient
                                        Run on both types of                                  of dummy variables (such
                                           employees or                                          that comparison is              Differential
                       Source        one of (exempt/nonexempt)                                white female/white male or      due to differences in
   Regression      # of employees           employees           Explanatory variables          black male/white male)      occupational distributions
                                                           Personal characteristic   Job
                                                                 variables         variables  Females    Blacks    Other      Females       Blacks  
I.
Regression (1)  Plaintiffs' Exhibit 504-N       both                    none                  -.5676[**]  -.3653[**]  -.2057[**]
                504-E (1,991)
Regression (2)           " (1,991)              both         set "D"                 none     -.3286[**]  -.2043[**]  -.1110[**]
                                                                                                                              -.2570        -.1006
Regression (3)           " (1,765)              both         set "D"                 set "b"  -.0716[**]  -.1037[**]  -.0564[**]
II.
Regression (4)  Plaintiffs' Exhibit           nonexempt                 none                  -.0382[**]  -.1507[**]  -.0764[**]
                504 and "Appendix I" (1,229)
Regression (5)           " (1,229)            nonexempt      set "D"                 none     -.0629[**]  -.1273[**]  -.0692[**]
                                                                                                                              -.0676        -.0781
Regression (6)           " (1,151)            nonexempt      set "D"     set "b" but without   .0047    -.0492[**]  -.0167
                                                                         dummy variables for
                                                                         "bank officer" or
                                                                         "other exempt"
III.
Regression (7)           " (762)               exempt                   none                  -.4207[**]  -.4184[**]  -.1889[*]
Regression (8)           " (762)               exempt        set "D"                 none     -.2777[**]  -.1329    -.1261
                                                                                                                              -.1596        -.0625
Regression (9)           " (614)               exempt        set "D"     set "b" but without  -.1181[**]  -.0704    -.0708[*]
                                                                         dummy variable for
                                                                         "other exempt"
IV.
Regression (10) Plaintiffs' Exhibit 1308        both                    none                  -.5754[**]  -.3585[**]  -.1967[**]
                           (1,757)
Regression (11)          " (1,757)              both         set "B2"                none     -.2907[**]  -.1901[**]  -.0934[**]
Regression (12)          " (1,757)              both         set "C2"                none     -.2125[**]  -.1973[**]  -.0981[**]
V.
Regression (13) Plaintiffs' Exhibit 1309        both         set "B2"                none      -.291[**]   -.190[**]
                (Table 1) [See also Testimony
                of Dr. Janice Madden at 67-68
                & 70; compare with Regression
                (11) above; section VIII (A) (1) (iii),                                                                        -.236         -.084
                infra]
Regression (14)          "                      both         set "B2"                set "b"   -.065[**]   -.106[**]


*297
                                                                                      1978                          Coefficient
                                                  Run on both types of                                         of dummy variables (such
                                                      employees or                                               that comparison is                 Differential
                          Source               one of (exempt/nonexempt)                                     white female/white male or        due to differences in
   Regression         # of employees                   employees               Explanatory variables           black male/white male)        occupational distributions
                                                                          Personal characteristic   Job
                                                                                variables        variables   Females    Blacks    Other        Females       Blacks   
Regression (1)     Plaintiffs' Exhibit 504-O             both                          none                  -.5659[**]  -.3573[**]  -.2091[**]
                   504-E (2,012)
Regression (2)              " (2,012)                    both             set "D"                  none      -.3236[**]  -.1985[**]  -.0943[**]
                                                                                                                                               -.2451        -.1037
Regression (3)              " (1,806)                    both             set "D"                  set "b"   -.0785[**]  -.0948[**]  -.0459[**]
II.
Regression (4)     Plaintiffs' Exhibit                 nonexempt                       none                  -.0402[**]  -.1425[**]  -.0668[**]
                   504 and "Appendix I" (1,238)
Regression (5)              " (1,238)                  nonexempt          set "D"                  none      -.0632[**]  -.1175[**]  -.0573[**]
                                                                                                                                               -.0581        -.0778
Regression (6)              " (1,-86)                  nonexempt          set "D"       set "b" but without  -.0051    -.0397[**]  -.0120
                                                                                        dummy variables for
                                                                                        "bank officer" or
                                                                                        "other exempt"
III.
Regression (7)              " (774)                     exempt                         none                  -.4248[**]  -.3801[**]  -.1405
Regression (8)              " (774)                     exempt            set "D"                  none      -.2850[**]  -.1602[*]   -.0868
                                                                                                                                               -.1584        -.0885
Regression (9)              " (620)                     exempt            set "D"       set "b" but without  -.1266[**]  -.0717    -.0743[*]
                                                                                        dummy variable for
                                                                                        "other exempt"
IV.
Regression (10)    Plaintiffs' Exhibit 1308              both                          none                  -.5656[**]  -.3504[**]  -.1916[**]
                              (1,798)
Regression (11)             " (1,789) (?)                both             set "B2"                 none      -.2900[**]  -.1844[**]  -.0788[**]
Regression (12)             " (1,798)                    both             set "C2"                 none      -.2030[**]  -.1915[**]  -.0851[**]
V.
Regression (13)    Plaintiffs' Exhibit 1309              both             set "B2"                 none      -.290[**]   -.184[**]
                   (Table 1) [See also Testimony
                   of Dr. Janice Madden at 67-68
                   & 70; compare with Regression
                   (11) above; section VIII (A) (1) (iii),                                                                                                -.218         -.088
                   infra]
Regression (14)             "                            both             set "B2"                 set "b"   -.072[**]   -.096[**]
*298 Set "A"
-education (highest grade completed)
-age (in years) (as a proxy for general labor market experience)
-age (squared)

See Plaintiffs' Exhibit 501, at 1; Plaintiffs' Exhibit 504 at Table 1; Testimony of Dr. Janice Madden at 12; Testimony of Dr. Francine Blau at 17-19.
Set "B1"
-education (years)
-dummy variable (for attaining exactly 12 years of education)
-dummy variable (for attaining exactly 16 years of education)
-dummy variable (for attaining postgraduate education)
-age in months minus 72 (as a proxy for work experience)
-age in months minus 72 [squared]

See Plaintiffs' Exhibit 1308, at Table 1 and 1-2; Testimony of Dr. Janice Madden at 19-23.
Set "B2"
-Identical with Set "B1" with following change:
"Rather than indexing work experience ... by age, total possible months of prebank experience ... were determined by subtracting months of education, months of bank experience, and 72 from age in months. As before, this experience term is squared ... to test for decreases in the salary effects of large amounts of prebank experience."

See Plaintiffs' Exhibit 1308, at 3.
Set "C1"
-identical with set "B1" plus additional variable:
-sex-experience interaction term (as one way of remedying overestimation of actual experience for women when using age as a proxy for experience)

See Plaintiffs' Exhibit 1308, at Table 1 and 3; Testimony of Dr. Janice Madden at 25-26.
Set "C2"
-identical with set "B2" plus additional variable:
"an interaction term ... which for this period is the interaction between sex and prebank experience .... The estimates of salary differences attributed to sex and race ... assume that the point effects of sex and prebank experience are entirely due to the overestimation of female prebank experience. As above, this results in unreasonably conservative estimates of the effects of race and sex on salary."

See Plaintiff's Exhibit 1308, at 3.
Set "D"
-identical with set "A" plus two additional variables:
-bank experience (in years) [using data on actual Republic National Bank hire dates]
-bank experience [squared]

See Testimony of Dr. Francine Blau at 20-21 & 88-92; Plaintiffs' Exhibit 504-A.
Set "a"
-bank officer (dummy variable equaling 1 if employee is a bank officer, and 0 otherwise)
-other exempt (dummy variable equaling 1 if employee is in another exempt category and 0 otherwise)

See Plaintiffs' Exhibit 501, at 1; Plaintiffs' Exhibit 504, at Table 2; Testimony of Dr. Janice Madden at 12-13 & 15; Testimony of Dr. Francine Blau at 24.
Set "b"
-identical with set "a" plus one additional variable:
-Hay points-Hay points associated with employee's occupational category

See Plaintiffs' Exhibit 504-B. See also Testimony of Dr. Janice Madden at 68 (re Hay point conversions used for non-exempts) & 96-97; Testimony of Dr. David Morgan (re assigning of midpoint of Hay range applicable to grade of nonexempt employees).


*299 2. The Bank's Models.

Dr. Stoikov[90] does provide a multiple regression model to explain salary differentials at Republic, whose underlying approach has been set forth at section VI(C), supra. See, e. g., Defendant's Exhibit 32 and 32'. Her analysis differed in several major ways from the plaintiffs' analysis. First, Dr. Stoikov sought only to explain salary differentials at a point in time in one year (1978) rather than, as plaintiffs did, at a point in time in each of the years from 1969 to 1978. Second, she used samples[91], and drawing in part on information from a questionnaire distributed to those employees which she chose to use as her sample for her regressions, Dr. Stoikov included more explanatory variables in her model than Drs. Blau and Madden, such as "relatedness of education, prior experience, [and] interests in assuming additional responsibilities". See Defendant's Exhibit 32, at 16; Testimony of Dr. Judith Stoikov at 95 & 98-101. A third major distinction between Dr. Stoikov's model and the models Dr. Blau and Madden rely on is that Dr. Stoikov believes that, as indicated by a "Chow" test, there are two different pay structures at the Bank, one for professionals and the other for non-professionals. Thus, with certain adjustments, Dr. Stoikov believed separate regressions for professionals and non-professionals were called for, and so ran separate regressions. See Testimony of Dr. Judith Stoikov at 127-30. (Drs. Blau and Madden did run separate regressions for exempts and nonexempts, on an arguendo basis, stating that to do this would introduce other problems.)[92] Fourth, her dummy variable-type comparisons were generally between the females and males, and between the blacks and whites; in contrast, the plaintiffs use "white males" as the base for comparison purposes in all their regressions, measured against white females and black males. Dr. Stoikov's other comparisons involve not dummy variables but "Chow" tests to determine whether the male and female pay structures and black and nonblack pay structures are equal. See Defendant's Exhibit 32, at 21; section VI(E), supra. Fifth, and foremost, in none of her group status regressions does Dr. Stoikov explicitly control for job characteristics (other than by running separate regressions for professionals and nonprofessionals).
The variables in Dr. Stoikov's pay regressions were listed in her testimonial exhibit 12, reproduced in part below. As can be seen, in addition to the salary and group status dummy variables, there are six collections of explanatory variables. See Testimony of Dr. Judith Stoikov at 106-22 & 135.


*300 VARIABLES USED IN PAY REGRESSIONS

Variable Description
A. Salary as of Year-End 1978 (in natural logarithms and includes shift differential)
B. Sex = Female or Male
C. Race = Black or Nonblack
D. Length of Service at RNB (in months, measured in natural logarithms)
E. Highest Level of Education (either at time of hire or as of year-end 1978)
Less than high school
High school graduate
Post high school technical training
Some college
Associates degree
Some college or Associates degree
Bachelors degree (at hire)
Bachelors degree (by year-end 1978)
Masters degree (at hire)
Masters degree (by year-end 1978)
Bachelors degree or higher (at hire)
Bachelors degree or higher (by year-end 1978)
F. Specialized Areas of Study for Bachelors Degree or Higher
Banking
Law
G. Prior Experience
1. Months of Prior Experience by Type
Technical
Clerical supervisory
Secretarial
Other clerical (except secretarial)
Managerial or professional (except banking)
Bank-related professional or managerial
Total non-RNB work experience
At time of hire
As of year-end 1978 (includes work experience during service breaks)
Total clerical nonsupervisory
2. Months not Working and Not in School or Military
At time of hire
As of year-end 1978
H. Investments in RNB Career
a. Completed AIB Certificate, offsite skill development or management training program since hire
b. Frequency of overtime worked (measured in six ranges)
I. Average Number of Occasions Absent from Work
J. Measures of Career Motivation
Career Motivation I (for Professional Regressions):
Importance of career (measured in six categories)
Willingness to accept promotion if it entails:*
1. taking on additional duties
2. working weekends or nights
3. assuming a high risk/large reward job
4. additional performance pressures
Career Motivation II (for Nonprofessional Regressions):
Willingness to accept promotion if it entails: *
4. additional performance pressures
5. travel away from Dallas
Aspires to higher skill level at RNB Job requested at time of hire (measured in two skill categories) (no request was made)
Plans to attend skill development courses during Bank time
Nonbank commitments have influenced time available to advance career
* Promotion characteristics 1 through 4 were used as a single measure in the professional regressions. This measure is labeled "career motivation" or "willing to accept promotion under specified conditions" on the tables and "MOTIVATE" on the computer printout. Promotion characteristics 4 and 5 were the only Career Motivation measures used in the male/female nonprofessional regressions.
We summarize below the results of Dr. Stoikov's regressions wherein she used dummy variables to denote group status. Interpreting *301 these tables is not an involved process. In the first table below, for instance, the left half of the table looks at professionals at time of hire whereas the right portion of the table looks at professionals as of 1978 (which group of course includes those who were hired as nonprofessionals but became professionals by 1978). Cf. Testimony of Dr. Judith Stoikov at 131. As a first step, Dr. Stoikov regressed pay on sex alone, obtaining a negative coefficient equivalent to a negative 30.44% male/female pay disparity (when using the stratification at hire). But after adding all her other variables to the regression equation, she found that the male/female pay disparity was a statistically insignificant negative 6.99 percent.


                            Stratified by Status         Stratified by Status
                                  at Hire                  at Year-end, 1978 
Status/Variables            Female/Male                  Female/Male
   Controlled               Pay                          Pay
  Cumulatively            Disparity    t-Statistic       Disparity      t-Statistic
                           (Percent)                     (Percent)
                              (1)           (2)             (3)              (4)
Status=Professional
Sex (1=Female), Unadjusted    -30.44      -5.87[**]     -25.75          -5.03[**]
Sex (1=Female), Adjusted for:
  Service at RNB              -21.98      -4.10[**]     -18.20          -3.42[**]
  Highest Educational Level   -17.16      -3.25[**]     -13.68          -2.71[**]
  Banking Major               -15.13      -2.78[**]     -11.10          -2.12[*]
  Length and Type of Prior
    Experience                -12.79      -2.22[*]      - 8.65          -1.58
  Nonproductive Time Out of
    Labor Force               -12.27      -2.05[*]      - 8.00          -1.40
  Investments in RNB Career   - 8.91      -1.49            - 4.22          -0.75
  Career Motivation I         - 6.99      -1.22            - 2.53          -0.46
Sample Size                     119                           135
Number of Females                54                            57
Status=Nonprofessional
Sex (1=Female), Unadjusted    -10.14      -2.72[**]       2.96            0.89
Sex (1=Female), Adjusted for:
  Service at RNB              -13.53      -4.53[**]      -2.08           -0.73
  Highest Educational Level   -12.94      -4.43[**]      -2.13           -0.75
  Banking Major               -10.59      -3.53[**]      -1.69           -0.60
  Banking Major in Professional
    Status[93]             - 9.51      -3.19[**]        --              --
  Length and Type of Prior
    Experience                -10.09      -3.35[**]      -1.57           -0.57
  Nonproductive Time Out of
    Labor Force               - 9.41      -3.02[**]      -1.87           -0.67
  Investments in RNB Career   - 6.75      -2.15[*]       -1.26           -0.44
  Avg. Ann. Absence Occasions
    from Work                 - 4.99      -1.51             -0.69           -0.23
  Career Motivation II        - 3.53      -1.03             -0.00           -0.00
Sample Size                     148                          131
Number of Females                98                           95



*302
                            REPUBLIC NATIONAL BANK
     MULTPLE REGRESSION ANALYSIS OF SEX-RELATED CURRENT PAY DISPARITIES
                  SAMPLE = EMPLOYEES WITH NO MISSING DATA
      DEPENDENT VARIABLE = NATURAL LOGARITHM OF SALARY AT YEAR-END, 1978
                                   Table 1
(Part of Testimonial Exhibit 14, Defendant's Exhibit 557.)



                                 Stratified by Status           Stratified by Status
                                      at Hire                    at Year-End, 1978 
Status/Variables              Black/Nonblack                  Black/Nonblack
   Controlled                 Pay                             Pay
  Cumulatively                Disparity       t-Statistic    Disparity        t-Statistic
                              (Percent)                       (Percent)
                                (1)              (2)            (3)               (4)
Status=Professional
Race (1=Black), Unadjusted     -22.35          -2.04[*]     -23.21           -2.41[**]
Race (1=Black), Adjusted for:
  Service at RNB               -10.99          -1.06           -14.46           -1.55
  Highest Educational Level    -11.45          -1.21           -13.34           -1.58
  Banking Major                -13.49          -1.47           -13.18           -1.59
  Length and Type of Prior
    Experience                 -11.31          -1.23           -11.39           -1.40
  Time Out of Labor Force      -12.04          -1.31           -12.16           -1.50
  Investments in RNB Career    -11.10          -1.26           -10.93           -1.41
  Career Motivation I          - 6.26          -0.72           - 6.14           -0.80
Sample Size                      119                             135
Number of Blacks                  10                              12
                            REPUBLIC NATIONAL BANK
      MULTIPLE REGRESSION ANALYSIS OF RACE-RELATED CURRENT PAY DISPARITIES
                          OF PROFESSIONAL EMPLOYEES
                   SAMPLE = EMPLOYEES WITH NO MISSING DATA
        DEPENDENT VARIABLE = NATURAL LOGARITHM OF SALARY AT YEAR-END, 1978
                                   Table 2
(Part of Testimonial Exhibit 17, Defendant's Exhibit 557.)



*303
                                      Stratified by Status               Stratified by Status
                                             at Hire                         at Year-End, 1978 
   Status/Variables                  Black/Non-black                     Black/Non-black
     Controlled                      Pay                                 Pay
    Cumulatively                      Disparity     t-Statistic           Disparity     t-Statistic
                                     (Percent)                           (Percent)
                                       (1)            (2)                  (3)           (4)
Status=Nonprofessional
Race (1=Black), Unadjusted           -11.70         -3.57[**]           -10.71      -3.47[**]
Race (1=Black), Adjusted for:
  Service at RNB                     - 9.28         -3.13[**]           - 8.87      -3.17[**]
  Highest Educational Level          - 9.67         -3.28[**]           - 9.00      -3.24[**]
  Banking Major                      - 9.46         -3.17[**]           - 8.21      -2.92[**]
  Length and Type of Prior
    Experience                       - 7.84         -2.45[**]           - 6.05      -2.00[*]
  Time Out of Labor Force            - 8.15         -2.54[**]           - 6.11      -1.99[*]
  Investments in RNB Career          - 7.53         -2.30[*]            - 6.03      -1.90[*]
  Avg. Ann. Absence Occasions
    from Work                        - 7.44         -2.29[*]            - 5.95      -1.88[*]
  Career Motivation II               - 3.45         -1.01                  - 3.44      -1.06
 Sample Size                           98                                     95
 Number of Blacks                      46                                     45
                           REPUBLIC NATIONAL BANK
         MULTIPLE REGRESSION ANALYSIS OF RACE-RELATED CURRENT PAY DISPARITIES
                        OF NONPROFESSIONAL EMPLOYEES
               SAMPLE = FEMALE EMPLOYEES WITH NO MISSING DATA
         DEPENDENT VARIABLE = NATURAL LOGARITHM OF SALARY AT YEAR-END, 1978
                                  Table 3
(Part of Testimonial Exhibit 18, Defendant's Exhibit 557.)



                                        Stratified by Status               Stratified by Status
                                             at Hire                        at Year-End, 1978 
   Status/Variables              Black/Non-black                    Black/Non-black
      Controlled                 Pay                                 Pay
     Cumulatively                    Disparity       t-Statistic        Disparity       t-Statistic
                                     (Percent)                          (Percent)
                                        (1)              (2)               (3)              (4)
Status=Nonprofessional
Race (1=Black), Unadjusted           -28.08             -4.55[**]       -13.58            -2.92[**]
Race (1=Black), Adjusted for:
 Service at RNB                      -19.32             -3.86[**]       -11.37            -3.08[**]
 Highest Educational Level           -14.94             -3.12[**]       - 9.92            -2.87[**]
 Banking Major                       -14.39             -3.04[**]       - 9.98            -2.86[**]
 Banking Major in Professional
   Status                            -13.44             -2.84[**]         --                --
 Length and Type of Prior
   Experience                        -12.50             -2.32[*]        - 6.28            -1.59



*304
                                      Stratified by Status               Stratified by Status
                                             at Hire                         at Year-End, 1978 
   Status/Variables                  Black/Non-black                     Black/Non-black
     Controlled                      Pay                                 Pay
    Cumulatively                      Disparity     t-Statistic           Disparity     t-Statistic
                                     (Percent)                           (Percent)
                                       (1)            (2)                  (3)           (4)
Status=Nonprofessional
Race (1=Black), Adjusted for:
 Investments in RNB Career           -13.20          -2.48[**]         - 5.90        -1.41
 Avg. Ann. Absence Occasions
   from Work                         -13.06          -2.43[**]         - 5.46        -1.30
 Career Motivation II                - 9.56          -1.63                - 4.11        -0.92
Sample Size                             50                                  36
Number of Blacks                        19                                  18
                             REPUBLIC NATIONAL BANK
       MULTIPLE REGRESSION ANALYSIS OF RACE-RELATED CURRENT PAY DISPARITIES
                          OF NONPROFESSIONAL EMPLOYEES
                SAMPLE = MALE EMPLOYEES WITH NO MISSING DATA.
         DEPENDENT VARIABLE = NATURAL LOGARITHM OF SALARY AT YEAR-END, 1978
                                    Table 4
(Part of Testimonial Exhibit 19, Defendant's Exhibit 557.)


C. Applying the Law to the Data.

Having described at length human capital theory and ordinary least squares regressions in section VI, supra, and the applicable legal standard for wage discrimination in section VII(A), supra, we can be more succinct in applying the law to the mass of regression numbers presented on wage discrimination, as summarized in tabular form in section VII(B), supra.
If the Equal Pay Act requirement of "equal work" applies to sex-based Title VII wage claims, the female plaintiffs have not established a prima facie case of wage discrimination among females as to any year. Even the most complete of the plaintiffs' many multiple regressions designed to test for wage discrimination control for work only in terms of Hay points, and officer or other exempt status. These limited controls mean that the plaintiffs' multiple regressions would suggest non-productivity-based differentials if a female with a given portfolio of productivity characteristics at a given job "A" (with "Y" Hay points and particular officer or other exempt status) were paid differently from a male with the same portfolio of productivity characteristics at a given job "B" (with the same Hay points and particular officer or other exempt status as job "A"). If the Equal Pay Act applies, as more fully discussed in section VII(A), supra, it would require that a plaintiff alleging wage discrimination on account of sex and relying on multiple regressions use comparisons with males in the same or substantially equal jobs.[94]
*305 Even were the female plaintiffs, both exempt and nonexempt, not restricted by the "equal work" requirement, and thus entitled to the full measure of protection from wage discrimination under Title VII, they would have failed to establish a prima facie case as to any year. This is due to the problems in the proxies for experience used by the plaintiffs, described at section VII(C)(5)(ii) below; we do not reach the issue of the validity of the Bank's other objections to the models when used for sex discrimination, except as incidentally discussed in this section.
For blacks there is no such restriction to substantially equal work when making comparisons. In the period 1973-78, at least one of the regressions offered by the plaintiffs controls for comparable jobs by way of Hay points as well as education and experience; for the period 1969-72, the plaintiffs, at best, control only for officer and exempt status.
The plaintiffs' 1969-72 models with respect to blacks must be rejected for the reason that they do not control for comparable jobs (through use of Hay points or grades as explanatory variables).[95] The multiple regressions offered by the plaintiffs for this period illustrate what two hypothetical individuals, one black and one white and identical in portfolio of productivity characteristics, would earn if they had the same officer or other exempt status. The regressions for 1969-72 do not purport to determine what pay differentials would exist for such twins who work at comparable jobs. A white bank guard may have the same nonofficer and nonexempt status as a black maintenance man; though it may be that they have the same portfolio of productivity characteristics, and thus perhaps should be working at the same job level, one cannot properly infer wage discrimination (in contradistinction to initial placement or promotion discrimination) from the fact that they are being paid differently.
On the other hand, the plaintiffs offer regressions for the period 1973-78 which control for Hay points (and as to these regressions, generally control for officer or other exempt status as well through the use of dummy variables). The choice of the explanatory variables and the form of the model were fully supported by the human capital theory, as testified to by Drs. Madden and Blau. And, as seen earlier, even the most complete of the plaintiffs' regressions indicates a large and statistically significant pay differential as a result of being black; the coefficients of the dummy variables for race are large, negative, and statistically significant. See D. Baldus & J. Cole, supra n.55, ง 9.41, at 317-18 (distinction between "practical" and "statistical" significance). Cf. Rowe v. General Motors Corp., supra n.59, at 354 ("Discriminations come in all sizes and all such discriminations are prohibited by the Act."). We find that a prima facie case of wage discrimination is *306 established as to blacks, both exempt and nonexempt, for 1973-78, discussing some of the specifics of this finding in the context of our analysis of the Bank's attempted rebuttal. The Bank attempts to rebut this finding, not by asserting that there were market reasons which account for differential payment of those in comparable jobs,[96] but by a series of arguments which challenge the plaintiffs' finding of differential payment of blacks, all of which fail. (See arguments (1)-(9), infra.)
In determining whether a prima facie case is established, or if established, rebutted, we keep in mind the fact that "no model in the social sciences ever meets all the requirements for a perfect regression analysis," and the crudity of modeling permitted plaintiffs by law on the prima facie case. See D. Baldus & J. Cole, supra n.55, ง 8.2, at 266; section IX(C), infra. Cf. Johnson v. Uncle Ben's, Inc., at 423 & 425 (5th Cir. Oct. 17, 1980). To challenge a model successfully, the defendant must do more than point to possible sources of error which would bias the modeling results against him. Several conditions must hold for defects to cause the modeling results to lose probative effect.
First, unless the error is an egregious one, it must be systematic. For instance, if the data is imperfect in a random way, accuracy of the signal of whether or not group status is being taken into account by the employer will not be affected.[97]
Second, one must ask if there is actuallyโ and not merely theoretically-a bias. For instance, an employer charged with sex discrimination who argues that omission from the model of the ability to supervise others biases the econometric results against it because men tend to be better supervisors must justify the proposition that the men at its plant are likely to be better supervisors. Even were men better supervisors in society generally, no bias in econometric results occurs if the supervisory ability of the men at its plant is the same as the supervisory ability of the women at the plant.
A defendant could of course seek to show through direct, accurate testing of the employees at its plant that there is indeed a difference in supervisory ability cutting across race or sex lines. Alternately, for the purposes of the econometric analysis-and solely for such purpose[98]-the defendant *307 may seek to infer the characteristics of the groups at its plant from the characteristics of various groups in society in general. For such inference to be valid, the defendant must start by offering clear, empirically โbased evidence on the asserted differences among groups. Mere "stereotyped" impressions about group characteristics are not enough. Cf. Los Angeles Department of Water & Power v. Manhart, supra, n.98, 435 U.S. at 707, 98 S.Ct. n.98, at 1374. After showing that there is indeed a difference among groups as to the characteristic at issue, the defendant must show that the societal patterns are a reliable guide to the actual employee characteristics at his plant: unreliability may stem from two sources. First, his number of employees may be small, and second, his employees may not be reflective of societal patterns as to that characteristic. For instance, if the employer had only two employees, one male and one female, it would be foolhardy for the employer to infer from general societal characteristics of men and women the characteristics of his one male and one female, especially as to less pronounced tendencies in society. Moreover, even were he to have not two, but two thousand, employees, societal patterns as to the characteristic at issue may not be present at the employer's plant: the "pools" may be different. For instance, while one group may have greater spatial coordination than another group in society overall, it is not necessarily true that among the machinists at the employer's plant such group differences are present to the same extent, or present at all.
Finally, if the employer shows that there is a systematic bias, and that with the actual employees at the employer's plant such a bias is present (either shown directly or through the two-step procedure of inference from general societal patterns discussed in the preceding paragraph), he must show that the model is "sensitive" to the bias, and that the sensitivity of the model together with extent of the bias interest such that the modeling results as to the presence or absence of discrimination are likely to be unreliable. In essence, this final stage is the "how much" stage, wherein the defendant must show that the net effect of the bias on the model's results affect the very validity of the model's finding of the presence or absence of discrimination (as distinguished from the extent of discrimination indicated by the model).
In the following discussion, we focus on blacks; we discuss women in section VII(C)(5)(ii), infra, showing that because of the inaccuracy of proxies used by plaintiffs for women, even assuming no Equal Pay Act barrier, no prima facie case of wage discrimination is established.[99]
(1) "Defendant's compensation regression is a superior model ...." Defendant's Post-Trial Brief at 465).
The Bank argues that its model is superior to the plaintiffs' models in "completeness, accuracy, and reliability," and that the *308 court must accept the "conclusion" from the Bank's model that sex and race were not statistically significant influences on pay during the period covered by the suit. Quite aside from the fact that defendant's wage regression analysis covered only one year (1978), and, without reaching the issue of whether there are other problems associated with her model, we find Dr. Stoikov's model not probative because it does not control at all for work performed. It cannot rebut the plaintiffs' regressions. There is no explanatory variable included in Dr. Stoikov's model for the job the individual is performing (such as would be done in crude fashion through the use of Hay points); nor does Dr. Stoikov run separate regressions for each of a series of "equal" or "substantially equal" jobs. Comparison of individuals in the same jobs occurs only in the sense that professionals are compared with other professionals, and nonprofessionals with other nonprofessionals. In the absence of more finely-tuned controls for work, the regressions tell us too little about whether wage discrimination is occurring. Thus Dr. Stoikov's pay model fails to rebut plaintiffs' prima facie case for reasons analogous to the reason for which the plaintiffs' models fail to establish a prima facie case for blacks for 1969-72.
(2) "Plaintiffs have provided no proof of compensation discrimination within the subclasses ordered by the Court." (Defendant's Post-Trial Brief at 413).
This argument can be divided into three parts: (i) the Bank urges that because the court split the work force into exempt and nonexempt subclasses, the plaintiffs should have run their regressions separately for exempts and nonexempts. (ii) The Bank complains that "[p]laintiffs chose to determine that a separate subclass of nonblack minorities existed, and withdrew all members of this subclass from the pool of nonclass members with whom class members were statistically compared, and even ran a separate analysis with this subclass." Defendant's Post-Trial Brief at 415. (iii) The Bank complains that "[p]laintiffs' regressions are not consistent with the certification in terms of the arbitrary exclusion of black males from any male/female comparisons and the exclusion of white females from any black/white comparisons." Id. at 416.
Argument (i) is an argument about the legal, not the statistical, argument on pooling of exempts and nonexempts.[100] The statistical argument on pooling is raised elsewhere (and we shall not here deal with it). See section VII(C)(4). The argument must be rejected even assuming arguendo (i) that it is proper to run separate regressions for exempts and nonexempts and (ii) that it is legally necessary for plaintiffs to provide separate evidence for exempt and nonexempt discrimination. Though for statistical reasons plaintiffs believed it unwise to do so, they nevertheless ran such regressions, and statistically significant negative pay differentials were still found for each year from 1973 to 1978 for black nonexempts, and for 1973 and 1975 for black exempts. Negative, though statistically insignificant (at the 5% level) pay differentials for black nonexempts were found for the other years (1974, 1976, 1977, and 1978).
Separating nonexempts from exempts in this fashion causes a downward bias in any finding of pay differential for nonexempts. See Testimony of Dr. Janice Madden at 138-39; Testimony of Dr. Francine Blau at 44. That the findings indicate discrimination for nonexempts, even after separate regressions are run, is all the more impressive.
For black exempts, plaintiffs concede that to run such a separate regression equation would bias upward any finding of pay differential. See Testimony of Dr. Janice Madden at 139.[101] However, there is no *309 indication that the upward bias is any more than de minimis; the Bank provides no quantitative estimate of the error. Despite the small sample[102] of black exempts, there were statistically significant negative pay differentials for blacks in 1973 and 1975, and negative pay differentials for each of the other years. In the absence of any sort of estimate of how much this bias affects results, and because the results of models pooling exempts and nonexempts are consistent with the exempt-only results, a prima facie case is established as to exempts. See section VII(C), infra (introduction) (re challenging prima facie case).
Arguments (ii) and (iii) are directed toward the concept of comparing the wages of (white) females (or (male) blacks) against those of white males. White males are presumptively not discriminated against, and are a proper yardstick for determining if there had been discrimination.[103] If "other minorities," or blacks or females are included in the yardstick, any discrimination against any of these groups will itself affect the yardstick said to measure discrimination. That arguments (ii) and (iii) are being offered by the Bank is somewhat surprising for another reason: Dr. Stoikov herself, in one of her promotion analyses for blacks, compared blacks not with whites, both male and female, but with white males alone. The reason for this, Dr. Stoikov asserted, was that the preferential treatment of white females was causing an indication of discrimination against blacks where there was none. See Testimony of Dr. Judith Stoikov at 175-177.[104] Thus Dr. Stoikov herself acknowledges the possible tainting of the discrimination yardstick if the comparison base includes members of a group who may be being treated differently.
(3) "Plaintiffs' model holds the Bank responsible for discrimination effected by sources outside the Bank." (Defendant's Post-Trial Brief at 420) and "The Plaintiffs' model is not fully specified: omitted variables bias the results in favor of Plaintiffs." (Defendant's Post-Trial Brief at 424.)
*310 The Bank here argues that certain explanatory variables are omitted from the plaintiffs' equations, and since these omitted variables are race- and sex-correlative (such that white males are more highly possessed of these characteristics), the omission biases the results against the Bank. Thus they assert:
[Plaintiffs' regression model] attributes to the Bank discrimination which was effected by persons other than the Bank or by society at large. Current economic literature shows that females and blacks bring to their employment lower career ambitions, values, and motivations than do white males. Plaintiffs' model attributes all the race and sex differences in these factors to discrimination on the part of the Bank.
Defendant's Post-Trial Brief at 420. The Bank continues:
[E]stimates derived from a multiple regression analysis are invalid or biased if the model is not fully specified. The Plaintiffs' model here is not fully specified, as it omits key variables of potential productivity. The omission of these key variables heavily biases Plaintiffs' estimates of compensation discrimination against the Bank and renders the model, as specified, an unreliable measure of actual discrimination.
Defendant's Post-Trial Brief at 424-5. It goes on to assert that "[t]he human capital literature is replete with studies showing differences in the productivity factors blacks and females bring to their work, outside the influence from their employer, compared with whites and males." Id. at 425-26. The Bank proceeds to list some of these asserted race- or sex-correlative potential productivity factors, "as reported in the human capital literature," including such categories of factors as "career motivation" and:
(vi) Miscellaneous productivity influences: analytical ability, creativity, supervisory capability, research ingenuity; physical strength; preferences and importance of physical characteristics of the work-place; childhood environmental influences such as educational level of parents; ability and intelligence as measured by performance on intelligence tests; and maturity.
Id. at 431-32 (footnotes omitted).
These arguments fail for a number of reasons in the face of plaintiffs' regressions which simultaneously control, in a rough way, for education, experience, and job characteristics. First, stripped of the jargon of race- or sex-correlativeness, when the Bank is discussing, for instance, "miscellaneous productivity influences", the Bank is asserting that due to influences extraneous to the Bank, females and blacks have less analytical ability, creativity, research ingenuity, physical strength, and so forth than white males. This court cannot accept as fact, on the basis of a few social science articles, assertions so close in nature to expressions of bias, which assertions are so clearly susceptible to challenge. Any asserted group differences in characteristics should have been the subject of fuller proof and more focused adversary contest at trial.
Moreover, there is little evidence in the record that the overall effect of omission of various productivity-affecting factors is such as would bias the results against the Bank. While there was some evidence that in the general population, women and blacks may possess lower amounts of certain omitted productivity characteristics (and that suggestion is reasonable), the move from this "proof" to a blanket assertion that the overall effect of omitted variables is to bias results against the Bank was not documented in a careful step-by-step analysis such as to allow the court to make such a multi-fact-dependent assumption. Compare Testimony of Dr. Judith Stoikov at 440 with Testimony of Dr. Francine Blau at 83.
In addition, Republic has relied heavily on the assumption that its black and female employees relative to its white male employees reflect the same patterns of distribution of omitted characteristics as blacks and females relative to white males in society *311 in general. In terms of the analytical construct we discussed just prior to our analysis of the Bank's objections to the plaintiffs' multiple regressions, the Bank has not shown that the number of employees is sufficiently large that societal patterns provide a reliable basis for inferring what the Bank's patterns are like, nor has the Bank shown that its selection procedures and criteria are such as to lead its pool of employees to mirror society as to the distribution patterns of the omitted characteristics across sex and race.
The Bank did attempt to look directly at the characteristics of its employees. The only direct[105] evidence on actual employee qualifications appears to be that provided in Dr. Stoikov's Testimonial Exhibits 15 and 16. See Tables 5 and 6. However, those exhibits look only at some productivity characteristics and only at one point in time.


                               REPUBLIC NATIONAL BANK
                  COMPARISON OF MEAN EMPLOYEE CHARACTERISTICS OF
            MALES, FEMALES, NONBLACKS AND BLACKS HIRED AS PROFESSIONALS
                                    (Year-End 1978)
       Employee Characteristics       Males     Females     Nonblacks   Blacks
    -------------------------------------------------------------------------------
                                      (1)        (2)          (3)         (4)
  Length of RNB Service
   (Months, Natural Logarithm)       4.3879     3.5179      4.0620       3.2419
  Some College or Associates
   Degree                            0.1692     0.2222      0.2018       0.1000
  Bachelors Degree                   0.4154     0.4074      0.3853       0.7000
  Bachelors Degree or Higher
   with Banking Major                0.7077     0.5741      0.6239       0.9000
  Masters Degree or Higher           0.2923     0.2593      0.2844       0.2000



*312
       Employee Characteristics       Males     Females     Nonblacks   Blacks
    -------------------------------------------------------------------------------
                                        (1)        (2)          (3)         (4)
  Length and Type of Prior
  Experience (in Months):
    Total Non-RNB Work Experience      83.8462    62.8333     74.9128      67.7500
    Bank Officer or Professional       22.6154    12.6574     19.5826       1.9000
    Non-Bank Officer or
      Professional                     22.8615    16.7593     20.5459      15.1500
    Technician                         13.8154     4.5370      9.7982       7.5000
    Total Clerical                      6.0000    23.4444     14.2202      10.6000
    Months Out of Labor Force           1.3692    21.6852     11.5229       0.4000
    Offsite Training or AIB
      Certificate                       0.6154     0.2778      0.4771       0.3000
    Frequency of Overtime Worked        4.1692     3.6852      3.9633       3.8000
    Career Importance:
      Extremely Important               0.6000     0.5370      0.5872       0.4000
      Very Important                    0.2154     0.3333      0.2661       0.3000
      Unimportant                       0.0308     0.0185      0.0275       0.0000
  Willing to Accept Promotion
      Under Specified Conditions        0.2923     0.1667      0.2385       0.2000
  Sample Size                               65         54         109           10
                                        TABLE 5
Testimonial Exhibits and Appendices of Judith Stoikov, Testimonial Exhibit 15
(Defendant's Exhibit 557).



                                 REPUBLIC NATIONAL BANK
                       COMPARISON OF MEAN EMPLOYEE CHARACTERISTICS OF
                           MALES, FEMALES, NONBLACKS AND BLACKS
                               HIRED AS NONPROFESSIONALS
                                    (Year-End 1978)
                                                                Females                  Males 
Employee Characteristics            Males      Females    Nonblacks    Black    Nonblack   Black 
                                     (1)         (2)        (3)         (4)       (5)       (6)
Length of RNB Service
 (Months, Natural Logarithm)        3.5010     3.7957      3.9485       3.6231    3.7633    3.0729
Less than High School               0.0200     0.0102      0.0192       0.0       0.0000    0.0526
High School Graduate                0.4800     0.6020      0.6346       0.5652    0.4516    0.5263
Bachelors Degree or More
 with Banking Major                 0.1600     0.0204      0.0385       0.0       0.2258    0.0526
Banking Major with Professional
 Status                             0.1200     0.0         0.0          0.0       0.1935    0.0000



*313
                                                                Females                  Males 
Employee Characteristics            Males      Females    Nonblacks    Black    Nonblack   Black 
                                      (1)        (2)         (3)           (4)       (5)       (6)
Length of Type of Prior
Experience (in Months):
Total Non-RNB Work Experience       67.3800   49.1429       51.8365      46.0978   84.5806    39.3158
Bank Officer or Professional         0.4800    0.0           0.0          0.0       0.7742     0.0
Non-Bank Officer or Professional    13.6600    2.3010        4.3365       0.0      21.4516     0.9474
Technician                           3.5000    2.9490        4.3654       1.3478    2.6129     4.9474
Clerical Supervisor                  2.6800    4.2908        4.5673       3.9783    3.8710     0.7368
Secretarial                          0.0       9.9898       15.8269       3.3913    0.0        0.0
Other Clerical                      25.9200   19.9388       18.2404      21.8587   33.3871     13.7368
Months Out of Labor Force            7.2800   37.0918       44.6538      28.5435    4.5484     11.7368
Offsite Training or AIB
 Certificate                         0.1800    0.0612        0.1154       0.0       0.2581      0.0526
Frequency of Overtime Worked         3.5600    3.0306        2.8654       3.2174    3.6774      3.3684
Average Occasions of Absence
 from Work                           2.0950    3.8946        3.8638       3.9293    2.1559      1.9956
Willing to Accept Promotion if:
 Additional Pressure                 0.5000    0.4490        0.4615       0.4348    0.6129      0.3158
 Additional Travel                   0.4800    0.2041        0.1731       0.2391    0.4839      0.4737
Sample size                              50        98            52           46        31          19
                                           TABLE 6
Testimonial Exhibits and Appendices of Judith Stoikov, Testimonial Exhibit 16
(Defendant's Exhibit 557).

More fundamentally, these two exhibits do not necessarily indicate that the fact that males at Republic possess more of the various characteristics listed than females or blacks mean that omitting such variables from the regression equation would result in omission biases detrimental to the Bank. Because males are generally in positions at the Bank higher than blacks and females, to compare the mean male with the mean female or mean black as to various productivity characteristics is only a self-fulfilling exercise. We leave an illustrative proof of this fact to the margin.[106]
The third reason the Bank's argument fails is that only certain job criteria are "so manifestly job-related" that the employer need not make an evidentiary showing of business necessity. See Smith v. Olin Chemical Corp., 555 F.2d 1283, 1287 (5th Cir. 1977). "As the cases have developed, once discriminatory effect is shown, the employer must carry a heavy burden of proof to show business `necessity' for the employment practice." Id. at 1286. The Bank has made an effort to show the business necessity of only some of the many productivity variables it asserts are omitted.
The fourth reason is that including all relevant explanatory variables increases the dangers of overinclusion, including the possibility of multicollinearity. See, e. g., Testimony of Dr. Francine Blau at 16; section *314 VI(D)(3), supra. As discussed earlier, if multicollinearity is present, the probability is increased that no statistically significant differential will be found despite actual differences in treatment. See D. Baldus & J. Cole, supra n.55, at 275. "The concerns about multicollinearity and over-fitting both argue generally for restraint in selecting variables to be included in a regression." Id.
The fifth reason is that a plaintiff is not required to produce a perfectly designed model in order to make out a prima facie case. See, e. g., Finkelstein, The Judicial Reception of Multiple Regression Studies in Race and Sex Discrimination Cases, 80 Colum.L.Rev. 737, 744-45 (1980). Cf. D. Baldus & J. Cole, supra, at 275-76. Plaintiffs offer, and we rely on as establishing the prima facie case, a model which controls, in a rough way, for experience, education, and job characteristics. By way of analogy, even though motivation may indeed have a bearing on who gets hired, to establish a prima facie hiring case, there is no need to look at the "highly motivated" segment of the relevant geographic/skill market for availability comparisons.
Finally, the Bank does not provide any quantitative estimate as to how much the omitted variables bias would affect the plaintiffs' modeling results. For instance, if the model showed a 25% pay differential due to group status, even if 5% of that were attributable to omissions bias, there would still be 20% worth of discriminatory behavior.
(4) "The pooling of exempt and nonexempt employees in the same pay regression is a specification error which biases Plaintiffs' estimates against the Defendant." (Defendant's Post-Trial Brief at 438).
We have already discussed this problem at (2). See also n.90, supra. Here the Bank's experts' statistical reasons for separating exempts from nonexempts are offered. The plaintiffs' experts contended that such separation may cause "truncation bias", and that Dr. Stoikov's assertion that truncation problems from separation can be avoided is subject to challenge. Because the controversy here appears to center on an issue on the frontier of econometrics, and there seems, at least to a court unschooled in the intricacies of econometrics, to be genuine conflict between the experts as to the proper approach, we do not decide the issue. See, e. g., Heckman, Sample Selection Bias as a Specification Error, 47 Econometrica 153 (1979); Cain, The Challenge of Segmented Labor Market Theories to Orthodox Theory: A Survey, 14 J.Econ. Lit. 1215, 1246-47 (1976). More to the point, this court is not institutionally competent to resolve such a mathematical dispute.
Even assuming arguendo that separate regressions are called for, the Bank's challenge fails. The plaintiffs have indeed performed separate regressions, although the plaintiffs state that their separate regressions suffer from truncation bias as a result. While such problems would indeed detract from the reliability of the plaintiffs' separate regressions, the results from such separate analyses are consistent with what one would expect from the results of the pooled analyses. Moreover, there is no indication of the extent to which error of non-separation, or separation using the plaintiffs' methodology, would actually affect the discrimination indicated by the plaintiffs' models. See section VII(C), supra (introduction).
(5) The productivity factors in Plaintiffs' model are improperly specified" (Defendant's Post-Trial Brief at 445).
(i) "Measurement of Education Years" (Defendant's Post-Trial Brief at 445).
"The Bank asserts that some of the plaintiffs' proxies for education are subject to two errors, both of which overstate the educational attainment of women relative to that of men. Whatever the validity of this criticism, it does not relate to blacks.
(ii) "Age as a Measure of Work Experience" (Defendant's Post-Trial Brief at 447).
The Bank asserts that "[p]laintiffs' experts, in both of their original reports, used age as a measure of prior experience or *315 total work experience (PX 501, 504). Plaintiffs' experts were well aware that age is a biased measure inflating the actual work experiences of women and blacks." Defendant's Post-Trial Brief at 447.
There was little evidence, based even on general societal patterns, that the age proxy is detrimental to the Bank in the case of blacks. The Bank implicitly concedes as much, stating as a footnote, after discussing various social science studies not specific to the employees at the Bank:
In fact, evidence adduced at the trial positively demonstrates that age is a bad and inflated measure for the work experience of women at the Bank (PX 557, TE 21).
Defendant's Post-Trial Brief at 448 n.183. Moreover, for the Bank to merely point out the general inaccuracy of such a proxy in the plaintiffs' prima facie model is not enough; it would have to show how much the results may be affected. Finally, keeping in mind our earlier discussion of stereotyping, no effort appears to have been made to show whether the age proxy is bad for the black employees actually at the Bank.
On the other hand, for women, there is sufficient evidence of both the existence of such actual work patterns among Bank employees as to render the age proxy being bad, and the severe impact on the plaintiffs' own regression results due to such an inaccurate proxy.
First, Dr. Stoikov prepared an analysis illustrating how age could be a bad proxy for experience for females at the Bank. See Table 7. From this analysis, she showed, using 1978 samples, that while the proxies used in the initial report indicate that females have 93.81% or 89.99% (depending on the measure used) as much experience as males, the actual work experience percentage was only 70.7%. Dr. Stoikov testified:
And so what happens in their [plaintiffs'] measures of experience and their proxies for experience, is they're severely under-measuring the prior experience of men relative to that of women. And that manifests itself in a dampening [sic] of the sex correlation between experience and pay so that if prior experience is a legitimate determinative [sic] of pay they're underestimating it and hence they're overestimating the discrimination coefficient because of that.
Testimony of Judith Stoikov at 111. Dr. Stoikov also determined that the average male was out of the labor force and not in school for about four months whereas the average female was out of the labor force and not in school for 30 months. This is presumably measured from the time the individuals first enter the labor force or leave school. See Table 7; Testimony of Judith Stoikov at 112.
While Dr. Stoikov's analysis was performed only on a 1978 sample, the plaintiffs do not seriously challenge the notion that to use age as proxy for experience for women would cause serious errors if applied generally to societal figures. For instance, Dr. Blau testified:
Q. Okay. You've never heard of any studies which show comparisons of experience, actual experience for blacks and for sex.
A. Compared to the actual experience against age, I think Mary Corcoran may have had some comparison of women versus men, may have reported the means in some of her studies. You know there is no question there is a disparity.
Q. Is it a significant or substantial disparity?
A. Yes, although I couldn't give a figure at present.
Testimony of Dr. Francine Blau at 199. Moreover, while plaintiffs do not concede that age[107] would be a poor proxy for experience for women at the Bank, they offer no evidence contradicting Dr. Stoikov's analysis for 1978 or offering any reason *316 why its applicability should be limited to 1978. See, e. g., Testimony of Dr. Francine Blau at 19-20.
It is difficult to determine from the regressions the effect these proxies for experience would have on the dummy coefficients and their statistical significance. Intuitively, we surmise that, because the only explanatory variables used by the plaintiffs (other than controls for job level) are those based on education, general labor market experience (as proxied by age and variations of age), and (in some regressions) "bank experience", a serious error in the proxies for general experience would greatly bias the results.


               COMPARISON OF AGE AND VARIOUS MEASURES OF WORK EXPERIENCE
                                 FOR MALES AND FEMALES
                                                                       Female as
                                                                        Percent
                                          Total     Male    Female     of Male 
                                                                       (Percent)
                                           (1)      (2)       (3)         (4)
1. Age in 1978 (in years)
     (Dr. Blau's measure)                33.840   35.078    32.908      93.81
2. Age in 1978 Minus Assumed
     Age at Completion of
     Education
      (in months)                       165.090  175.096   157.568
      (in years)                         13.758   14.591    13.131      89.99
     (Dr. Madden's measure)
3. Months of Total Work
   Experience at Hire
     (NERA's measure)                    62.802   75.400    53.333      70.73
4. Years Out of School at
     Hire                                 7.567    7.278     7.784     106.95
5. Months Not Working and Not
     in School at Time of Hire           18.881    3.939    30.111     764.43
     Sample Size                            268      115       153
                                       TABLE 7
Testimonial Exhibits and Appendices of Judith Stoikov, Testimonial Exhibit 21
(Defendant's Exhibit 557).

Given the seriousness of this proxy error, the plaintiffs should have used actual measures of general experience, or the plaintiffs should have met the burden of showing that such a serious proxy error did not sufficiently affect their assertions of discrimination. Plaintiffs did neither. They offered no regression equation which controlled for job level by way of Hay points or grades and in which actual general labor market experience (as distinguished from bank experience) was used. And the plaintiffs offered no legally cognizable[108] regression wherein a "sex experience" interaction term was added to an equation which controlled for Hay points or grade used.
Nor did the plaintiffs provide convincing evidence of the unimportance of the bad proxy on the dummy coefficients. If anything, the regressions which the plaintiffs *317 offered seemed to show the opposite. Below, in the first row we list the dummy coefficients for females reported in column 2 of Tables 1-10 of Plaintiffs' Exhibit 1308. The second row consists of coefficients from column 3 of Tables 1 through 10 of Plaintiffs' Exhibit 1308.


     1969    1970    1971    1972    1973    1974    1975    1976    1977    1978
   -.3892  -.3601  -.3773  -.3739  -.3755  -.3633  -.3397  -.3009  -.2907  -.2900
   -.1453  -.1432  -.1677  -.1521  -.2981  -.2935  -.2518  -.2264  -.2125  -.2030
                                      Table 4

The first row consists of coefficients of female dummy variables, controlling for education [in the way detailed in section VII(B), supra] and experience, proxied by age or variants thereof [in the way detailed in section VII(B), supra], but not controlling for job level. The most important difference between this first regression and the regression of the second row is that in the second regression is now included a variable which in effect culls out the effect of overestimation of experience due to the use of age as proxy.[109] As can be seen from Table 4, the difference between coefficients when age is used as a proxy and when actual work experience measures are used can be striking. (Detracting somewhat from the dramatic differences in coefficients is the fact that the regressions generating such estimates seriously overcompensated for errors in the use of age as proxy for experience. It was "assumed that all variance in salary which would be attributed either to sex or to our constructed experience variable is attributed entirely to experience." Plaintiffs' Exhibit 1308, at 3. See also id. at 4.)
(6) "The model is not an appropriate measure of equal pay" (Defendant's Post-Trial Brief at 451).
Defendants offer a variety of objections under this heading:
(i) "The measure of equal work used in these reports is totally inconsistent with measures of equal work required by the courts, and for that reason alone must be rejected as a plausible inference of unequal pay." (Defendant's Post-Trial Brief at 451).
(ii) "There are three additional specification errors in the Plaintiffs' compensation regressions which include Hay points among the explanatory variables. Each of these specification errors has a consequence akin to that of omitting a relevant explanatory variable; namely, estimated coefficients are biased and inconsistent.
.   .   .   .   .
(1) Hay points are not homogeneous units across exempt and nonexempt jobs at the Bank.
.   .   .   .   .
(2) Hay points and grades are inaccurately measured by Dr. Blau and Dr. Madden for nonexempt employees. Their most egregious error is to assign an average Hay point value to all employees of the same grade.
.   .   .   .   .
(3) There is a qualitative change in the relationship between Hay points and pay at at [sic] least three levels of pay at the Bank.... Failing to recognize that there are shifts in the pay/Hay point relationship leads to a biased coefficient for Hay points." (Defendant's Post-Trial Brief at 456-57).
(iii) "[T]here are two sources of heteroskedastic disturbances in the Blau/Madden measures of Hay points. The presence of heteroskedasticity means that the standard tests of significance do not apply, and calculated standard errors and confidence intervals are wrong." (Defendant's Post-Trial Brief at 457-58).
*318 Argument (i) is based on the assumption that the equal work standard of the Equal Pay Act applies in Title VII cases to blacks as well as to females. As discussed in section VII(A), supra, we disagree.
Argument (ii)(1) can be answered in three ways: First, the Bank does not explain why this argument cannot be answered by the plaintiffs' use of officer and "other exempt" dummy variables used in the relevant years (1973-78). Cf. R. Wonnacott & T. Wonnacott, supra n.74, at 100-03 (example of value of dummy variable for war or peace in purchase of government bonds). Second, the plaintiffs have run separate regression equations for exempts and nonexempts, and the results are consistent with pooled regressions. Third, the extent of any bias is not shown.
As for argument (ii)(2), the Bank points out that "[t]o the extent there are sex- and race-related differences in actual Hay point distributions within a grade, the sex and race coefficient estimates are biased." Even assuming arguendo that the Bank has shown that there are such differences, it has not shown in which direction such estimates would be biased or given any indication of how much bias would result.
The three levels referred to in argument (ii)(3) refer to two shifts in Hay point structure: "[t]he pay associated with an increase in Hay point[s] shifts between nonexempt and exempt jobs and between exempt jobs with fewer or greater than approximately 1,500 Hay points." Defendant's Post-Trial Brief at 457. Our comments in discussing argument (ii)(1) are applicable here as well. As for the third step, the associated differential treatment of those over 1,500 Hay points, the Bank has failed to provide convincing evidence on the extent of bias caused by ignoring this difference.
The Bank asserts in argument (iii) that various Hay point distribution patterns "will manifest themselves as heteroskedasticity if a research fails to find the correct function specifications for Hay points." Defendant's Post-Trial Brief at 458. This argument fails. Heteroskedasticity does not cause biased estimates; for our purposes, it only produces error in confidence intervals and increases the variability of the group-status coefficient to a degree not indicated by the ordinary standard error for that term. See R. Pindyck & D. Rubinfeld, supra, at 96; D. Baldus & J. Cole, supra n.55, ง 8A.42, at 285. Given that fact, the fact that the confidence interval for the coefficient of the dummy variable is larger than calculated does not necessarily mean that that coefficient is no longer statistically significantly negative. Moreover, even were it not significant, the sign of the coefficient of the dummy variable may nevertheless be probative. The Bank has not shown that the relevant regressions offered by plaintiffs suffer from heteroskedasticity to such an extent that the coefficient of the dummy variable of those regressions is non-probative. The Bank has done little more than offer Dr. Stoikov's non-calculation-based comments on plots of residuals performed by Dr. Madden as to some of Dr. Madden's regressions. See Testimony of Dr. Judith Stoikov, at 436-39; Plaintiffs' Exhibit 1410.
(7) "Plaintiff's cohort regression must be rejected as nothing more than a sample of their compensation regressions." (Defendant's Post-Trial Brief at 460).
The court does not rely on the plaintiffs' cohort analysis for its finding of discrimination. The cohort analysis does not control in any way for job level, and so cannot be used in support of a wage discrimination claim.
(8) "The Plaintiffs Database." (Defendant's Post-Trial Brief at 61).
This has already been discussed in section V(B)(2), supra.
(9) "Dr. Odell's Critique of the Blau-Madden Method for Detecting and Estimating Salary Differentials Due to Discrimination." (Defendant's Exhibit 556).
Dr. Odell's critique at trial of the Blau-Madden method was not raised in the Bank's lengthy post-trial brief. No doubt this was due in great part to Dr. Stoikov's having dealt with most of the same objections to the Blau-Madden method elsewhere. *319 Both experts discussed, for instance, the question of omitted variables. We deal with some of the other issues here. First, as Dr. Odell properly points out, and as was conceded by the plaintiffs, Dr. Blau's multiplication of the coefficients of dummy variables by a factor of 100 to obtain percentage pay differentials involved a rough approximation.[110] The actual percentage pay differentials indicated by Dr. Blau's models were less than the approximated percentage pay differentials offered. However, our entire analysis has been concerned with the dummy coefficients themselves, not with the percentage approximations. Moreover, we are not concerned with magnitude, but only direction (e. g., was the influence of race or sex negative). Thus our analysis is not tainted by Dr. Odell's valid objection.
Dr. Odell criticized the use of the natural logarithm of salary rather than salary itself. Dr. Blau has testified that the natural logarithm form was chosen because an enormous amount of empirical literature suggests that this is the correct approach. Dr. Stoikov herself uses the natural logarithm of salary in her wage regression analysis.
Dr. Odell quite properly asked why the "dummy coefficient" mode of determining inequality of treatment of twins was used, rather than the more general method wherein it would not be necessary to assume that all the coefficients for the explanatory variables are exactly the same for the two groups being compared. See section VI(E), supra (comparison of running separate regressions, comparing sets of coefficients, with using dummy variables). Dr. Blau has testified that she had a "very strong guess" that running separate regressions for males and females would not affect her results. Testimony of Dr. Francine Blau at 67. No evidence contradicted this empirical judgment by an expert with a background in labor economics who was familiar with the relevant empirical work. Dr. Odell is an impressive expert as to statistical issues, but not as to labor economics issues.

D. Summary

The plaintiffs have failed to establish a prima facie case of compensation discrimination for females, either exempt or nonexempt, for any period of time.
The plaintiffs have established a prima facie case of compensation discrimination as to blacks, both exempt and nonexempt, but only from 1973 on. This prima facie case is not successfully rebutted by the Bank by its own modeling or by its challenges to the reliability of the plaintiffs' modeling. A Phase I finding of liability therefore exists for blacks, both exempt and nonexempt, employed at the Bank at any time in the interval from January 1, 1973, to the end of the class period.

VIII. Initial Placement and Promotion Analysis.


A. The Data Presented.

As in the compensation case, the parties rely primarily on statistical evidence in their initial placement and promotion analyses.[111] As the Fifth Circuit recently *320 stated, "[i]n racial discrimination cases, statistics often demonstrate more than the testimony of many witnesses, and they should be given proper effect by the courts." Swint v. Pullman-Standard, 624 F.2d 525, 529 (5th Cir. 1980). At the same time, we must and do consider the anecdotal evidence. See section V(D), supra. The court turns once again to the numbers.
An individual's position within a hierarchy at any point in time is necessarily defined by where he started (initial placement) and the opportunities for upward movement he experienced (promotion). We refer to this positional discrimination, regardless of which of the two sources was responsible, as "placement" discrimination, "vertical" discrimination, and "occupational distribution" discrimination. The hiring case involves the statistical analysis of various subgroups within the exempt and nonexempt groups. Necessarily, such analysis is peripherally affected by any placement discrimination. Yet the questions of initial placement and promotion are separate from that of hiring. In evaluating hiring, we are concerned with whether various groups are being accepted into the organization. Initial placement focuses on the question of whether, when the person enters the organization, she is being placed where a race- or sex-blind employer would place her. Similarly, promotion looks at the individual after she has entered the organization and seeks to determine if she faces the same opportunity for advancement as her male (or white) counterpart.

1. Plaintiffs' Data.

(i) Plaintiffs' initial placement and promotion case is heavily dependent on the compensation models discussed in section VII, supra. Dr. Blau asserts that there are two major mechanisms for salary discrimination:
(1) UNEQUAL PAY FOR EQUAL WORK. Pay differentials among individuals with similar qualifications that are due to pay differences within occupational categories.
(2) UNEQUAL WORK. Pay differentials among individuals with similar qualifications that are due to differences in occupational distributions (i. e., differences in access to occupations on the basis of race and/or sex). Such discrimination in access to occupations may be due discriminatory hiring and/or promotion policies of the firm.
Report of Dr. Francine Blau at 2 (Plaintiffs' Exhibit 504) (emphasis supplied). In the compensation case, we described the plaintiffs' regressions which sought to determine pay differentials after controlling for type of work performed. For instance, looking at regression (3) of the 1973 table in section VII(B)(1), supra, blacks earned approximately 9.97% less than whites, a "differential due to pay differences within occupations." See id. at 6. To calculate what discrimination blacks faced as a result of placement discrimination-what Dr. Blau refers to as "unequal work"-she relies on the hypothesis that the total differential due to discrimination is equal to the sum of that due to "unequal pay for equal work" and that due to "unequal work." This, according to Dr. Blau, is indicated by the pay differential just before controlling for job level (the "total differential due to discrimination") (16.77%, from regression (2) of the same table). She then calculates "unequal work" by subtracting the "unequal pay for equal work" from this total differential due to discrimination. She subtracts 9.97% from 16.77% to obtain a 6.8% "discriminatory difference" due to "unequal work." The same sort of calculation is performed as for other years and for women, and discriminatory *321 differences are generally found[112] elsewhere as well (and of course are more meaningful if both the minuend and the subtrahend are statistically significant).
This difference represents the degree of (vertical) occupational segregation. It does not purport to state how much vertical discrimination is separately attributable to initial placement discrimination and to promotion discrimination. Thus in the general context of questioning concerning the vertical discrimination, Dr. Blau testified:
Q. (By Ms. Peters): Well, other than promotion is there evidence where you can check or feel you can concerning the hiring and placements of white females โ
A. Okay.
Q. -or black females?
A. Let me say with respect to the specific finding regarding white females it could arise in 2 fashions and this finding does not distinguish which fashion it arises in or any combination. One is that the-the white females and males hired into the nonexempt category are roughly similar in their qualifications but the white males are more likely to be promoted.
It could also be the case though it's a hiring issue where as a woman with-white female with relative [sic] high qualifications is placed in the nonexempt category where is a similar male may be placed directly in the exempt category and I can't on the basis of my finding distinguish which of those it is.

Testimony of Dr. Francine Blau at 60-61 (emphasis supplied).
Dr. Blau's calculation of the salary differential due to (vertical) occupational distributions is straightforward: she determines the difference between the salary differential adjusted for personal characteristics and the salary differential adjusted for personal characteristics as well as job level. As a result, if either of the two differentials used in making the calculation is dubious, then the figure for occupational distribution discrimination is itself dubious. We found in section VII, supra, that job level controls such as officer or other exempt status are not a sufficient control for jobs. At the same time, we found with the addition of Hay points that there was sufficient control. As a result, those regressions before 1973 could not be used for determination of "unequal pay for equal work" (in the Blau sense). Similarly then, the pre-1973 occupational distribution differentials based on these regressions are of little probative value.[113]See section VII(B)(1), supra, (tables, at extreme right-hand column).
As described earlier, the age proxy used for experience, when applied to females, misrepresents the experience of females and seriously affects the estimates of discrimination generated by the regressions. Thus neither the differential adjusted for personal characteristics nor the differential adjusted for personal characteristics and job level are accurate for females. Hence the difference-the indicator of occupational distribution discrimination-is unreliable for females.
(ii) Dr. Madden has offered a "cohort analysis," wherein data for the same group of people are analyzed at different points in time. Cf. Haworth, Gwartney, & Haworth, Earnings, Productivity, and Changes in Employment Discrimination During the 1960's, 65 Amer.Econ.Rev. 158, 163 (1975). See Plaintiffs' Exhibit 1308; Testimony of Dr. Janice Madden at 42 et seq. and 212 et seq. *322 In this cohort analysis, Dr. Madden asserted that she followed "all those employees that were hired in 1969 and who are on board at the bank January 1, 1969" through time. See Testimony of Dr. Janice Madden at 43-45. She describes the procedure as follows:
For each year, 1970 thru 1978, the effects of race and sex group on salary for those individuals hired in 1969 is derived by: (a) regressing sex and race on the natural logarithm of salary in current year to determine the unadjusted differential (column 1, Table 13); (b) regressing sex, race, experience, and education on the natural logarithm of salary in current year to determine the pay differential adjusted for worker's productivity characteristics (column 2, Table 13); (c) regressing sex, race, experience, education and a sex/experience interaction term on the natural logarithm of salary in current to determine the pay differential adjusted for worker's productivity characteristics and attributing all the joint variation in salary due to both experience and sex only to experience. (column 3, Table 13).
Plaintiffs' Exhibit 1308, at 5-6. Table 13 of Dr. Madden's is reproduced in part below as Table 8:


                                                        Adjusted Pay
                                                        Differential
                                   Adjusted Pay         Attributing Joint
                                   Differential for     Sex-Experience
                                   Personal             Variance to
                                   Characteristics      Experience
Black Males
(relative to
 white males)
                        1970            -.231                -.228
                        1971            -.275                -.274
                        1972            -.250                -.247
                        1973            -.279                -.288
                        1974            -.284                -.300
                        1975            -.256                -.266
                        1976            -.323                -.333
                        1977            -.355                -.361
                        1978            -.350                -.366
White females
(relative to
 white males)
                        1970            -.213                -.153
                        1971            -.268                -.181
                        1972            -.281                -.161
                        1973            -.337                -.199
                        1974            -.390                -.277
                        1975            -.449                -.357
                        1976            -.331                -.266
                        1977            -.429                -.332
                        1978            -.545                -.331
                               Table 8
             Dummy Coefficients, Employees Hired in 1969

See also Testimony of Dr. Janice Madden at 47. As discussed earlier, use of the sex/experience term in column 3 is intended to overcompensate for any overmeasurement for experience of females by use of the age proxy. Id. at 49-50.[114]
*323 Table 8 shows that the pay differentials for black males and white females hired in 1969 increase relative to white males over the 1970 through 1978 period, using either of the personal-characteristics-adjusted columns.[115] Dr. Madden believes that an increasing pay differential over time in a cohort analysis of this sort could hypothetically stem from two causes:[116]
(a) Lower rates of promotion and/or pay increase for black men, white women ...
(b) differential turnover rates by race and sex such that the most productive black men, white women ... leave the bank while the most productive white men stay.
Plaintiffs' Exhibit 1308, at 6. Dr. Madden believes that if there was indeed differential turnover [i. e., (b) above] "it would likely be the result of differences in incentive structures by race and sex;" Id. Dr. Madden claims to show, however, that the explanation is not (b), but (a). Explanation (a), in turn, is, according to Dr. Madden, due either to discriminatory promotion, or deleterious initial placement:
There is [sic] really 2 things that could be accounting for the tremendous growth we are seeing in the wage differential overtime. One is these people are being placed in jobs that are dead end and that is [sic] not consistent with their qualifications because we are adjusting for education and experience, or they're not being promoted out of jobs at the same rate as white males or there is something going on with datas so that productive white males leave the bank and productivity females-let's see.
.   .   .   .   .
Q. Vice versa.
A. Such that let's say the productive white males stay and the productive females and blacks leave.
.   .   .   .   .
Q. We know that typically if you look at a secretary that's been employed for job [sic] 5 years and look at her boss who say-employed in the job for 5 years, typically the work experience of the executive gets rewarded with higher percentage increase that that of the secretary. It's what we call in economics flat adjusting professions versus steep age earnings professionals. Some jobs over time that increases the faster with respect to age than other jobs. And so the job placement and what kind of ladder you get into with respect to further prospects have much effects percentage increases. There are jobs clear, it's known there are some jobs that experience lower growth with age than other jobs. Lawyers have faster increasing income over time than brick layers do.
Testimony of Dr. Janice Madden at 51-52 & 57.
Dr. Madden attempts to show that it is explanation (a) rather than (b) by examining the characteristics of the 1969 hires still active in the Bank in each succeeding year. See Plaintiffs' Exhibit 1308 ("Table 14"). She offered the following table:


                1970   1971   1972   1973   1974   1975   1976   1977
Number          498    282    215    151    135    126    107    95
                                                                 (or 91)
Mean Years of
Education       13.4   13.5   13.5   13.7   13.7   13.8   13.6   13.7



*324
                1970   1971   1972   1973   1974   1975   1976   1977
Percent
  College
    Graduate     17     22     21     20     19     19     16     19
  Post
    Graduate      5      6      7      9     10     12      9      9
  Female         66     63     61     61     60     60     65     65
  Black           9      8      8     13     11     11     13     13
  Other           9      8      8      5      4      3      4      5
  Officers        3     10     16     23     31     34     30     34
Mean Haypoints   -      -      -     368    414    441    410    447
                                Table 9
                     Profile of 1969 Cohort, 1970-1978
(The number for 1977 is either 95 or 91. See Testimony of Dr.
Janice Madden, at 60-61.)

In examining Table 9, Dr. Madden noted: From 1970 to 1974, women quit at a higher rate than men (i. e., their share of the employment pool drops from 66 to 60 percent), while after 1975, the men quit at a higher rate. From 1970 thru 1978, blacks have lower turnover rates than whites (i. e., they increase from 9 to 13 percent of the pool), while other minority groups have higher turnover rates. The turnover rate by educational level shows no consistent turnover patterns. Officers become an increasing share of the pool due to internal promotion. All in all, nothing in the pattern reported in Table 14 reveals a differential turnover pattern which would explain the large increases in pay differentials by race and sex. Plaintiffs' Exhibit 1308, at 6-7 (emphasis added).
(iii) Utilizing the concept outlined in section VIII(A)(1)(i), supra, Dr. Madden used new specifications of salary to derive numbers for occupational distribution discrimination for 1969-78. See Plaintiffs' Exhibit 1309, at 1-2, and Table 1. As her explanatory variables for personal characteristics, she used education, bank experience, and other work experience in the form described by Plaintiffs' Exhibit 1308. Though her report is not clear on this point, her testimony appears to suggest that in this study there was not a separate sex-experience interaction term in order to adjust for overestimation of experience by use of an age-based work experience proxy for females.[117]*325 See Testimony of Dr. Janice Madden at 68 ("pay differential adjusted for personal characteristics that includes education, bank experience and other work experience") & 70; see also section VII(B)(1), supra (tables). In the period from 1969 to 1972, job placement was defined "only according to being in professional versus nonprofessional and officer versus nonofficer categories." Testimony of Dr. Janice Madden at 68. On the other hand, for the period 1973 through 1978, she used Hay points for exempts and Hay point conversions for nonexempts. Id. Her results, in part, were as follows:


                       (1)               (2)               (3)
                                    Pay Differential
                  Pay Differential  Adjusted for
                  Adjusted for      for Personal       Pay Differential
                  Personal          Characteristics    Due to Job
                  Characteristics   and Job Placement  Placement       
Black Males
(relative
 to white
 males)
             1969       -19.7              -15.2             -4.5
             1970       -22.6              -17.2             -5.4
             1971       -20.7              -15.9             -4.8
             1972       -22.4              -16.8             -5.6
             1973       -15.6              -10.2             -5.4
             1974       -18.8              - 9.8             -9.0
             1975       -19.7              -10.5             -9.2
             1976       -19.0              - 9.9             -9.1
             1977       -19.0              -10.6             -8.4
             1978       -18.4              - 9.6             -8.8
White
Females
(relative
 to white
 males)
             1969       -38.9              -16.8             -22.1
             1970       -36.0              -13.4             -22.6
             1971       -37.7              -14.3             -23.4
             1972       -37.5              -13.6             -23.8
             1973       -37.5              - 7.9             -29.6

The caveats of the analysis in section VIII(A)(1)(i), supra, apply here as well. The controls for job level for the pre-1973 regressions are almost useless, and hence all pre-1973 indicators of occupational distribution are suspect. And, as discussed earlier,


*326
                       (1)               (2)               (3)
                                    Pay Differential
                  Pay Differential  Adjusted for
                  Adjusted for      for Personal       Pay Differential
                  Personal          Characteristics    Due to Job
                  Characteristics   and Job Placement  Placement       
White
Females
(relative
 to white
 males)
             1974       -36.3              - 7.4              -28.9
             1975       -34.0              - 7.9              -26.1
             1976       -30.1              - 6.7              -23.4
             1977       -29.1              - 6.5              -23.6
             1978       -29.0              - 7.2              -21.8
                                Table 10
          "Percentage"[118] Salary Differentials by Race and Sex
Plaintiffs' Exhibit 1309, Table 1 (as corrected).[119] See
Testimony of Dr. Janice Madden at 69. All the coefficients
in the first two columns are statistically significant at the 1%
level. See id. at 70.

the results for females are suspect because of the age-for-experience proxy problem.
(iv) Dr. Madden also ran regressions which seek to determine the effect of race or sex on the Hay points of employees active in 1973 through 1978. See Plaintiffs' Exhibit 1309, at 2 and Table 2. In Table 2 she showed the Hay point level of blacks and females relative to white males in column 1, unadjusted for any employee characteristics (more technically, column 1 consists of the race and sex coefficients when Hay points are regressed on race and sex dummy variables). Thus, we see, for example, that the average black male held a job that was valued at 125 fewer Hay points than that held by the average white male. Column 2 controls for certain personal characteristics and various measures for education and experience. Dr. Madden did not clearly testify as to just what explanatory variables were used, making only a general statement on the nature of the variables and giving a citation to a computer printout. See Testimony of Dr. Janice Madden at 71-72. It appears from the court's independent analysis of the lengthy computer printout that the non-group-status explanatory variables included were: months in bank squared; months with Republic; highest grade completed; age in months; and age in months squared. See Plaintiffs' Exhibit 110 (Printout 10), at 100, 296. Compare id. at 227 with Plaintiffs' Exhibit 1309, Table 2, at column 2 (1973). No sex-experience interaction term appears to have been included. Her results from Table 2 of Plaintiffs' Exhibit 1309 are listed below:


                         (1)                   (2)
                                        Haypoint Differential
                     Total Haypoint     Adjusted for Personal
                     Differential       Characteristics      
Black Males
(relative to
 white males)
               1973       -125             -32
               1974       -145             -56



*327
                         (1)                   (2)
                                        Haypoint Differential
                     Total Haypoint     Adjusted for Personal
                     Differential       Characteristics      
Black Males
(relative to
 white males)
               1975       -162             -68
               1976       -156             -69
               1977       -161             -69
               1978       -162             -75
White Females
(relative to
white males)
               1973       -398             -227
               1974       -376             -223
               1975       -389             -220
               1976       -380             -207
               1977       -384             -215
               1978       -375             -211
                          Table 11
          Hay Point Differentials by Race and Sex,
               All Active Employees, 1973-1978

It appears that the only non-statistically-significant Hay point differential is the "-32" for black males in column 2 for 1973. Compare Plaintiffs' Exhibit 1309 (Table 2) with Testimony of Dr. Janice Madden at 78.
(v) Effects of sex and race on job placement within nonprofessional categories.
Dr. Madden also examined the effects of sex and race on job placement within nonexempt jobs. Job grades were available to rank nonexempt jobs for 1969 through 1978; Hay points were available to rank nonexempt jobs after 1973.
In separately analyzing nonexempts, the plaintiffs believe that they are subjecting their own analysis to truncation bias, described earlier in section VII, supra. The plaintiffs appear to have asserted that as a result of this self-imposed truncation bias, their statistical proof of discrimination will be made more difficult: any indication of discrimination will be downwardly biased.[120]See Testimony of Dr. Janice Madden at 79-80. Any indication of discrimination found using this method will be all the more impressive.[121]
Tables 3 and 4 of Plaintiffs' Exhibit 1309, designated below as Tables 12 and 13, report the effects of race and sex on job grades for 1969 thru 1972 and for 1973 thru 1978, respectively. Column 1 reports unadjusted grade differential for each race/sex group in various years. Column 2 reports the grade differential apparently adjusted for education by way of years of education and dummy variables for attaining (i) exactly 12 years (HS), (ii) exactly 16 years, and (iii) postgraduate education, experience, for dummy variables indicating date of hire, for months of education, for 72 subtracted from age in months, and for those months squared. See Plaintiffs' Exhibit 1309, at 3; Plaintiffs' Exhibit 1308, at 1-2. Column 3 *328 reports the grade differential when a variable is added whereby all variance in grade which could be statistically attributed to either sex or experience is attributed to experience. See Plaintiffs' Exhibit 1309, at 3. Tables 12 and 13 are reproduced in part below. All the coefficients in both tables are statistically significant at the 1% level:


                                                      Total Adjusted
                                     Total Grade      Grade Differential
                                     Differential     Attributing Joint
                                     Adjusted for     Sex Experience
                       Total Grade   Personal         Differential Due
                       Differential  Characteristics  to Experience    
Black Males
(relative to
 white males)
                1969     -3.209        -3.148             -2.799
                1970     -3.144        -2.867             -2.741
                1971     -2.924        -2.392             -2.316
                1972     -3.075        -2.660             -2.620
White Females
(relative to
 white males)
                1969     -2.377        -2.188             -3.014
                1970     -2.278        -2.126             -2.494
                1971     -1.632        -1.501             -1.773
                1972     -1.356        -1.152             -1.302
                                Table 12
           Grade[122] Differentials by Race and Sex,
                 Nonexempt Employees, 1969-72



                                                          Total Adjusted
                                      Total Grade         Grade Differential
                                      Differential        Attributing Joint
                                      Adjusted for        Sex Experience
                        Total Grade   Personal            Differential Due
                        Differential  Characteristics     to Experience   
Black Males
(relative to
 white males)
                1973     -1.500        -1.330              -1.338
                1974     -1.248        -1.143              -1.153
                1975     -1.215        -1.174              -1.183
                1976     -1.102        -1.100              -1.112
                1977     -1.132        -1.076              -1.083
                1978     -1.086        -1.024              -1.031
White Females
(relative to
 white males)
                1973     -.759         -.837               -.694
                1974     -.451         -.523               -.350



*329
                                                          Total Adjusted
                                      Total Grade         Grade Differential
                                      Differential        Attributing Joint
                                      Adjusted for        Sex Experience
                        Total Grade   Personal            Differential Due
                        Differential  Characteristics     to Experience  
                1975      -.350         -.415              -.206
                1976      -.372         -.511              -.323
                1977      -.554         -.700              -.595
                1978      -.542         -.632              -.538
                                Table 13
            Grade[123] Differentials by Race and Sex,
                   Nonexempt Employees, 1973-78

Table 5 of the report, reproduced below in part and designated as Table 14, corresponds to the analysis of Table 13 for the 1973 through 1978 nonexempt employees, but using Hay points rather than job grades. Just as with Tables 12 and 13, statistically significant negative job placement differentials are indicated for both blacks and females.


                                                      Total Adjusted
                                     Total Haypoint   Haypoint Differential
                                     Differential     Attributing Joint
                                     Adjusted for     Sex Experience
                     Total Haypoint  Personal         Differential
                     Differential     Characteristics  to Experience   
Black Males
(relative to
 white males)
                1973      -33.3        -29.5               -29.6
                1974      -28.4        -26.0               -26.2
                1975      -27.6        -26.6               -26.8
                1976      -25.8        -25.7               -26.0
                1977      -26.7        -25.4               -25.6
                1978      -25.6        -24.2               -24.3
White Females
(relative to
 white males)
                1973      -20.2        -21.9               -19.6
                1974      -13.4        -15.1               -11.8
                1975      -11.1        -12.6               - 8.1
                1976      -10.6        -13.8               - 9.6
                1977      -14.9        -18.3               -15.8
                1978      -14.0        -16.2               -13.7
                                Table 14
                Haypoint Differentials by Race and Sex,
                    Nonexempt Employees, 1973-78

*330 Dr. Madden noted the fact that in the regressions for nonexempts, placement discrimination seemed less for females than for blacks, while in regressions for all employees [in section VIII(A)(1)(iv), supra], blacks seemed to suffer less placement discrimination. Thus, Professor Madden interprets that:
[W]ithin the nonexempt grades black males or that race placys [sic] at the significant much more significant role in job placement within the nonexempt category whereas the nonexempt category whereas for females it was the difficulty of getting a job in the exempt category that was primarily their job placements problem at Republic.
.   .   .   .   .
[F]or this group of white females it's the promotion from nonexempt into exempt, the placement in nonexempt rather than exempt that accounts for so much of their job place-the-the job place-the fact they had them on their job placements policy, whereas for black males there was job placement effects within the nonexempt categories. There was job placement in being placed in nonexempt rather than exempt but that's not as large for the black males relative to white females.
Testimony of Dr. Janice Madden at 86-87.
(vi) Utilizing a sample of employees hired from 1969-78 used in Defendant's Exhibit 458, as "corrected," Dr. Madden offered a multiple regression analysis which assertedly shows that a female or black is less likely to be hired as a professional, and less likely to be a professional in 1978. See Plaintiffs' Exhibit 1410 (Table 3). The coefficients for race and sex were statistically significant as to both analyses, both before and after controlling for various personal characteristics. We reproduce the table below:


                            Dependent Variable:                        Dependent Variable:
                            Hired as Professional                     Professional in 1978
Independent
Variables                      1              2                3                 4            5  
Female                      -.5926[**]  -.2569[**]  -.6317[**]  -.4324[**]  -.4386[**]
Black                       -.2379[**]  -.1485[*]   -.2742[**]  -.1814[**]  -.1710[**]
Master's Degree                             .1475                         .1743[*]    .1965[*]
Bank Related Major                          .4098[**]                                 .1666[*]
Female with Bank
  Related Major                            -.2516[*]                                  .0318
Natural Log of Total
  Related Pre-RNB
  Experience                                .0368[**]
Bachelor's Degree                                                         .1175[*]    .1311[*]
Natural Log of Total RNB
  Experience                                                              .0942[**]   .0949[**]
Natural Log of Total
  Pre-RNB Experience                                                                     .0295[*]
                                 Table 15



*331 2. The Bank's Data.

The Bank offers three sets of analyses on promotional opportunities [sections VIII(A)(2)(i), (ii)(b), and (iii), infra] and two sets on initial placement [sections VIII(A)(2)(ii)(a) and (iii), infra].

i. Promotions: Work Force Comparison.
Dr. Patrick Odell[124] compared promotions in each "job family"[125] against the representation of blacks and females in that job family. Actual proportions of promotions for blacks and females in each two-digit[126] job family were compared against their representation in the work force for that job family for the period 1969-78. This analysis was performed using the Z-score[127] statistical testing. "Promotion" was defined, roughly speaking, as any upward move in job grade or Hay points, a move from nonexempt status to exempt status, and/or an election to officership or move upward in officer level. See Defendant's Post-Trial Brief at 258; Defendant's Exhibit 550; Testimony of Thomas Barksdale. That is, under this definition, even a small upward move in Hay points would be counted in the same fashion as the election of an employee to officership. Dr. Odell prepared for each job family two charts, one for blacks and one for females, such as the following for job family 310 for blacks.


Job Family-530
Nonexempt
                                                    Blacks    Total
1969      # Promoted                                     0       20
          # In Workforce                                 0       92
          Z Score                                   0.0000        *
          Randomness Range                         0  -  0        *
1970      # Promoted                                     1       32
          # In Workforce                                 1      107
          Z Score                                   0.0000        *
          Randomness Range                         0  -  1        *
1971      # Promoted                                     0       19
          # In Workforce                                 5      105
          Z Score                                   0.0000        *
          Randomness Range                         0  -  2        *
1972      # Promoted                                     2       22
          # In Workforce                                 3       92
          Z Score                                   0.0000        *
          Randomness Range                         0  -  2        *
1973      # Promoted                                     1       22
          # In Workforce                                 7       98
          Z Score                                   0.0000        *
          Randomness Range                         0  -  3        *
1974      # Promoted                                     1       21
          # In Workforce                                18      111
          Z Score                                   0.0000        *
          Randomness Range                         1  -  6        *



*332
Job Family-530
Nonexempt
                                                    Blacks    Total
1975      # Promoted                                     0       14
          # In Workforce                                 8       89
          Z Score                                   0.0000        *
          Randomness Range                         0  -  3        *
1976      # Promoted                                     1       22
          # In Workforce                                 7       83
          Z Score                                   0.0000        *
          Randomness Range                         0  -  3        *
1977      # Promoted                                     0       17
          # In Workforce                                 8       79
          Z Score                                   0.0000        *
          Randomness Range                         0  -  3        *
1978      # Promoted                                     4       30
          # In Workforce                                12       86
          Z Score                                   0.0000        *
          Randomness Range                         2  -  7        *
                                 Table 16
                   Comparison of Promotions Against Workforce
                         Representation by Job Family

From Defendant's Exhibit 7. See Defendant's Exhibits 7-10.
The Bank asserts that this promotions analysis generally showed no underutilization of either blacks or females. For instance, with respect to the subclass of black nonexempt employees, the analysis revealed one instance of statistically significant disparity adverse to blacks [job family 570 (clerk-typist) in 1973], and seven or eight observations of statistically significant disparities favorable to blacks. All other promotion counts for black nonexempts fall within the "randomness range."[128]
These comparative analyses are of little value for at least two reasons. First, in many of the units of observation (e. g., 1971 for job family 310 for blacks) zero promotions falls within the randomness range. As noted in section IX(G), infra, when this occurs, results falling within the randomness range show nothing: even were there rampant promotion discrimination, resulting in no black or female promotions whatsoever, such a test would not indicate discrimination. Furthermore, the disaggregation itself causes fragmentation problems, see n.210, infra; no reason appears to the court why the units of observation needed to have consisted of so few individuals that such statistical difficulties arose. Indeed, the Bank's own expert, Dr. Stoikov, performed calculations, described in section VIII(A)(2)(ii), infra, involving aggregation across years and jobs in support of her initial placement and promotion analysis. More aggregation would have been helpful. Second, the proxy for promotion is quite crude. Election to officership is treated the same way as a small upward move in Hay points. Cf. section VII(B), supra (Dr. Spalding's analyses).
ii. Dr. Stoikov presented analyses intended to answer two questions. First, do men and women (or blacks and nonblacks), possessing equal productive value, who come to work for the Bank have the same probability of being hired into professional job families? Second, do men and women (or blacks and nonblacks), who are hired *333 into nonprofessional job families and have equal productive value, share the same probability of being promoted to professional job families? See Defendant's Exhibit 458.
(a) Dr. Stoikov's answer to the first question-that concerned with initial placement-is that females are positively advantaged by their sex while blacks are neither advantaged nor disadvantaged by their color. The data used in this analysis include all people in the sample used for Dr. Stoikov's compensation analysis who were hired during the period 1969-78.
Because the probability of being hired into a professional job family is measured in dichotomous form (i. e., an employee either is or is not hired into a professional job family), Dr. Stoikov believed it advisable to corroborate probability estimates obtained with a "linear" technique (such as ordinary least squares) with probability estimates obtained with techniques such as "probit" and "logit" analysis. J. Stoikov, Supplementary Report on Probabilities of Being Hired or Promoted into Professional Job Families at Republic National Bank of Dallas 2 (1979) (Defendant's Exhibit 458).
Using various variables for employee educational and prior experience characteristics at time of hire which Dr. Stoikov considered indicative of productivity, she sought to determine how sex and race affected the probability of being hired as a professional. She obtained the following results from the ordinary least squares methodology:


Variable                        Sex Regression           Race Regression
                                  Coefficient              Coefficient  
Sex (1 = Female)                   .0917[**]
Race (1 = Black)                                             -.1078[**]
Hire Year                          .0069                      .0096
Less than High School             -.3261                     -.2874
High School Graduate              -.1337[**]              -.1146[**]
Degree in Banking                  .7750[**]               .7441[**]
Degree in Nonbanking Subject       .4937[**]               .4976[**]
Masters Degree                     .1159[*]                .1014[*]
Time Out of Labor Force           -.0007[*]               -.0005
Prior Months of Experience as:
     Bank Profess. or Officer      .0009                      .0008
     Nonbank Profess. or Manager   .0003                     -.0001
     Technician                    .0020                      .0018
     Clerical Supervisor           .0007                      .0004
     Secretary                    -.0007                     -.0007
     Other Clerical                .0013[*]                .0013
Total Experience                   .0003                      .0002
Constant                          -.4618                     -.5554
This table is derived from Testimonial Exhibit 22 (Defendant's
Exhibit 557).
                                Table 17
                Regression Coefficients Relating Professional
                  Status at Hire to Employee Characteristics

Thus, Dr. Stoikov found that the estimated value of being female is a statistically significant +.0917 on the probability of being hired as a professional. However, Dr. Stoikov found that there was a race coefficient of -.1078 which was also statistically *334 significant. This, Dr. Stoikov asserted, was due not to discrimination against blacks but to the fact that females were benefited by affirmative action. Among the evidence offered to justify this interpretation was the fact that when compared with white males, the regression shows no statistically significant race coefficient. See table following.


                           Regression
                          Coefficient
Race (1=Black)               -.0165          (t-statistic of race
Hire Year                     .0035            coefficient -.43)
Less than High School        -.1142
High School Graduate         -.0289
Degree in Banking             .8150[**]
Degree in Nonbanking          .3692[**]
Masters                       .1484[**]
Time out of Labor Force      -.0004
Prior Months of
  Experience as:
     Bank Profess. or
       Officer                .0009
     Nonbank Profess.
       or Manager            -.0007
     Technician               .0004
     Clerical Supervisor     -.0009
     Secretary               -.0007
     Other Clerical          -.0002
     Total Experience         .0005
Constant                     -.22039
This table is derived from Testimonial Exhibit 24 (Defendant's
Exhibit 557).
                              Table 18
               Regression Cofficients Relating Professional
                Status at Hire to Employee Characteristics

Dr. Stoikov also performed a "probit" analysis which generated useful results. Probit analysis has an advantage over ordinary least squares in dealing with probabilities: with the latter, but not with the former, it is possible the regression will predict greater than a 100% probability of being hired as a professional or smaller than a zero probability of being hired as a professional. Testimony of Dr. Judith Stoikov at 178.
Using the same variables as in her ordinary least squares analysis, Dr. Stoikov's probit analysis generated one statistically significant coefficient in favor of females and one nonstatistically significant coefficient in favor of blacks. Id. at 180; Testimonial Exhibit 25 (Defendant's Exhibit 557).[129] Thus the probit analysis differed from the least squares analysis in that it showed that being black does not handicap one's chances of being placed into a professional job family.[130]
Dr. Madden challenged the sample used by Dr. Stoikov for these "placement as professional" analyses on technical grounds. *335 Dr. Stoikov conceded that criticism as to these analyses had some merit, and so made the appropriate corrections. See Defendant's Exhibit 573. The "corrected" linear regression coefficient for sex was negative and insignificant, while that for race was positive and insignificant. Id. at 1. The "corrected" probit analysis indicated positive, but what appear to be insignificant, influences for race and sex on professional status. Id. at 2.
(b) On the second issue, Dr. Stoikov also performed an analysis on the influence of sex or race on the probability of being promoted from a nonprofessional to a professional position, using as her sample all sample members hired into nonprofessional job families. See J. Stoikov, supra, at 2 (Defendant's Exhibit 458). Using a variety of variables intended to control for productivity, she obtained the following regression coefficients:


                                 Sex Regression         Race Regression
Variable                         Coefficient            Coefficient    
Sex (1 = Female)                  -.0638 (t-statistic
                                            of -1.12)
Race (1 = Black)                                        -.0774 (t-statistic
                                                                of -1.64)
Service at RNB                     .0782[**]          .0716[**]
Less than High School             -.2543                -.2353
High School Graduate               .0215                 .0205
Degree in Banking Subject          .3152[**]          .3153[**]
Prior Months of Experience as:
     Technician                   -.0015                -.0016
     Clerical Supervisor           .0028[*]           .0027[*]
     Bank Profess. or Officer      .0197                 .0200[*]
     Nonbank Profess. or Manager  -.0015[*]          -.0016[*]
Time Out of Labor Force           -.0012[**]         -.0014[**]
AIB Courses or Training            .2922[**]          .2883[**]
Average Occasions of Absence      -.0262[**]          .0288[**]
Overtime Frequency                 .0238[*]           .0267[*]
Willing to Accept Promotion if:
     Additional Pressure           .0875[*]           .0750[*]
     Additional Traveling Required .1323[**]          .1436[**]
Career Importance                 -.0385                -.0194
Constant                          -.1898                -.1786
This table is derived from Testimonial Exhibit 27 (Defendant's
Exhibit 557).
                             Table 19
                  Regression Coefficients Relating Professional
                   Status in 1978 to Employee Characteristics

A probit analysis using the same variables generated results similar to the least squares estimates: the race and sex coefficients were statistically insignificant. However, whereas both group status coefficients were negative with least squares here, the race coefficient was negative while the sex coefficient was positive. See Testimonial Exhibits 28 & 29, Defendant's Exhibit 557 (comparison for males/females, nonblacks/blacks with specified employee characteristics).

iii. Work force utilization analysis.
Dr. Odell also performed "work force utilization" analyses of the sort offered by Dr. Stoikov as to the hiring case. See Defendant's Exhibits 24-27. These analyses compared the rate at which the Bank filled its *336 jobs, either through external hire or internal promotion, against actual labor market availability of persons qualified to hold such jobs (as defined by the Bank). See Defendant's Post-Trial Brief at 265.
According to the Bank,[131] "utilization has been measured in terms of jobs `filled' by blacks and females, as reflected in the number of individuals by race and sex hired into or promoted into a particular job family in a given year." Id. at 268 (emphasis supplied). This measure of utilization was then compared with black and female availability (as defined by the Bank) in a certain geographical area (the Bank chose the Dallas SMSA for nonexempts and what it designated the "SOUTHWEST STATES SMSA (LA, AR, OK, NM, TX)"[132] for exempts). This analysis consisted of a series of pages off the following sort:


Job Family-310
Nonexempt
                                             Blacks      Total
1969          # Jobs Filled                       6         59
              % of Total Filled                10.2      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range             0  -   5          *
1970          # Jobs Filled                       1         14
              % of Total Filled                 7.1      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range               0 -  2          *
1971          # Jobs Filled                       0          9
              % of Total Filled                 0.0      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range              0  -  1          *
1972          # Jobs Filled                       0         15
              % of Total Filled                 0.0      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range              0  -  2          *
1973          # Jobs Filled                       3         24
              % of Total Filled                12.5      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range              0  -  3          *
1974          # Jobs Filled                       2         19
              % of Total Filled                10.5      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range              0  -  2          *
1975          # Jobs Filled                       1         20
              % of Total Filled                 5.0      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range              0  -  2          *



*337
Job Family-310
Nonexempt
                                             Blacks      Total
1976          # Jobs Filled                       1         33
              % of Total Filled                 3.0      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range             0  -   3          *
1977          # Jobs Filled                       6         53
              % of Total Filled                11.3      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range             0  -   4          *
1978          # Jobs Filled                       4         57
              % of Total Filled                 7.0      100.0
              % of SMSA Representation          3.5      100.0
              Z Score                        0.0000          *
              Randomness Range             0  -   4          *
                               Table 20
               WORKFORCE UTILIZATION BY RACE (1969-1978)
               Compared Against Dallas-Fort Worth SMSA
                            By Job Family
  From Defendant's Exhibit 24.

The Bank states that the results "directly controvert Plaintiffs' ... claim that the bank discriminates in promotion and job assignment." Defendant's Post-Trial Brief at 266.
There are at least two reasons why this work force utilization analysis is not probative on the issues of initial placement or promotion. First, there is too high a level of disaggregation: zero is often in the randomness ranges[133] and fragmentation errors[134] arise. Second, the geographic markets used are inappropriate. Dallas-Fort Worth SMSA figures were used for non-exempts. We reject the use of this geographic market for non-exempts in the hiring case, for reasons described in section IX(G), infra; no substantial reason has been offered as to why this geographic market ought to be here used, and so we reject it here as well. The Bank similarly offered no substantial justification, either at trial or in a report, for the use of the five-state SMSA geographic market for the exempt analysis. See, e. g., Defendant's Preliminary Statistical Brief at 61. Cf. id. at 26; Defendant's Final Statistical and Trial Brief at 65-66. The Bank should have made more clear the connection between the fact that exempts tend to have more diverse geographical backgrounds and something as specific as the selection of five Southwestern states as a labor market. See section IX(F) & n.205, infra. In contrast to the hiring studies done by Dr. Stoikov, in which she at least used the Bank's actual recruiting patterns to establish geographic markets, Dr. Odell offered no justification for the use of the five-state "SMSA" instead of any other facially reasonable market.


*338 B. Applying the Law to the Data.

Courts have recognized the variety of statistical evidence usable in proof regarding initial placement and promotion discrimination. First, it appears that the plaintiff need not separate an employer's discrimination in initial placement from discrimination in promotion. As discussed earlier, an employee's present position on the hierarchy ("placement") is a result of his initial placement and subsequent promotions. Present placement has been used by courts both as proof of initial placement discrimination, and as evidence of promotion discrimination. In determining that the lower court had committed error in its finding that the defendant employer had "at no time made initial job assignments (either to departments or the specific jobs) on the basis of an employee's race," the Fifth Circuit has looked at evidence that the average job class for blacks was lower on the hierarchy than the average class for whites. See James v. Stockham Valves & Fittings Co., supra n.95, at 321 & 326-27 (emphasis supplied). In Davis v. Califano, 613 F.2d 957, 965 (D.C.Cir.1979), the D.C. Circuit looked at disparities in GS-level between males and females, in the context of a white female plaintiff's allegations of promotion discrimination. Similarly, the Fifth Circuit in Fisher v. Procter & Gamble Manufacturing Co., supra n.99, at 543-44, looked at existing placements as evidence of discrimination in promotion. See also Johnson v. Uncle Ben's, Inc., 628 F.2d at 423 & n.2 (5th Cir. 1980).
Second, the Fifth Circuit has at least implicitly accepted, under appropriate circumstances, the use of groups' wages to proxy for their position on the hierarchy. Thus, as with the job allocation issue in Stockham Valves, supra, the court was influenced by a disparity in the average hourly earnings of rate between black employees and white employees as of September 1973. See id. at 321 & 327. Cf. Kirby v. Colony Furniture Co., 613 F.2d 696, 701 (8th Cir. 1980).
Third, the Fifth Circuit permits a prima facie case of promotion discrimination to be established by comparing the percentage of promotions obtained by blacks with their percentage representation in the relevant work force. See Watkins v. Scott Paper Co., 530 F.2d 1159, 1190-93 (5th Cir.), cert. denied, 429 U.S. 861, 97 S.Ct. 163, 50 L.Ed.2d 139 (1976).
Fourth, it appears that a particular analytical construct must be used when business necessity is used to justify standards or procedures which operate to disparately deny promotions to blacks. As described by the Fifth Circuit in Swint v. Pullman-Standard, supra, at 535-36:
[W]e derived from Griggs v. Duke Power Co. three factors for determining whether promotional standards and procedures arise from a legitimate business necessity so as not to constitute a Title VII violation: "(1) The standards and procedures were not shown to be significantly related to successful job performance. (2) The procedures operated to disqualify a substantially higher rate of Blacks than Whites. (3) The jobs had formally [sic] been filled by whites as part of a long-standing practice of discrimination ..." Rowe [457 F.2d] at 355.
.   .   .   .   .
[F]or a practice, which is not intentionally discriminatory or neutral but perpetuates consequences of past discrimination, to be justified by business necessity, the practice must "not only foster safety and efficiency, but must be essential to that goal ... and there must not be an acceptable alternative that will accomplish the goal "equally well with a lesser differential racial impact." Parson v. Kaiser Aluminum & Chemical Corp., 575 F.2d 1374 (5th Cir.), cert. denied, 441 U.S. 968, 99 S.Ct. 2417, 60 L.Ed.2d 1073 (1979).
Applying these standards, we find the evidence presented by the plaintiffs establishes a prima facie case of discrimination in initial placement and in promotion for blacks and female non-exempts for all years and black exempts from 1973. They do not establish a case for female exempts for any year. We also find that the Bank *339 does not successfully rebut any prima facie case. See section IX(C) and IX(F); infra (re standards for prima facie case and rebuttal).

1. Black and Female Nonexempts.

The court rests its finding on the establishment of the prima facie case as to black nonexempts and female nonexempts primarily on the studies controlling for personal characteristics and adding the sex/experience interaction term, discussed in section VIII(A)(1)(v), supra. While such separate regressions may suffer from some truncation bias, such bias would make it more, not less, difficult for plaintiffs to prove discrimination. Moreover, the results from these regressions are consistent with those from other regressions. Cf. Testimony of Dr. Janice Madden at 89 & 91.
As has been described more fully in section VIII(A)(1)(v), supra, Dr. Madden examined the effects of sex and race on job placement within nonexempt jobs, and found, both before and after controlling for various productivity characteristics and adjusting for age as a bad proxy for the experience of females,[135] that for each race-sex group for each year there was a statistically significant coefficient for group status when the dependent variable was job placement (as measured by grades and/or Hay points) in nonprofessional jobs. As shown by Table 3, for the 1969-72 period, on a 17-grade system, black nonexempt males tended to be placed approximately three grades lower than white males. When non-group-status explanatory variables are included, black nonexempt males were placed 2 to 3 grades lower. For white females, the unadjusted grade differentials were approximately 1 to 2 grades, while when non-group-status explanatory variables are included, grade differentials ranged from about 1 grade to 3 grades. In Table 4, for 1973-78, on a 10-grade scale, the corresponding differentials for blacks were each about one, while for women ranged from .350 to .759, and .206 to .694.[136]
Dr. Madden also performed the same analyses as in Table 4, except with Hay point conversions, with similar results.
These analyses establish a prima facie case. Much of the discussion of the Bank's objections to plaintiffs' analyses in the prima facie compensation case are applicable here as well. We discuss these arguments seriatim:
(i) "The primary weakness of Dr. Madden's analyses is her lack of understanding of the Bank's use of Hay Points in its salary administration. This ignorance has resulted in an analysis that relies on a dependent variable which is not an accurate measure of employee skill at time of hire." (Defendant's Post-Trial Brief at 151).
This description of the "primary weakness" in Dr. Madden's analysis is unclear; indeed, the passage above is the full extent of the Bank's elucidation of this argument. We can do no more than cite it.

*340 (ii) "[D]r. Madden's use of Haypoints and/or nonexempt job grades in her rebuttal analysis does not measure the employer's points at time of hire, but at year end. Thus, she does not differentiate between Hay points at time of hire, and at time of last job at year end.... Her dependent variable, therefore, is not an accurate measure of skill or qualifications at time of hire, and, accordingly, is completely invalid as a measure of initial job placement." (Defendant's Post-Trial Brief at 152).
Unquestionably, Dr. Madden's analysis cannot assert that there was initial placement discrimination. After all, she is examining people at their then current positions, and such positions are the result of initial placement and subsequent promotions (if any).
But, as discussed earlier, under Fifth Circuit doctrine, a plaintiff can use existing position as evidence of initial placement. See, e. g., James v. Stockham Valves & Fittings Co., supra n.95, at 321 & 326-27. And, as the D.C. Circuit held in Davis v. Califano, supra, existing placement can also be used to establish promotion discrimination.
(iii) "[D]r. Madden's analysis suffers from measurement problems with respect to her dependent variable. Dr. Madden chose ... to rely on job grades for nonexempt employees and, for exempts, on whether the individual was an officer. But these measures were not shown to adequately differentiate between the skills and qualifications required for the myriad of exempt and nonexempt positions at the Bank. Dr. Stoikov's testimony established that such measures are far too crude to serve as skill level measures in regression analyses, and that Dr. Madden's analysis is accordingly invalid." (Defendant's Post-Trial Brief at 152).
The Bank itself used the grade structure as a hierarchy; that the grade contains a motley group of jobs (involving the use of different skills and qualifications) does not detract from this assertion. Since initial placement/promotion analysis is directed at the issue of position within the hierarchy, the use of grades here is acceptable. The Fifth Circuit used job classes to analyze initial placement in James, supra; we see no reason why we should not similarly do so in an initial placement and promotion context.
(iv) With regard to the 1973-78 data, "[a]ll nonexempt employees in each job grade ... were given a single Haypoint value, by virtue of a conversion factor, irrespective of the fact that the Bank assigns a range of Hay points for each nonexempt job grade and individual employees are assigned Hay points throughout that range (Barksdale testimony).... [T]his mismeasurement results in coefficients which are seriously biased in favor of a finding of discrimination." (Defendant's Post-Trial Brief at 153)
The quick answer is that assuming arguendo the validity of this assertion, it does not answer the results of Tables 3 and 4-the grades analysis.
(v) "Hay points at the Bank do not represent an ordinal index of internal skill differentiation (i. e., a job valued at 800 Hay points is not worth twice as much as one valued at 400 Hay points). Econometric theory, as Dr. Stoikov testified, is violated when a nonordinal measure is used as a dependent variable in a regression analysis of this type." (Defendant's Post-Trial Brief at 154)
This argument is ambiguous, in part because the Bank appears to have meant "cardinal" when it used "ordinal." As worded in the Bank's brief, neither sentence makes sense. If Hay points are not an ordinal index, presumably the Bank means to assert that they are a cardinal one, in which case a job valued at 800 Hay points would indeed be worth twice as much as one valued at 400 Hay points. If econometric theory is violated when a nonordinal measure is used as a dependent variable, as the brief says Dr. Stoikov testified, then her own salary estimation regression is invalid.[137]
*341 We shall assume then that the Bank intended to use the word "cardinal" where it used "ordinal." This does not, however, help the Bank here. Assuming arguendo that Hay points are an ordinal measure, and that to use ordinal measures as variables violates least squares, the study would not lose its probative effect. When discussing her "grades" analysis on direct examination, Dr. Madden made a point of mentioning that her "grades" dependent variable was an ordinal measure; she made no mention of how this could deleteriously affect any conclusions of the presence or absence of discriminatory placement. Implied by this is that she felt that such effect would be de minimis for the limited purpose of detecting the presence of discrimination. Keeping in mind the inevitable imperfection of social science modeling, discussed in the compensation case, and the role of the prima facie burden, it was up to the Bank to show not only that there was an imperfection in modeling and the magnitude of its impact of such imperfection on the plaintiffs' indicators of proxy. The Bank has done little more than point out the possible existence of a violation of the assumptions of least squares analysis.
(vi) "[T]he evidence at trial plainly showed that Hay points are not evenly distributed among Bank employees .... And, Bank Vice President Thomas Barksdale testified that RNB Hay points range as high at 6,000; Dr. Madden assumed the range was 1,500 (Madden testimony).
The effect of this lack of even distribution makes it inappropriate to use the actual value of such a variable in a linear regression model. Dr. Madden admitted at trial that this measurement problem existed; her only response was that it would take an expensive calculation to rectify the misspecification. Dr. Stoikov established, however, that this is simply not true; calculating a nonlinear functional from such as the natural logarithm of a variable is no more complicated than using the actual value in regression. Moreover, failure to use a nonlinear transformation of, rather than the actual value of, a variable which is not normally distributed across the population results in completely invalid estimates." (Defendant's Post-Trial Brief at 155).
This argument is puzzling. First, the argument seems to mistake the nature of a normal distribution. An "even distribution" of Hay points would consist of a line parallel to the horizontal axes; a normal distribution would not. Second, we do not understand that Dr. Stoikov testified-or that Dr. Madden conceded-that it is necessary that the dependent variable in a least squares analysis be normally distributed. See, e. g., G. Maddala, supra n.49, at 75. While it is true that the sum of random effects embodied in the disturbance term must be distributed normally," in the `bell curve' generally characteristic of the distribution of the sum of independent random effects,"[138] the Bank has not explained how this normality assumption requires that the dependent variable also be normally distributed.[139]
(vii) "[A]ll exempt employees without Hay points are excluded from the analysis.... Dr. Madden failed to estimate separate placement regressions for exempt employees ..." (Defendant's Post-Trial Brief at 155).
Whatever the validity of these criticisms, they are not directed at the nonexempt analysis.
(viii) "Dr. Odell ... demonstrated the invalid results that obtain from improperly measuring variables in actual value rather than in natural log form, and vice versa (DX556)." (Defendant's Post-Trial Brief at 155-56).
To say the least, the Bank is unclear as to whether Dr. Madden used natural logarithms where she should actual value, or *342 vice versa. In the grade and Hay point analysis, Dr. Madden used actual values; Dr. Odell did not testify or state in Defendant's Exhibit 556 that this was improper. Cf. section VII(C)(9), supra.
(ix) "Dr. Madden's analysis fails because she has omitted important and quantifiable variables from her analysis that are effective human capital predictors of both pay and job placement." (Defendant's Post-Trial Brief at 156).
The Bank's omission bias argument is here quoted in its entirety. We reject it here for reasons analogous to those dictating its rejection in the compensation case.
(x) "[Dr. Madden's] cohort analysis (PX 1309) fails as an estimation of the Bank's initial job placement practices." (Defendant's Post-Trial Brief at 156).
Whatever the validity of the cohort analysis, we do not rely on it here as to the black nonexempts case.

2. Female Exempts.

We find that the plaintiffs have not established a prima facie case as to female exempts: that is there is not enough evidence probative of discrimination in the initial placement or promotion of the females with exempt positions.[140] All of the evidence marshalled in support of the prima facie case suffers from fatal flaws.
The plaintiffs rely, as they may under James v. Stockham Valves, supra n.95, at 321 & 326-27, on salary (among other things) as an indicator of placement on a hierarchy rather than on some more direct measure, such as Hay points. Moreover, they may rely on existing placement as probative of both initial placement discrimination and promotion discrimination. See id.; Davis v. Califano, supra, at 965; Fisher v. Procter & Gamble Manufacturing Co., supra n.99, at 543-44. Thus, the analytical construct of comparing present salaries after culling out the effects of salary discrimination and differential productivity is a legally cognizable way of gauging the presence or absence of initial placement and promotion discrimination.
The analyses we discussed more fully in sections VIII(A)(1)(i) and (A)(1)(iii), supra, are intended to so gauge this sort of salary difference. The analysis in section VIII(A)(1)(iv), supra, is intended to determine more directly the current placement of an employee. These three sets of analyses, including all of the relevant analyses outlined in the tables of section VII(B)(1), supra, all fail for females for the same reason: with those regressions which include a Hay point variable, the use of age (or age-based) proxies rather than actual general experience (and "uncorrected" by use of a sex-experience interaction term) causes the relative experience of females to be so overestimated that the results are likely to be biased to such an extent that those regressions cannot be used to establish a prima facie case. Our discussion in section VII(C)(5)(ii), supra, on age as proxy for general experience for females is applicable here in most respects, and so we will not go through that exercise again.
The cohort analysis in section VIII(A)(1)(iii), supra, suffers from a number of serious problems, perhaps none of which alone may not be enough to invalidate the analysis, but which together will cause us to reject it. First, the number of females in the sample who were exempts when hired in 1969 was probably very small, probably something on the order of six, out of the 66 females in that cohort.[141] The *343 results for females generated by the cohort analysis would be reflective primarily of the experience of the nonexempt females, and, for all we know, the six females we presume to have been hired as exempts may have left the Bank in 1972, and no women may have been promoted to exempt status thereafter: we would be following a cohort solely of nonexempt females.
Second, the widening pay differentials which the cohort analyses take to be probative may be attributed, as Dr. Stoikov herself stated, to either:
(a) lower rates of promotion and/or pay increase for black men, white women, black women, and other minority females; (b) differential turnover rates by race and sex such that the most productive black men, white women, black women, and other minority women leave the bank while the most productive white men stay.
Plaintiffs' Exhibit 1308, at 6. Dr. Madden has not shown that explanation (b) does not account for the widening differential. Of the sample of 498 employees hired in 1969 and still at the Bank in 1970, only 91 remained in 1977. See Plaintiffs' Exhibit 1308 (Table 14); Testimony of Dr. Janice Madden at 60. The percentage of females and blacks fluctuated. Id. Merely because the mean years of education of the cohort as a whole did not change very much (the number fluctuates within a narrow band-13.4 to 13.8 years-in the period 1970-77) does not mean that the mean years of the females or of blacks did not fluctuate or change in a systematic way.[142]

3. Black Exempts.

We rely on the occupational distribution discrimination implied by both the separate regression for exempts from 1973 on as well as other, pooled regressions under the section VIII(A)(1)(i) analysis, supra, to hold that a prima facie case of initial placements and promotion discrimination is established for black exempts from 1973 on. Through our discussion in that section, in section VII, supra, and in our discussion of the black and female nonexempts, section VIII(B)(1), supra, we have covered all the objections the Bank offers to the black exempts prima facie case. The only objection not earlier discussed-that the Blau/Madden analysis omitted exempt employees without Hay points from the analysis, "without a valid demonstration that the remaining employees constitute a representative sample of the population"-is not well taken. See Defendant's Post-Trial Brief at 158. The Bank made no showing as to the extent of the effect of missing Hay points on the plaintiffs' modeling results.
The evidence presented by the Bank as to both initial placement and promotion does not rebut any of the prima facie cases established.
The work force comparisons, discussed in section VIII(A)(2)(i), supra, must be rejected for the reasons therein discussed.[143] Especially serious is the crudity of the promotion definition: it does not take into account the fact that the overall effect of the promotion policies is determined not only by the number of promotions but by the size of each. Cf. Smith v. Union Oil Co., 17 Empl. Prac. Dec. ถ 8411 (N.D.Cal. 1977) (at discussion of first table in section *344 E). The zero in the randomness range problem and overfragmentation problem are technical problems, which might possibly be remedied if this court, on its own, aggregated results and performed independent calculations. Given the number of calculations necessary and the general limitations of institutional competence, we decline to do so.
Sections VIII(A)(2)(ii)(a) and (b), supra, are only concerned with very narrow questions of initial placement and promotion. They, unlike the plaintiffs' analysis, do not look at overall patterns of initial placement and promotion. Even if otherwise valid-which issue we do not decide[144]-these studies can only show that only a certain narrow band of employees may not have been discriminated against in a certain manner.
Section VIII(A)(2)(ii)(a), supra, is concerned with whether one who comes to the Bank has a probability of being hired into professional job families untainted by group-status consideration. By definition, this question is not relevant to any claims of initial placement discrimination of those hired into the exempt category, even as to blacks or females who would have been placed higher within the exempt category had they been white males. The question is also not relevant to any claims by those in the nonexempt category who claim not that they have been discriminated against in being placed in the nonexempt category rather than the exempt, but that they should have been placed higher within the nonexempt category.
The study described in section VIII(A)(2)(ii)(b), supra, is concerned with the influence of sex or race on the probability of being promoted from a nonprofessional into a professional position. The study is limited to only one year (1978), leaving unclear the answer to this question in other years. And again, the promotional opportunity examined is one which concerns not exempts at all, or even most nonexempts, but only those in the nonexempt category who had a realistic chance of crossing the dividing line between exempts and nonexempts. The analysis does not examine the opportunities available to employees of various group statuses to cross all the dividing lines on the hierarchy at Republic. In short, nonexempt-to-exempt is not the only promotional possibility.
The analysis described in section VIII(A)(2)(iii), supra, must be rejected because of the sample size problems and the lack of proof for the relevance of the particular geographic markets chosen.[145]

C. Summary.

The plaintiffs have failed to establish a prima facie case of initial placement or promotion discrimination as to female exempts for any year.
The plaintiffs have established prima facie cases of initial placement and promotion discrimination as to black and female nonexempts for all years and black exempts from 1973. These prima facie cases are not successfully rebutted by the Bank by its own modeling or by its challenges to the reliability of the plaintiffs' modeling. A Phase I finding of liability therefore exists for (i) blacks and females employed at the Bank in nonexempt positions at any point in the class period; and (ii) blacks employed at the Bank in exempt positions at any time in the interval from January 1, 1973, to the end of the class period.

IX. Hiring

The parties' proof on the question of discrimination in hiring consists of a series of comparisons between the Bank's workforce or its hires during the class period and appropriate segments of the general labor market. To understand these analyses we must first explore the techniques of statistical inference, which must be used extensively by the court in evaluating the results *345 of the statistical studies. Only then may we turn to the studies themselves, looking at the data, the law, and the results obtained when the law is applied to the data.
At the outset, the relationship of the hiring analysis to other parts of the case, principally the placement/promotion analysis, must be made clear. The scope of the hiring case is commensurate with the scope of the applicant subclasses, see section I, supra: only those rejected altogether by the Bank may complain of "hiring" discrimination. Those who were hired, but who were placed in an inappropriate position as a result of discrimination, complain instead of initial placement discrimination, discussed in section VIII, supra.

A. Statistical Inference.

The starting point for statistical proof of hiring discrimination in a Title VII class action is a simple comparison requiring no statistical sophistication: does the proportion of a racial or sexual group in the relevant segment of the labor force exceed the proportion which that group represents among those hired by the defendant for a particular job or jobs? While the focus remains on deliberate discrimination, such a "community composition comparison approach," see Shoben, Probing the Discriminatory Effects of Employee Selection Procedures with Disparate Impact Analysis Under Title VII, 56 Tex.L.Rev. 1, 8-19 (1977), provides an important insight into the existence of such discrimination:
Statistics showing racial or ethnic imbalance are probative in a case such as this one only because such imbalance is often a telltale sign of purposeful discrimination; absent explanation, it is ordinarily to be expected that nondiscriminatory hiring practices will in time result in a work force more or less representative of racial or ethnic composition of the population of the community from which employees are hired. Evidence of long-lasting and gross disparity between the composition of a work force and that of the general population thus may be significant even though Section 703(j) [42 U.S.C. ง 2000e-2(j)] makes clear that Title VII imposes no requirement that a work force mirror the general population.
International Brotherhood of Teamsters v. United States, 431 U.S. 324, 340 n.20, 97 S.Ct. 1843, 52 L.Ed.2d 396 (1977).
A disparity between the availability of racial or sexual groups and the proportions of those groups among a defendant's hires does not, however, always result from discrimination. Like any series of uncertain events, the hiring of a number of employees may through random fluctuation result in more or fewer than the expected number of hires of members of a specific group. Just as the flipping of 100 coins may result in more or fewer than 50 "heads," the hiring of 100 employees may result in a number of black or female hires larger or smaller than that predicted by the proportion of blacks or females in the relevant labor pool. Baldus and Cole have identified "four main sources of random instability and variability that can influence the outcomes of a selection process:" the possibility that the sample of the defendant's hires chosen for study (e. g., those hired during a given year) may be unrepresentative; the possibility of error in the measurement of individuals' characteristics and qualifications (by the employer or by the statistical analyst); the use of a random selection process, as in jury selection; and "random shocks from unknown or unknowable factors operating in the selection process," such as the effect of subjective race- and sex-neutral preferences among small numbers of applicants. D. Baldus & J. Cole, supra n.55, ง 9.11, at 298-99. It is the purpose of statistical testing to offer insight into the likelihood that a given disparity was produced by such random factors rather than by systematic bias.
The mathematical underpinning for such statistical tests lies in certain elementary concepts of probability. The first of these is known as the "product rule." That rule states that the probability of occurrence of a given series of events, each of which is independent of the others, may be obtained by multiplying the probabilities that each *346 event will occur. See G. Wadsworth & J. Bryan, Applications of Probability and Random Variables 11-12 (2d ed. 1974). Thus the probability of two consecutive "heads" tosses of a coin is the probability that the first toss will be "heads" (ฝ) multiplied by the probability that the second toss will be "heads" (ฝ), that is, ผ. Similarly, the probability that a card drawn at random from a deck of cards will be the ace of hearts equals the probability that the card will be a heart (ผ) multiplied by the probability that the card will be an ace (1/13), that is, 1/52.
The second such concept, while traveling under the formidable name of combinatorial theory, is equally simple. Combinatorial theory is used to calculate the number of different combinations or sequences which may result when a series of events (in our case individual hiring decisions) takes place, each of which may take place in more than one way (e. g., hiring of a female, hiring of a male). Referring to our earlier example, the probability of two consecutive "heads" is ฝ times ฝ, or ผ, because there is only one sequence in which that event may occur. Suppose, however, that we were interested in the probability that two tosses might produce one "heads" and one "tails." Applying the product rule, we would multiply the probability that one coin would be "heads" (ฝ) by the probability that the other coin would be "tails" (ฝ), and would obtain ผ. This time, however, there would be two combinations or sequences in which this event could occur: the first coin might be "heads" and the second "tails," or the first "tails" and the second "heads." Multiplying the probability obtained from application of the product rule (ผ) by the number of sequences in which that probability may arise (2), we obtain a probability of one "heads" and one "tails" of ฝ.
Where the numbers of events are small, the number of possible sequences can be determined by common sense. When dealing with larger groups, however, a common-sense listing of possible sequences becomes extremely laborious. Thus while it is intuitive that there are two sequences in which a "one-heads-one-tails" result may occur when two coins are tossed, it is by no means intuitive that there are 252 different sequences of five heads and five tails which may occur in a group of 10 tosses. A mathematical formula is available to calculate the number of different sequences in which a given number of events of a particular type (e. g., female hires) may occur in a given number of events (e. g., total hires).[146]
These two concepts conjoin to enable the statistician in a Title VII case to aid the court in determining whether a particular hiring outcome is attributable to chance rather than to discrimination. Using these concepts, the statistician can calculate the probability that a given outcome, i. e., a given number of hires of members of a racial or sexual group out of a given total number of hires, would occur by operation of random factors in the case of a nondiscriminatory employer. If a result which appears to suggest discrimination might *347 frequently occur by chance even in the absence of discrimination, that appearance cannot be relied upon to prove that discrimination actually occurred.
A concrete example may be helpful. Suppose that in a given labor market blacks represent 25% of those qualified for a particular task. Suppose further that an employer has hired four persons during the relevant time period to fill jobs involving that task, and that none of those hired were black. What is the probability that a nondiscriminatory employer, choosing his employees without respect to race from among those qualified, would have failed to choose any blacks for the four positions? What is the distribution of probabilities for hiring of blacks, i. e., what is the likelihood that a particular number of blacks, between zero and four, would be hired by such an employer?
Applying the product rule, the probability of hiring nonblacks in all four positions is the product of the probabilities of hiring a nonblack into each such position. Since nonblacks represent 75% of the qualified population, the probability that any one position would be filled with a nonblack is .75, and the probability that all four such positions will be so filled is .75 times .75 times .75 times .75, or .3164. Similar probabilities for the hiring of three nonblacks (and one black), two nonblacks (and two blacks), and so forth, can be computed using the product rule and the number of sequences as outlined above.[147] This process of using the product rule and combinatorial theory results in a set of probabilities known as a "binominal distribution."
Thus we observe that, while on the average a nondiscriminatory employer faced with a 25% availability of blacks would hire one black in filling four positions, there is a 31.64% probability that in such a situation no blacks would be hired. Here a facially discriminatory outcome-no black hires despite 25% availability-is revealed by the statistical test to be an outcome which appears frequently even in the absence of discrimination, and consequently an outcome which cannot be relied upon to support an inference of discrimination.
In evaluating the likelihood that a given hiring result might occur in the absence of discrimination, it is helpful to look not merely at the probability that that particular result might occur, but at the probability that a result at least as extreme as the observed result might occur. Thus in evaluating a hiring outcome in which, despite a 50% female availability, only two of 10 hires were female, we may wish to examine the probability that zero, one, or two hires might occur if only chance were operating.[148] To make such an examination, we would add the probability of zero female hires occurring, the probability of one female hire occurring, and the probability of two female hires occurring. The resulting probability is designated a "significance level." If the probability that an apparently discriminatory result will occur by chance is, for example, 5%, it is customary to state *348 that the result is "statistically significant at the .05 level."[149]
A test of statistical significance does not determine the probability that any particular result in fact occurred by chance.[150] Such an evaluation, depending upon an evaluation of all the facts and circumstances, many of which are not susceptible of objective measurement, is a task for the court. While evidence of statistically significant disparities can aid the court in that task, it cannot be used to state with certainty the probability that discrimination has actually occurred. Cf. section V(D), supra (role of anecdotal evidence).
This being the case, what level of statistical significance should be required by the court before the inference may be allowed to arise that discrimination has occurred? It has become a convention in social science to accept as statistically signficiant values which have a probability of occurring by chance 5% of the time or less, i. e., values which are statistically significant at the .05 level. E. g., D. Baldus & J. Cole, supra n.55, ง 9.221, at 308 n.36; N. Nie, C. Hull, J. Jenkins, K. Steinbrenner, & D. Bent, Statistical Package for the Social Sciences 222 (2d ed. 1975).[151] Statistical significance at the .05 level is consistent with the "two or three standard deviation" approach employed by the Supreme Court in Castaneda v. Partida, 430 U.S. 482, 496 n.17, 97 S.Ct. 1272, 1281, n.17, 51 L.Ed.2d 498 (1977), see D. Baldus & J. Cole, supra n.55, ง 9.221, at 308 n.35a, and has been widely used by the courts. Id. at 308 n.36; Beyond the Prima Facie Case, supra n.57, at 400 & n.58. See Hallock, The Numbers Game-The Use and Misuse of Statistics in Civil Rights Litigation, 23 Vill.L.Rev. 5, 13 (1977-78) (recommending use of 5% level in employment discrimination cases). While the courts may properly rely on the level of significance employed in other fields, the judgment as to what is or is not statistically significant "is a legal determination properly made by the court and not by an expert." D. Baldus & J. Cole, supra n.55, ง 9.221, at 308.
Regardless of the level of significance chosen by the court, it must be remembered that statistical significance is not an "either/or" proposition. As the probability of an event's occurrence in a nondiscriminatory environment decreases, the significance of the results increases in a continuous fashion. Hence it is no surprise that the courts have recognized that a result falling just short of the chosen level of significance is nearly as important as a result just surpassing the chosen level. See Reynolds v. Sheet Metal Workers Local 102, 22 Empl.Prac.Dec. ถ 30,739, at 14,814 (D.D. C.1980); Pennsylvania v. Rizzo, 466 F.Supp. 1219, 1231 (E.D.Pa.1979); Watkins v. Scott Paper Co., 6 Empl.Prac.Dec. ถ 8,912, at 5880 (S.D.Ala.1973); aff'd on this point, 530 F.2d 1159, 1187 n.40 (5th Cir.) (".1 significance level might be acceptable;" emphasis in original), cert. denied, 429 U.S. 861, 97 S.Ct. 163, 50 L.Ed.2d 139 (1976); cf. United States v. Georgia Power Co., 474 F.2d 906, *349 915 & n.11 (5th Cir. 1973) (requirement of 29 C.F.R. ง 1607.5(c)(1) that employment tests be shown to be job-related at the .05 level of significance "must be read as setting a desirable goal and not a prerequisite"). Indeed, it has been suggested that the best approach is to evaluate statistical significance on a sliding scale, giving great weight to highly significant results and little weight to slightly significant results, rather than to establish any cutoff level of significance. D. Baldus & J. Cole, supra n.55, ง 9.221, at 308 n.36.
When dealing with small numbers, statisticians prefer to use the binomial distribution in calculating statistical significance levels, inasmuch as that distribution enables a precise calculation of the probabilities that extreme results could occur due to random fluctuation.[152] As the sample size (in our case the number of hires in a relevant period) increases, the mathematical calculations necessary to determine significance levels using the binomial distribution become extremely onerous even with the aid of digital computers.[153]
In such cases, use is made of the fact that for large sample sizes the binomial distribution may be accurately approximated by using a distribution known as the "normal distribution."[154]See G. Wadsworth & J. Bryan, supra, at 132-33. The normal distribution has the advantage that statistical significance levels may be determined using widely published tables, see, e. g., R. Burington, Handbook of Mathematical Tables and Formulas 257-60 (2d ed. 1946), and a simple transformation of the data. This transformation consists of subtracting the number of hires from a particular group expected in the absence of discrimination from the number of hires from that group actually observed, and dividing that result by the standard deviation.[155] The result of this mathematical manipulation is an important and useful statistic known as the Z-statistic or Z-score.[156]
The value in Z-scores lies in the fact that they may be easily correlated with statistical significance levels. Thus, for example, if the court adopts a .05 level of significance using a two-tailed test, reference to tables reveals that such a level corresponds to a *350 Z-score of -1.96.[157] R. Burington, supra, at 258. That is, results which deviate from the average by 1.96 or more standard deviations may be expected 5% of the time when dealing with the normal distribution. See Rich v. Martin Marietta Corp., 467 F.Supp. 587, 601 (D.Colo.1979); Caulfield v. Board of Education, 486 F.Supp. 862, 21 Empl. Prac.Dec. ถ 30,389, at 13,195 (E.D.N.Y.1979). The Z-scores for other levels of significance may be determined by reference to the tables.[158] The use of Z-scores aids in evaluation of statistical results and provides a convenient benchmark for comparison with the Castaneda "two or three standard deviation" test.[159]
Finally, a concrete example may be in order. Returning to our earlier example in which no females were hired in four positions despite a 25% female availability, and calculating the Z-score using the formula contained in n.162, infra, we obtain a Z-score of -1.155. This corresponds to a significant level of .2482. R. Burington, supra, at 258. The fact that this significance level is somewhat different from the actual significance level of .3203, computed using the more exact binomial distribution,[160] demonstrates that the normal distribution is not a good approximation for the binomial distribution where the numbers are small. For larger groups, however, the use of Z-scores based on the normal distribution will provide an accurate approximation of statistical significance levels.

B. Plaintiffs' Case: The Data.

Plaintiffs' hiring case consisted primarily of the testimony and report of Dr. Paula England.[161] Dr. England's studies of hiring at the Bank consisted of a series of statistical comparisons, relating the proportion of blacks and females in the overall Bank population and in various subgroups of that population to similar proportions in various segments of the society. One additional study compared the proportions of females hired into exempt positions by the Bank with a number of availability measures.
Dr. England did not attempt to identify the most appropriate geographical labor market with which to compare the Bank's hiring performance. Instead, she chose to present data for two such labor markets, the Dallas/Fort Worth Standard Metropolitan Statistical Area ("SMSA") and the City of Dallas. These areas were chosen based on the assumption that the Bank recruits its work force in part from the City of Dallas and in part from outlying areas of the Dallas/Fort Worth SMSA. Dr. England did not attempt to identify the respective proportions of the Bank's work force contributed by each of these areas, or to ascertain whether the Bank draws from other areas as well. SMSA and city work force area were derived from reports from the Texas Employment Commission.
Dr. England compared the proportion of blacks in the overall Bank work force with the availability of blacks in the City and SMSA. She also compared proportions of *351 blacks and females in selected relatively high paid groups at the Bank. These groups consisted of employees designated as "officials and managers" by the Bank in its annual EEO1 reports, employees designated as "professionals," and employees designated as "exempt." These groups were compared to the two geographical labor forces and to the proportions of blacks and females in the overall Bank work force. The latter comparison was based on the assumption that the Bank's overall work force represents a potential availability pool from which it could draw employees for these elite groups. Data regarding the Bank's work force were derived from EEO1 reports and from other data supplied by the Bank.
The results of these tests are set forth in the margin.[162] The tables show an underutilization *352 of blacks or females, statistically significant at the .05 level using a two-tailed test, in cases where the Z-score is negative and exceeds 1.96 in magnitude. The results show such a disparity in all cases except for the comparison of blacks at the Bank with blacks in the SMSA for 1974-78, the comparison of black "professionals" at the Bank with availability figures for 1969-70, and the comparison of female "professionals" at the Bank with availability figures for a variety of years.
Dr. England also performed comparisons between the Bank's subgroups and similar subgroups at other banks. These comparisons did not proceed on the assumption that the Bank actually recruits in large measure from other banks, but rather on the assumption that hiring of blacks and females by other banks in a given proportion indicates that similar results could have been been achieved at Republic. Data for all banks filing EEO reports were obtained from EEOC reports. Comparisons were also performed using data from banks studied by the Council on Economic Priorities. See T. Simcich, Shortchanged: Update 29, 32, 44, and 47 (1976). These data pertain to the three largest banks, by deposit, in Atlanta, Chicago, Detroit, New York, Philadelphia, and Washington, D.C. Comparisons with CEP banks were performed using proportions of "minorities" rather than proportions of blacks, since the CEP data was so structured. The results of these studies are presented in the margin.[163] Most such *353 results show disparities for blacks and females; some are statistically significant at the .05 level.
The foregoing comparisons involve so-called "stock" figures, i. e., figures relating to the number of employees at the Bank in a given category, rather than the number hired in any given year. Dr. England also performed one series of comparisons involving "flow" figures: she compared the proportion of females hired into exempt positions by the Bank to labor force proportions, Bank work force proportions, and the proportions of females receiving bachelor's degrees. These results are shown in the margin.[164] Disparities for females which *354 were statistically significant at the .05 level were achieved in all but three such comparisons.
Finally, Dr. England compared the proportion of minorities hired by the Bank and the proportion of minorities interviewed by the Bank with the proportion of minorities who applied for positions at the Bank. The results of this study are shown in the margin.[165] These results show racial disparities which are generally significant at the .05 level.

C. Plaintiffs' Case: The Legal Standards.

We begin with the now universally recognized proposition that a plaintiff in a Title VII class action may establish a prima facie case of discrimination in hiring by showing a gross disparity between the proportion of minorities or females in the relevant labor market and the corresponding proportion in an employer's workforce or among its new hires.[166] In International Brotherhood of Teamsters v. United States, supra, 431 U.S. at 340 n.20, 97 S.Ct. at 1856 n.20, Justice Stewart referred to the probity of "[e]vidence of longlasting and gross disparity between the composition of a work force and that of the general population" in the search for purposeful discrimination. In Hazelwood School District v. United States, supra n.99, 433 U.S. at 307-08, 97 S.Ct. at 2741, the Court extended this principle, stating that "[w]here gross statistical disparities can be shown, they alone may in a proper case constitute prima facie proof of a pattern or practice of discrimination."
The degree to which the availability figures with which an employer's record is compared must reflect relevant educational and occupational qualifications necessarily varies on a case-by-case basis. In Teamsters, the Court noted that "evidence showing that the figures for the general population might not accurately reflect the pool of qualified job applicants would also be relevant." *355 431 U.S. at 340 n.20, 97 S.Ct. at 1856 n.20. In Hazelwood, this standard was applied in a concrete manner: the Court held that the defendant school district's work force should be compared not with general population figures but with the qualified public school teacher population. 433 U.S. at 308, 97 S.Ct. at 2741. See Mayor of Philadelphia v. Educational Equality League, 415 U.S. 605, 620-21, 94 S.Ct. 1323, 1333, 39 L.Ed.2d 630 (1974) (where city charter limited membership on panel to highest ranking officers of designated organizations, relevant population was group of highest ranking officers); Chance v. Board of Examiners, 330 F.Supp. 203 (S.D.N.Y.1971), aff'd, 458 F.2d 1167 (2d Cir. 1972) (pool for hiring as high school principals must meet teacher qualifications). But see Spurlock v. United Airlines, Inc., 475 F.2d 216, 218 (10th Cir. 1972) (prima facie case of discrimination in hiring of airline pilots by showing of raw disparity). Similarly, in Hester v. Southern Railway Co., 497 F.2d 1374, 1379 n.6 (5th Cir. 1974), the court held a comparison between incumbents of positions requiring a typing ability of 60 words per minute and general population figures to be of "questionable value," and indicated that a "more significant comparison" would be that between the defendant's hires and the proportion of the population who could meet the typing requirement.[167]
The refinement of availability figures based upon qualifications may be viewed as a spectrum, ranging from the use of undifferentiated general population figures at one end to the pool of applicants meeting an employer's challenged job requirements at the other. While the cases teach that general population statistics are not appropriate for comparisons for jobs whose skills are not possessed and cannot easily be acquired by the general public,[168] to require a plaintiff's proof to meet the opposite extreme would effectively eviscerate the "business purpose" requirement first set forth in Griggs v. Duke Power Co., 401 U.S. *356 424, 91 S.Ct. 849, 28 L.Ed.2d 158 (1971). Griggs requires that an employer demonstrate the job-relatedness of any employment criterion having a disparate impact on minorities; a facile transformation of such employment criteria into "job qualifications" which must be met by a plaintiff's availability pool would render this requirement a dead letter. While some refinement of raw figures is essential in some cases to render the resulting statistics meaningful, plaintiffs must be given the opportunity to mount a genuine challenge to job criteria with disparate impact which they believe to lack job-relatedness.
A standard of proof accommodating these competing interests is that set forth in dictum by this court in Davis v. City of Dallas, 487 F.Supp. 389 (N.D.Tex.1980). In that case, the court noted the necessity of refining availability estimates to encompass only the available minority population within the geographical area relevant to the employer's hiring. 487 F.Supp. at 392 n.1. It then went on to state:
Some refinement as to skill level may also be appropriate in cases where the jobs in question require a readily quantifiable skill whose validity is not open to serious question. Just as it makes no sense to judge a Maine employer by the minority population of Mississippi, it may make no sense to judge discrimination in the hiring of textbook writers against an availability pool including the illiterate.
Id. See D. Baldus & J. Cole, supra n.55, ง 6.21; cf. Thompson v. McDonnell Douglas Corp., 416 F.Supp. 972, 981 (E.D.Mo.1976), aff'd, 552 F.2d 220 (8th Cir. 1977) ("at least minimal qualifications"); Dobbins v. Local 212, IBEW, 292 F.Supp. 413, 436 (S.D.Ohio 1968) ("basic skill in the particular trade involved"). This standard ensures that the plaintiff's statistics will have at least minimal probative value, while resolving doubts concerning job qualifications in favor of the plaintiff at the prima facie stage.[169]
The appropriate degree of refinement for labor market statistics may also depend on the availability of statistical data with which comparisons are to be made. While the most relevant labor market for employees who operate a particular type of sophisticated and complicated machinery might be the proportion of the general work force who are trained in the operation of that machinery, Hester, supra, practical limitations on the availability of such data might require that comparisons instead be made with the proportion qualified to operate similar machinery or possessing skills similar to those required to operate the machinery. Likewise, where necessary, the number of persons qualified for a particular task may be proxied by the number of persons actually employed in that task throughout the labor force. See Shoben, supra, at 16-17.
Additional problems are presented where, as here, a variety of jobs, requiring widely differing levels and types of skills, are aggregated for statistical analysis. In such a case, the plaintiff must present availability figures refined to include those who possess the skills typically required by jobs within the mix. To require an overall skill level comparable to that required only for isolated high-level positions would significantly underestimate the availability for typical jobs of groups whose skills are average or less than average. Conversely, to reduce all job qualifications to the lowest common denominator would test an employer's hiring against a pool which includes a large number of persons unqualified for most positions.
*357 Just as availability figures must be refined to account for differential qualification levels, availability statistics must also reflect the appropriate geographical labor market. At the prima facie stage, this requirement is governed by a standard analogous to that applied to qualification levels: the plaintiff may not ignore geographical constraints on the employer's hiring where those constraints are not open to serious question. Although the geographical market chosen by the plaintiff for its prima facie showing may not coincide precisely with that ultimately deemed most relevant once a defendant brings forth evidence of its actual hiring patterns and the reasonableness of those patterns, plaintiffs, just as in the case of skill levels, must be afforded considerable latitude. Thus a showing of gross disparities using a geographical labor market not manifestly unrelated to the defendant's hiring needs and practices will suffice at least to shift to the defendant the burden of demonstrating that some other labor market is more appropriate.
Finally, it must be remembered that the burden of demonstrating that a prima facie showing constitutes a meaningful statistical comparison is upon the plaintiff. Miller v. Weber, 577 F.2d 75, 77 (8th Cir. 1978); James v. Wallace, 533 F.2d 963, 967 (5th Cir. 1976). But see Shoben, supra, at 19 (advocating contrary rule). While, as indicated, this burden does not require a showing of exact correspondence between the geographical, educational, and occupational characteristics of the plaintiff's proffered pool and the defendant's hiring requirements, it does require that the plaintiff produce statistics sufficiently meaningful to support the inference that discrimination has occurred. Once this standard has been met, the burden shifts to the defendant to show that such an inference of discrimination is unwarranted by virtue of flaws in the plaintiff's statistical proof, or to dispel the inference of discrimination by showing that observed disparities are in fact produced by the application of neutral and job-related standards.[170]See generally, Johnson v. Uncle Ben's, Inc., 628 F.2d 419, at 421-22 & 424 (5th Cir. 1980).

D. Plaintiffs' Case: Applying the Law to the Data.

Examination of the plaintiffs' prima facie showing must begin with a consideration of the extent to which the plaintiffs' proffered availability pools reflect appropriate skill levels. Since many different jobs, with diverse skill requirements, have been grouped together for analysis, it is necessary, as previously discussed, to consider the skill requirements typical of the mix. Because, however, the typical skill levels required for exempt and nonexempt positions at the Bank differ markedly, because the plaintiffs' analyses have in some cases been separately performed for exempt and nonexempt categories, and because the class has been subdivided along exempt/nonexempt lines, it is appropriate to consider skill requirements separately for these two groups.
The mix of skills required for nonexempt jobs at the Bank is not significantly different *358 from the mix of skills possessed in the general labor force or easily acquired by those in the labor force. The Bank's nonexempt job families include computer technicians, tellers, accounting clerks, general clerks, clerk/typists, secretaries, business machine operators, and security guards. See n.185, infra. While some of these job families (e. g., computer technician) undoubtedly require specialized skills not possessed by the work force at large, such job families account for only a small fraction of the total nonexempt work force.[171] The largest proportion of nonexempt jobs at the Bank is in lower-level clerical positions.[172] Skills for these positions are commonly possessed or easily acquired. Taking into account the mix of low-skill and moderate-skill jobs in the nonexempt category, the court is convinced that comparison of nonexempt hiring with labor force availability figures is sufficiently meaningful to meet the plaintiffs' burden at the prima facie stage.[173]
The Bank places considerable reliance on the case of EEOC v. United Virginia Bank, supra n.155, in asserting that general work force comparisons are inadequate for nonexempt hiring. In that case, the Fourth Circuit stated:
The EEOC, however, rigidly continues to argue that all the black local labor force is qualified for the office and clerical positions at UVB. This is simply not true. Tellers must be able to deal with the public, handle and account for money, and operate adding machines, typewriters and other office machines. The district court found that the entire percentage of black people in the local labor force would not provide an appropriate statistical group for comparison with UVB black employees. Since this determination was a factual one, it will not be disturbed unless clearly erroneous.
615 F.2d at 149-50 (footnote omitted). Insofar as the United Virginia Bank holding relies on factual findings as to the qualifications of the local labor force and the characteristics of UVB jobs, it is of course not controlling in this case. To the extent, however, that the court holds as a matter of law that general labor force figures do not reflect the range of skills (many of which are easily acquired) required in clerical positions in a bank, it reaches a result this court must respectfully decline to follow. Such a holding would set up a standard of refinement at the prima facie stage neither warranted under the cases no consistent with the availability of statistical data.[174]
The Bank's pool of exempt jobs is significantly different from its nonexempt jobs. The exempt jobs include such job family groups as officials and managers, accountants, securities analysts and economists, marketing personnel, commercial loan analysts, commercial loan representatives, data processing personnel, trust representatives, consumer loan representatives, internal services personnel, operative services personnel, and petroleum engineers. See n.185, infra. The skill mix required for these jobs is quite different from that possessed by the general labor force or easily acquired by it. Several job families, accounting for almost ผ of the exempt hires in 1970-78, require skill levels comparable *359 or superior to those possessed by individuals with college degrees.[175] Another 25% of exempt hires involves commercial loan analysts and representatives. Extensive testimony of bank officials and other experts in the banking industry convinces the court that a college degree in a business-related field is a requirement for these positions in most cases. See section IX(G), infra. In sum, the plaintiffs have failed to meet their burden of showing that their proffered comparison between exempt hires and general work force availability figures produces meaningful statistical results.[176]
We turn now to an application of these principles to the plaintiffs' hiring data. Dr. England did not perform a separate comparison for black nonexempt hiring.[177] Nevertheless, the data she presents for all black hiring and for black exempt hiring, see n.162, supra, together with numbers of black employees from the Bank's EEO1 reports (Plaintiffs' Exhibits 518-523) and numbers of black exempt employees from Plaintiffs' Exhibit 670, enables the court to disaggregate exempt and nonexempt hiring and to calculate Z-scores for nonexempt hiring. The results of these computations appear in the margin.[178] The data show a statistically significant overutilization of blacks in nonexempt positions when SMSA figures are used. The question thus becomes whether the City of Dallas is a relevant labor market on which the plaintiffs may predicate a prima facie showing as to black nonexempt hiring.
The Bank argues that availability data must reflect the actual geographical area from which it draws its employees. Regardless of the validity of this approach in comparing and contrasting plaintiffs' and defendant's hiring showings, see section IX(F)(2), infra, this argument loses sight of the standard applicable to a prima facie showing. Given the nature of the nonexempt jobs at the Bank, the court cannot state that comparisons with City of Dallas figures are so bereft of probative value as to deprive them of meaning. Most nonexempt applications are obtained through "walk-ins" at the Bank's offices in the central city. The City of Dallas is an area from which the Bank can, and to a large extent does, recruit its nonexempt work force. Hence comparisons with City of Dallas figures are sufficient to raise an inference of discrimination warranting rebuttal by the defendant.
*360 The Z-statistics for the City of Dallas comparison show varying levels of significance for the years 1973-78. The only clearly significant underutilization is that for 1973.[179] Since data for pre-1973 exempt hiring was not presented, thus preventing the court from disaggregating nonexempt hiring from total hiring for 1969-72, the results for 1969-72 must be extrapolated. Since the figures for total black hiring, see n.162, supra, show a sharp drop-off in significance levels between 1969-73 and 1974-78, consistent with the disparity between the 1973 and 1974-78 figures for black nonexempt hiring, we may infer that results for 1969-72, if available, would show significant underutilization as in 1973. That is, the significant 1973 result, together with the pattern of disparities shown by the total black hiring figures, raises the inference that the discrimination present in 1973 existed in 1969-72 as well. Hence a prima facie case has been made out for those members of the subclass of black nonexempt applicants who applied in 1969-73.
The Bank argues that the comparisons performed by Dr. England, constituting as they do "stock" comparisons, i. e., comparisons involving the Bank's labor force at a given time, rather than "flow" comparisons involving those hired during a given period, do not adequately segregate the effects of pre-Act discrimination from discrimination occurring after Title VII became effective. While without question flow statistics are a superior measure of hiring discrimination during a given time period, Movement for Opportunity & Equality v. General Motors Corp., 622 F.2d 1235, 1244-45 (7th Cir. 1980) [Movement v. GMC]; D. Baldus & J. Cole, supra n.55, ง 3.112[2]; Gwartney, Asher, Haworth, & Haworth, supra n.40, at 651-53, it does not follow that flow statistics are required to establish a prima facie case, Movement v. GMC, supra, at 1244-45. The evidence indicates that there is very high turnover in nonexempt positions in the Bank. Hence the effects of pre-Act hiring discrimination "frozen" into the Bank's work force or "stock" statistics are likely to be small.
Black applicants for exempt positions do not fare as well as nonexempt applicants. As previously stated, comparisons of exempt hiring with general labor force figures, as in n.162, supra, are virtually meaningless, and cannot be relied upon to establish a prima facie case. The only other statistical evidence relevant to black exempt hiring consists of comparisons between the Bank and other banks, see n.163, supra, and between black hires and interviews at the Bank and black applicant flow, see n.165, supra. Neither comparison establishes a prima facie case.
Dr. England's comparison between black officials and managers and professionals at the Bank and at banks filing EEO1 reports is flawed in several respects. First, data are presented for only three of the years covered by the class certification period. While this fact would not preclude gap-filling inferences of discrimination if the results for the years presented reflected a consistent discriminatory pattern, the results do not reflect such a pattern. No significance is revealed for 1969. While results for 1973 are significant, results for 1975 shows significance for officials and managers but not for professionals.[180] Since Dr. England does not state the exact levels of significance but only whether or not the results attain the .05 level of significance, the court is unable to determine whether the results are highly significant or only marginally significant. More fundamentally, *361 there has been no showing that the banks to which Republic is compared are comparable in size and structure to Republic.[181] Hence the use of the availability of blacks in the work force of EEO1 banks as a proxy for the availability of qualified blacks for Republic is a questionable assumption indeed.
Use of work force data from banks in the CEP study presents similar problems. For CEP banks, data for only two years are presented. Since the CEP study used racial minority data rather than data limited to blacks, the comparisons are made using minority proportions. The "officials and managers" and "professional, technical, and sales" groupings do not together correspond exactly to the "exempt" category. As with EEO1 data, the sample of CEP banks includes many banks in branch banking states with work force compositions different from Republic. Finally, the average minority proportion in the SMSAs from which the CEP banks draw is higher than that in the Dallas/Fort Worth SMSA.[182]
Balanced against these drawbacks, the plaintiffs argue that the exempt work forces of other banks represent extremely conservative estimates of availability of blacks for high-level banking positions, since other banks are likely to have discriminated against blacks as well. This may well be so. Indeed, the degree to which this factor favors the Bank may outweigh the drawbacks to use of interbank comparisons described above. Yet the presence of uncertainty and inaccuracy in each direction does not render the comparisons more reliable. In the absence of any evidence tending to show the cumulative effect of these factors, the court cannot state with any degree of reliability whether or not interbank comparisons are statistically meaningful. This being the case, the plaintiffs have failed to meet their burden to justify the use of such statistics.
The plaintiffs finally rely on a comparison between the proportions of minorities among those interviewed and hired and the corresponding proportion among applicants. These data fail to distinguish between those seeking exempt and nonexempt jobs, fail to account for requisite skill levels, do not differentiate between blacks and other minorities, and are based on incomplete applicant flow data. Hence they do not constitute an adequate prima facie showing.
Although comparison of female exempt hiring with general labor force figures is not appropriate, the plaintiffs supplied alternative comparisons which are adequate to establish, a prima facie case as to that group. These data, given in n.164, supra, are the results of a comparison between female exempt hires at the Bank (flow statistics) and the proportion of females among those receiving bachelors' degrees from American universities. Statistically significant disparities appear for each year from 1971 through 1978. The unanimity of the results also suggests that a prima facie case exists as to the years 1969 and 1970. While the Bank correctly points out that many of its exempt positions require skills not possessed by a typical liberal arts graduate, the skills possessed on the average by those with college degrees are sufficiently commensurate with the skill mix of exempt jobs at the Bank to render comparisons meaningful.
*362 In summary, the plaintiffs have established a prima facie case of hiring discrimination as to female applicants for exempt positions and as to black applicants for nonexempt positions prior to 1974. The court turns now to an examination of the extent to which the Bank has successfully rebutted this prima facie showing.

E. The Bank's Case: The Data.

The Bank's hiring case consisted primarily of the testimony and reports of Dr. Judith Stoikov.[183] Like Dr. England, Dr. Stoikov performed a series of statistical comparisons to detect the presence or absence of discrimination in the Bank's hiring. Dr. Stoikov's methodology, however, differed from that of Dr. England in several important respects. First, Dr. Stoikov performed her analyses using hiring data segmented into so-called "job families," representing groups of jobs at the Bank with similar content, compensation, and opportunity for advancement, while Dr. England's tests were performed using the entire Bank population or large, broadly-defined subgroups of that population. Second, Dr. Stoikov used availability estimates consisting of weighted averages reflecting the various educational levels and types of experience sought by the Bank when hiring for particular positions.[184] Her availability estimates were also weighted geographically to reflect the various geographical areas from which the Bank draws its employees.
The grouping of the Bank's 3500 job titles into job families was performed not by Dr. Stoikov, but by an EEO task force at the Bank. This task force consisted of four Bank officers at differing levels in the Bank's personnel administration, together with the Bank's lead counsel in this lawsuit. The task force used as its starting point the draft Compliance Manual of the Office of Federal Contract Compliance Programs. That manual identifies four factors to be used in assigning job titles to job groups: functions, pay, advancement, and sufficient number of incumbents to support meaningful statistical analyses. Apart from broad guidelines, such as the requirement that individual job families not include employees from different one-digit EEOC job categories (e. g., officials and managers, professionals, etc.), the manual does not provide detailed criteria for classifying particular jobs. This function was performed by the task force.
The results of this initial classification were evaluated and approved by the Bank's attorneys. Questionnaires were submitted to employees, and the job family classifications audited on the basis of the responses to the questionnaires. The audited results were submitted to Dr. Stoikov for her approval. Dr. Stoikov used five criteria in assessing the job family classifications: Are the geographical dimensions of the labor market identical for all jobs within a three-digit job family? Is the level of skill required homogeneous for jobs within a three-digit job family? Is the range of salaries smaller within than across three-digit job families? Are opportunities for career advancement similar for all jobs classified within the same job family? Are there sufficiently large numbers of people employed in each job family to support meaningful statistical analysis? After combination of some three-digit job families to obtain sufficient numbers of observations to support meaningful statistical analysis,[185]*363 *364 Dr. Stoikov found the Bank's job family classifications to be "valid and useful constructs for analyzing equal employment practices."
For each job family, Dr. Stoikov identified the proportion of employees who were hired based on their prior work experience and the proportion who were hired based upon their educational achievement. Job families were matched with Census Bureau occupational categories and levels and types of education, and a weighted average availability was computed based upon the proportions of hires bearing each such qualification and the availabilities of blacks or females possessing qualification. These results were further weighted for each family by the proportion of hires selected from *365 each of three labor markets: the Dallas/Fort Worth SMSA; the State of Texas; and the entire United States. Occupational availabilities were taken from the 1974 Current Population Survey for 1970-74 comparisons, and from the 1978 CPS for 1975-78 comparisons. Educational availability data were taken from 1973-74 data for Earned Degrees Conferred supplied by the Department of Health, Education and Welfare for 1970-74 comparisons, and 1975-76 data for 1975-78 comparisons.
The weighted average availabilities obtained were then compared for each job family with the proportion of females or blacks hired into that job family. Data were aggregated into two time periods (1970-74 and 1975-78) to obtain sufficient numbers for meaningful statistical analysis. For each such comparison, a "randomness range" was computed. This range represents the range of the number of female or black hires which could be expected to occur, in the absence of discrimination, 95% of the time.[186] An actual number of hires outside the randomness range thus indicates an underutilization or overutilization of blacks or females statistically significant at the 5% level using a two-tailed test.
The results of these statistical tests are shown in the margin.[187] The results show statistically significant underutilization of *366 *367 females in a few job families, and statistically significant overutilization of females or blacks in a few job families.
To meet a possible accusation of over-segmentation of the Bank's work force, Dr. Stoikov performed additional studies using *368 aggregations of job families.[188] Availabilities for these groups of job families were computed using weighted averages of individual job family availabilities. The results of comparisons using these aggregated job groups appear in the margin.[189] These results *369 show a few instances of statistically significant underutilization and several instances of statistically significant overutilization.

F. The Bank's Case: The Legal Standards.

In the rebuttal phase, it is the defendant's burden to show that the apparently discriminatory results shown in the plaintiffs' prima facie case are in fact explainable by non-discriminatory factors. In the present case, the Bank proffers its desire to hire qualified individuals as a non-discriminatory explanation for the racial and sexual disparities shown to exist in its workforce. This explanation is acceptable โ that is, the explanation is truly "nondiscriminatory" โ if and only if the Bank can show its hiring standards to be job-related. See generally sections VI(B) and (E), supra. Since these hiring standards are transformed in the trial setting into occupational, educational, and geographical weights, the court must determine the type and quality of the proof which must be presented to show job-relatedness in this context.[190]

1. Occupational and Educational Weights.

The bulk of the cases which have addressed the requirement that hiring criteria be shown to be job-related have arisen in the context of a specific and usually inflexible requirement that an applicant pass a particular test or possess a particular degree or other objective measure of qualification. E. g., Griggs v. Duke Power Co., supra n. 38; Scott v. City of Anniston, 597 F.2d 897 (5th Cir. 1979), cert. denied, 446 U.S. 917, 100 S.Ct. 1850, 64 L.Ed.2d 271 (1980); United States v. Georgia Power Co., supra. In such circumstances, the courts have required that an employer prove the job-relatedness[191] of the test, degree, or other requirement by means of evidence possessing high accuracy and reliability. The EEOC has developed guidelines for the validation of such requirements, which enable an employer to prove the validity of a requirement in any of three ways. See 29 C.F.R. งง 1607.1 et seq. First, an employer may demonstrate "criterion validity" by producing "empirical data demonstrating that the selection procedure is predictive of or significantly correlated with important elements of job performance," 29 C.F.R. ง 1607.5(B); see 29 C.F.R. ง 1607.14(B). Second, an employer may show "content validity" through presentation of "data showing that the content of the selection procedure is representative of important aspects of performance on the job for which the candidates are to be evaluated," 29 C.F.R. ง 1607.5(B); see 29 C.F.R. ง 1607.14(C). Finally, the employer may prove "construct validity" by means of "data showing that the procedure measures the degree to which candidates have identifiable characteristics which have been determined to be important in successful performance in the job for which the candidates are to be evaluated," 29 C.F.R. ง 1607.5(B); see 29 C.F.R. ง 1607.14(D).
These guidelines include a myriad of technical requirements, see 29 C.F.R. ง 1607.14, and performance of a study meeting these requirements may be enormously costly. See Gwartney, Asher, Haworth, & Haworth, supra n.40, at 643 (validation study for arithmetic test for machinists may cost $20,000-$100,000). Hence it is not surprising that the guidelines, although entitled to deference as an administrative interpretation of Title VII by its enforcing agency, Griggs v. Duke Power Co., supra n.38, 401 U.S., at 433-34, 91 S.Ct. at 854-855, *370 have not been regarded by the courts as absolute requirements for a showing of job-relatedness. In United States v. Georgia Power Co., supra, the Fifth Circuit stated:
We view the reference to the Griggs court to EEOC guidelines as an adjunct to the ultimate conclusion that such tests must be demonstrated to be job related. We do not read Griggs as requiring compliance by every employer with each technical form of validation procedure set out in 29 C.F.R., Part 1607. Nevertheless, these guidelines undeniably provide a valid framework for determining whether a validation study manifests that a particular test predicts reasonable job suitability. Their guidance value is such that we hold they should be followed absent a showing that some cogent reason exists for noncompliance.
474 F.2d at 913. Thus, for example, courts have held that in specific cases validation need not be by means of empirical evidence, Davis v. Washington, 352 F.Supp. 187, 190-91 (D.D.C.1972), rev'd, 512 F.2d 956 (D.C. Cir.1975), rev'd, 426 U.S. 229, 96 S.Ct. 2040, 48 L.Ed.2d 597 (1976); see Washington v. Davis, 426 U.S. 229, 249-52 & n.17, 96 S.Ct. 2040, 2052 & n.17, 48 L.Ed.2d 597 (1976) (expert opinion of personnel research psychologist, relying in part on empirical evidence, sufficient); cf. United States v. Jacksonville Terminal Co., 451 F.2d 418, 456 (5th Cir. 1971), cert. denied, 406 U.S. 906, 92 S.Ct. 1607, 31 L.Ed.2d 815 (1972) (validation "most often" by "positive empirical evidence"), and that it need not be performed by a "professional," Broussard v. Schlumberger Well Services, 315 F.Supp. 506, 512 (S.D.Tex.1970). See also Washington v. Davis, supra, 426 U.S. at 247 n.13, 96 S.Ct. at 2051 n.13 ("It appears beyond doubt by now that there is no single method for appropriately validating employment tests for their relationship to job performance.") The reason for this relaxation was recently articulated by the Second Circuit: "The danger of too rigid an application of technical testing principles is that tests for all but the most mundane tasks would lack sufficient validity to permit their use." Guardians Association of New York City Police Department, Inc. v. Civil Service Commission, 630 F.2d 79, at 90, 23 Empl.Prac.Dec. ถ 31,154, at 16,975 (2d Cir. 1980).
When the factual focus shifts from pencil-and-paper tests inflexibly required for low-level jobs to educational and experiential credentials used as flexible indicia of competence for skilled positions, the legal focus must shift as well:
Case law fashioned to deal with the problems of providing equal employment opportunity for employees who work with their hands rather than with people, paper, or ideas cannot be applied without alteration or adjustment to employment practices at the white collar and professional levels. The problems of selecting and evaluating workers whose success depends upon such intangibles as salesmanship or innovation necessarily are very different from the problems of selecting assembly line workers or craftsmen. They require different procedures and are deserving of a different standard of judicial evaluation.
Waintroob, supra n.69, at 46. Three differences between the heretofore typical case and cases such as the present compel a reexamination of job validation requirements: the greater difficulty of predicting competent performance in skilled positions, with a concomitant and unavoidable degree of subjectivity in the hiring process; the difference between arbitrary employer-created and employer-scored tests and the more objectively reliable measures of competence used here; and the willingness of the employer to rely on a variety of criteria rather than a single standard which must be met by all candidates for a given position.
The hiring of skilled individuals for managerial and professional positions is qualitatively different from the hiring of lower-level employees. "Usually, lower-level persons are hired to perform some discrete function, but upper-level persons often are hired for their intelligence, initiative, compatibility, experience or similar qualities, all of which resist measurement on an objective *371 scale." Hunt & Pazuniak, Special Problems in Litigating Upper Level Employment Discrimination Cases, 4 Del.J. Corp.L. 114, 124 (1978). Accord, e. g., Rogers v. International Paper Co., 510 F.2d 1340, 1345 (8th Cir.), vacated on other grounds, 423 U.S. 809, 96 S.Ct. 19, 46 L.Ed.2d 29 (1975) ("decisions about hiring and promotion in supervisory and managerial jobs cannot realistically be made using objective standards alone"); EEOC v. E.I. du Pont de Nemours & Co., 445 F.Supp. 223, 254 (D.Del.1978) (not "feasible to eliminate subjective criteria from the selection process" for upper-level positions); Frink v. United States Navy, 16 Fair Empl.Prac.Cas. 67, 69-70 (E.D.Pa.1977) ("subjective and technical factors necessarily must be considered" for promotion as naval architect), aff'd mem., 609 F.2d 501 (3rd Cir. 1979), cert. denied, 445 U.S. 930, 100 S.Ct. 1319, 63 L.Ed.2d 763 (1980); Kohn v. Royall, Koegel & Wells, 59 F.R.D. 515, 521 (S.D.N.Y.1973), appeal dismissed, 496 F.2d 1094 (2d Cir. 1974) ("hiring a professional requires weighing many subjective factors contributing to the applicant's qualifications as a whole, above and beyond the more objective academic qualifications").
The difficulty of quantifying and measuring the intangible skills which managerial and professional employees must bring to their jobs makes validation of a subjective decisionmaking process difficult. Gwartney, Asher, Haworth, & Haworth, supra n.40, at 641-42; Waintroob, supra n.69, at 67.[192] The EEOC guidelines recognize that the most common type of validation study โ a content validity study โ is not feasible in such a situation.[193] They also acknowledge that circumstances exist in which no validation (in the technical sense) is possible.[194] The difficulty of devising mathematical tests for subjective qualifications such as loyalty, ability, reliability, and aptitude suggests that formal validation of such measures is a virtual impossibility. See Hunt & Pazuniak, supra, at 123-31.
That the decision to hire a managerial or professional employee is necessarily subjective does not mean that it cannot or need not be based at least in large part on objective criteria. These criteria may include such factors as education, experience, *372 length of service, and possession of specific skillsโcriteria which though objectively measurable are weighed subjectively by the employer in making a hiring decision. The use of a subjective process which does not rely on objective (and hence reviewable) factors is condemned under Title VII because of the possibility for discrimination which such a system creates. Rowe v. General Motors Corp., supra n.59, at 358-59. See No-Alternative Approach, supra n.37, at 111-12 (process must be both reasonable and reviewable). See generally Underwood, supra n.35 (tension between standardized criteria and subjective judgment in a variety of legal contexts). And, naturally, the use of a particular objective criterion must be justified on business grounds. See section VI(E), supra.
In the present case, the Bank has chosen to rely on two objective inputs into its subjective hiring process as those which exert the greatest influence on its hiring decisions: education, measured both by quantity and by field of study; and experience, measured primarily by an employee's occupation prior to coming to the Bank. This being the case, what showing must be made that these factors are job-related? In particular, must the Bank conduct a formal validation study to justify the hiring of bank managers as bank managers, MBAs as credit analysts, and so forth?
The EEOC guidelines purport to apply not only to formalized tests but to all other measures of employee qualification. 29 C.F.R. ง 1607.2(B). Nevertheless, several courts have suggested that a lesser quality of proof is sufficient to establish the job-relatedness of college degrees for managerial, professional, and other white-collar positions. See Spurlock v. United Airlines, Inc., supra, at 219 (testimony of company officials adequate to establish job-relatedness of college degree requirement for airline pilots); Castro v. Beecher, 459 F.2d 725, 735 (1st Cir. 1972) (study by Presidential Commission shows job-relatedness of high school diploma requirement for police officers); Scott v. University of Delaware, 455 F.Supp. 1102, 1124-26 n.64 (D.Del.1978), vacated on other grounds, 601 F.2d 76 (3d Cir.), cert. denied, 444 U.S. 931, 100 S.Ct. 275, 62 L.Ed.2d 189 (1979) (Ph.D. requirement for university professors held job-related despite "surprisingly sparse" evidence); League of United Latin American Citizens v. City of Santa Ana, 410 F.Supp. 873, 900-02 (C.D.Cal.1976) (high school diploma requirement for police officers but not for firefighters; extent of evidence not fully stated). See generally Hunt & Pazuniak, supra at 134-44.[195]
One reason for this apparent willingness to relax stringent validation requirements may lie in the lesser potential for abuse of academic credential requirements compared to employment tests. Most such tests are employer-scored; many are employer-created. Usually no objective outsider has determined whether the tests accurately measure any particular skill, much less skills in the position for which the tests are being administered. In such a situation, the potential for deliberate use of the tests as racial or sexual screening devices is high. Even if no overtly discriminatory motive is present, the use of the tests by an employer not qualified to judge their relevance creates the risk that minorities or females may be needlessly excluded from positions for which they are actually qualified. In contrast, a college degree reflects the judgment of a comparatively neutral institution, better qualified and with greater opportunity than the employer to assess an individual's credentials, that the recipient has *373 reached a threshold level of competence in a given field. The risk still remains that an employer will erroneously require a greater degree of knowledge and skill than is appropriate; more objective measurement of competence nevertheless suggests that the need for formal validation is diminished.[196]
Finally, the present case differs from the traditional validation situation in that no one credential is inflexibly required for any position. Ample testimony establishes that the Bank has no formal degree requirements for any of its jobs. While in many job families college degrees are desired, including in some cases specific degrees in specific fields, Bank officials testified that each applicant is evaluated on his or her own merits. Thus, for example, while 51% of new credit analysts have MBA degrees, another 29% have only a business-related bachelor's degree. See n.212, infra. Some MBA applicants are regarded as unqualified, by virtue of poor academic performance or other deficiencies, while some applicants with business-related bachelor's degrees (or other educational backgrounds) are perceived as having strong points which compensate for their lack of training at the master's level. Similarly, some credit analysts are hired for their experience in banking or other fields in lieu of recent formal education.[197]
The Bank's hiring practices have been used to develop occupational and educational weights, which correspond to the proportions of individuals hired into each job family who possess given academic or experiential credentials. These weights proxy for the importance which the Bank attaches to a given qualification for a given job family.[198] Reliance on a single qualification as the sole reason a particular employee was hired is admittedly an over-simplification: many employees were hired for a variety of reasons and based on a variety of qualifications. The weighted average approach does, however, provide a rough measure of the employment criteria used by the Bank for each job family.
In this context, a showing of job-relatedness takes the form of a demonstration that the pool of qualified individuals possesses the same mix of occupational and educational credentials as the employer's hires โ that is, that the weights used by the employer are correct.[199] Thus, if an employer hires into a job family only those with a given degree or type of experience (i. e., a 100% weight), he must show that of those qualified for that job family, 100% have that credential, i. e., that the credential is essential to render an individual qualified. E. g., Griggs v. Duke Power Co., supra n.38 (high school diploma requirement).[200] If *374 60% of the hires in a job family possess a given credential, the defendant need only show that 60% of those qualified for that job family have that credential, the lack of which can be overcome by one who possesses skills which comprise the other 40% of the weighting.[201] And if education or experience of a particular type accounts for only 1% of the weighting, the employer need only show that an occasional qualified person possesses that type and level of education or experience, and that the bulk of the qualified pool possess other qualifications.[202] Throughout this analysis, the touchstone remains a showing that the defendant does what a nondiscriminatory employer would do, i. e., that he hires from particular groups in proportion to the extent those groups make up the pool of qualified individuals.[203] Proof that this is so cannot and *375 need not be accomplished through formal validation studies. Nevertheless, the Bank continues to bear the burden of showing that its educational and occupational preferences "have a manifest relationship to the employment in question," and that diplomas (and work histories) are indeed the "useful servants" the Supreme Court envisioned they could be. Cf. Griggs v. Duke Power Co., supra n.38, 401 U.S. at 432-33, 91 S.Ct. at 854.

2. Geographical Weights.

The choice of an appropriate geographical labor market has caused considerable difficulty for the courts. A variety of approaches to the problem has been employed.[204] While some courts have blindly accepted the use of the area in which an employer recruits, e. g., Detroit Police Officers Ass'n v. Young, 446 F.Supp. 979 (E.D. Mich.1978), rev'd on other grounds, 608 F.2d 671 (6th Cir. 1979), this practice is fraught with possibilities for undetected discrimination. For example, an employer located in or near a predominantly white suburb could recruit from that area even though economic factors such as availability of qualified employees and ease of commuting might dictate that a nondiscriminatory employer would recruit from a broader area. See Mays v. Motorola, Inc., 18 Empl.Prac.Dec. ถ 8,903 (N.D.Ill.1979) (such a case); Gastwirth & Haber, supra n.204, at 34. Alternatively, an employer might recruit nationally despite an abundance of qualified applicants in an area closer to home with greater than average minority representation.
Courts which have gone beyond the area-where-the-employer-recruits standard have adopted a variety of formulations for determining the appropriate geographical labor market. See, e. g., United States v. Ironworkers Local 86, supra n.204, at 551 n.19 (area "from which [employer] would most likely draw vast majority of workers"); Johnson v. Goodyear Tire & Rubber Co., 491 F.2d 1364, 1371 (5th Cir. 1974) (statistics must reflect mobility of labor force); Greenspan v. Automobile Club of Michigan, supra n.204, 495 F.Supp. at 1028, 22 Empl.Prac.Dec. at 15,166 (area "must bear a reasonable relationship to the scope of the employer's business and the area from which it draws applications and employees"). These standards represent variations on a common theme: the touchstone with geographical weighting, as with occupational and educational weighting, is the weighting which a nondiscriminatory employer would use. As the Supreme Court has recognized, this is a factual determination which must be made on a case-by-case basis. Hazelwood School District v. United States, supra n.99, 433 U.S. at 310-12, 97 S.Ct. at 2743. The relevant market will also vary according to the type of job in question. Where the court finds that geographical *376 hiring practices, having a disparate impact, are not justified by economic factors, it must require the use of the appropriate labor market statistics even if this means choosing an area other than that from which the defendant actually hires. League of United Latin American Citizens v. City of Santa Ana, supra, at 897.[205]

G. The Bank's Case: Applying the Law to the Data.

Analysis of the Bank's rebuttal effort must begin with an evaluation of the validity of the job family approach at the core of that effort. Although the plaintiffs raise numerous challenges to the use of job families, none can overcome the utility of the concept in the identification of requisite skill levels among a vast population of job titles, and the resulting facilitation of detailed statistical analysis. The theory of the use of job families cannot seriously be questioned. As the Office of Federal Contract Compliance Programs ("OFCCP") has recognized, a grouping of jobs by content, compensation, and opportunity for advancement unquestionably facilitates statistical analysis of employment discrimination. The utility of the job family approach has been recognized by other courts. Movement v. GMC, supra, at 1266; cf. Croker v. Boeing Co., 437 F.Supp. 1138, 1156 (E.D.Pa. 1977) (use of census standard occupational classifications). Perhaps recognizing the theoretical validity of the job family approach, the plaintiffs' objections to the use of job families focus on the way in which that concept was put into practice at the Bank.
While job titles were matched to job families by an EEO task force at the Bank, many of whom had no experience as personnel administrators outside the Bank, this function was not accomplished without objective checks on its accuracy. The guidelines of the OFCCP were followed, the results audited on the basis of a questionnaire survey of current Bank employees, and the final product evaluated by a labor economist from outside the Bank. Although the job family concept was admittedly developed with this litigation in mind, the Bank's expressed intention, already partially fulfilled, to utilize the job families as an integral part of its ongoing personnel administration mitigates incentives to distort results to aid the litigation effort. More importantly, when the focus is turned from incentives for distortion to actual evidence of distortion, the plaintiffs' criticisms founder. An intuitive check of the match between job titles and job families, presented in Defendant's Exhibit 402, confirms the reasonableness of the classifications. While some job families at the Bank contain (contrary to Dr. Stoikov's assertions) too few incumbents to support meaningful statistical analysis, this problem can be solved through aggregation of job families where necessary, and the Bank's statistical presentation recognizes this fact.
We turn now from job families to the statistical results obtained when such job families are used. The results of Dr. Stoikov's analysis for black nonexempt hiring appear in n.187, supra. The results for the 1970-74 time period show no statistically significant underutilization of blacks, and several families in which significant overutilization is present. These data are, however, subject to an important statistical flaw. In many job families, zero black hires falls within the randomness range, i. e., the range of the number of black hires in which no statistical significance at the 5% level is present. In such a situation, the *377 statistical test tells us nothing: even were discrimination rampant, resulting in no black hires whatsoever, the test would show no significance at the .05 level. Where zero hires is part of the randomness range, statistical significance can never be present. This fact indicates that job families in which this phenomenon is present contain, at least for present purposes, too few members (or in this case, hires) for meaningful statistical analyses.
This problem is largely eliminated when the aggregated data of n.189, supra, are used. In each of the aggregated job family groups, the number of black hires is nonzero, and in most such groups, the randomness range does not include zero. These results show statistically significant overhiring of blacks into all nonexempt aggregated job family groups. Hence the prima facie case of discrimination in black nonexempt hiring has been successfully rebutted if the assumptions underlying the availability figures are logically and legally valid.
The availability figures for blacks in nonexempt job families, as with all of Dr. Stoikov's availability figures, are weighted averages of the availabilities of blacks in various occupational groups in various geographical markets, and at various educational levels. The weights utilized by Dr. Stoikov for nonexempt job families are shown in the margin.[206] With exceptions in the computer technician job families, virtually all nonexempt availabilities were based on availabilities of blacks in specific occupational categories in the Dallas/Fort Worth SMSA.
The flaw in the weightings used for nonexempt hiring is that the Bank presented *378 no evidence to demonstrate the job-relatedness of the weights as hiring criteria. Thus, for example, there is no evidence that experience as a "computer and peripheral equipment operator" is, as the Bank's hiring record would suggest, a virtual prerequisite for the job of computer operator (job family OO).[207] While the court may have some intuitive idea that prior experience as a computer operator is helpful in performing that job, such intuition does not extend to validation of a practice of near-complete refusal to hire one who does not possess such experience. Intuition similarly suggests that a degree in computer science would make one a better computer operator, but that fact alone, absent a showing that such a degree is required in substantially all cases for such a job, would not justify a practice of recruiting computer operators almost completely from those with such degrees. This problem is exacerbated for job families (e. g., "production coordinator," job family PP) whose titles give little clue to their content. Even job family titles such as "general clerk" may be misleading: the content and levels of skill in such positions may be different from those which the title would imply.
In the clerical and service worker job families, the use of 100% weight for incumbency in various occupational categories has not been justified. For example, the Bank has not shown that only those in the labor market presently holding receptionist or typist jobs are qualified for clerk/typist positions (job families WW and XX), although a 100% weight has been assigned in those job families to those so employed, or that only those who are now working as protective service workers are qualified to be security guards (job family BBB) at Republic, although a 100% weight is employed there as well.[208] While the proffered availability pools for some job families (junior computer operator, job family NN, and lower-level general clerk, job family UU) include the inexperienced unemployed, there has been no showing that that group does not possess and cannot easily acquire the skills necessary to perform in at least some of the 13 other nonexempt job families. Nor has the Bank demonstrated that incumbents of one occupational category (e. g., bookkeepers) cannot readily perform or acquire the skills to perform in a different but related and roughly comparable job family (e. g., bank tellers).[209] In sum, the Bank has not shown that its proffered occupational and educational weights are accurate reflections of job-related skill requirements in nonexempt job families.
The Bank's failure to show the job-relatedness of its occupational and educational weights is alone sufficient to warrant rejection of its rebuttal evidence as to black nonexempt hiring. Added to this inadequacy, however, is the questionable use of the Dallas/Fort Worth SMSA as the relevant labor market for most nonexempt hiring. While it is unquestionable that most of the Bank's nonexempt labor force comes from *379 within the D/FW SMSA, use of SMSA statistics may significantly underestimate the true availability of blacks for nonexempt jobs. See n.205, supra. As previously noted, most nonexempt applicants are "walkins" at the Bank's headquarters in the central city. Economic literature suggests that workers in lower-level jobs are more likely to live near their workplace than are their higher-paid counterparts. See, e. g., Gastwirth & Haber, supra n.204, at 33 & n.7. Absent a showing that clerical workers are in fact recruited from throughout the Dallas/Fort Worth area, use of SMSA rather than city statistics is intuitively suspect.
Turning to Dr. Stoikov's results for female exempt hiring, the figures shown in n.187, supra, suffer from the same statistical drawback as those which relate to blacks: in most cases, zero hires is part of the randomness range, and in several job families the actual number of female hires is also zero. Use of the partially aggregated figures of n.189, supra, largely eliminates this problem. The results show statistically significant underhiring of females as officials and managers, and, for 1970-74, as operating services personnel and petroleum engineers. Significant overhiring is present in 1975-78 for internal services personnel. The balance of the results are mixed: during 1970-74, most job family groups show underhiring which is not significant at the .05 level (i. e., actual hires are within the randomness range, but below the midpoint of that range), while 1975-78 figures generally show nonsignificant overhiring (actual hires within the range but above the midpoint).
The results in this form are hard to interpret. While only scattered instances of statistical significance are present, this may be due in no small part to the "divide-and-conquer" aspect inherent in the fragmentation of the workforce into job families or even into broad job family groups.[210] It is fortunately possible to eliminate this potentially biasing factor while retaining the flexibility in determining availabilities which the job family approach provides. Dr. Stoikov aggregated job families by weighting the availabilities in n.187, supra, by the number of hires into the corresponding job families, to obtain the job family group availabilities in n.189, supra. A further aggregation of the group availabilities into a single weighted average availability figure for female exempt hires, using the same procedure, enables the court to evaluate the Bank's hiring performance across the exempt group as a whole.
The results of such an aggregation are shown in the margin.[211] The results for 1975-78 show statistically significant overhiring of females, while the 1970-74 figures show underhiring at the 6% level of significance (Z = -1.88). The effect of these results on the Bank's liability for female *380 exempt hiring turns on the extent to which the occupational, educational, and geographical weights used in the availability calculations have been shown to be job-related.[212]*381
*382 The principal testimony in this regard was presented by Thomas Hofstedt, Ph.D.[213] Dr. Hofstedt undertook a comprehensive review of the Bank's hiring policies and practices. This review consisted of a study of the formal systems of organization, job descriptions, *383 and other documentation used by the Bank, followed by interviews with 15-18 Bank employees and a review of summary hiring data. Interviews were also conducted with the Bank's senior management and with a number of graduates of the Bank's management training programs. Dr. Hofstedt then related his findings at Republic with his experience involving other large commercial banks and the commercial departments of large retail banks.
Dr. Hofstedt found Republic to be remarkably similar to other large commercial banks. In particular, he found the qualifications required by Republic for most exempt positions to be similar to those required by other banks. He discussed at length the mix of education and experience required for a variety of job families,[214] and *384 concluded that Republic's hiring policy and practice revealed "reasonable policies consistently applied." In his opinion, the Bank's qualifications for exempt jobs are realistic, not unreasonably high, not overly rigid in adherence to a particular degree requirement, tailored to a reasonable expectancy of satisfactory job performance, and sensitive to exceptional cases. Finally and most importantly, Dr. Hofstedt expressed his opinion that Republic's qualifications were necessary for the efficient conduct of its business, and that this was true across all exempt job families.
Although Dr. Hofstedt testified in some detail about the qualifications necessary for a variety of positions in banking, his principal testimonial focus, coinciding with his area of greatest expertise, was on requirements for the Bank's credit analysis program. He testified that commercial loan officers at a major bank require a mix of general training, specialized financial training, social skills, and mental orientation, of the type provided in a business-related undergraduate or graduate education. He testified that loan officers must deal on a day-to-day basis with corporate treasurers who have financial training, and that a large commercial bank's competitive edge lies in its loan officers' familiarity with national and international financial markets and their ability to construct intricate and innovative financial arrangements. He found Republic's requirement of productive work during the credit training program to be unusual, and estimated that it would require 1 to 1ฝ years of business-related education, exclusive of productive time, to train a liberal arts graduate in the technical skills necessary to be a loan officer. Dr. Hofstedt found a knowledge of microeconomics, finance, law, and taxation to be necessary to develop skills and vocabulary for the productive aspects of the credit training program.[215]
The impressive testimony of Dr. Hofstedt convinces the court that the occupational and educational weights used by Dr. Stoikov in her analysis of exempt hiring are job-related.[216] In particular, the Bank's heavy reliance on business-related experience or education in such fields as credit analysis and accounting has been shown to be justified in light of the demands of the jobs in those fields. Although Dr. Hofstedt did not testify specifically about the Bank's geographical recruiting patterns, no such testimony was needed: the geographical weights used for exempt positions are reasonable given the economic realities of recruitment for skilled professional positions, which were discussed in the testimony of several Bank officials.
The showing of job-relatedness renders the positive Z-score (+ 1.96) for 1975-78 female exempt hiring dispositive of liability for those years. The -1.88 figure for 1970-74 is more difficult to interpret. That result corresponds to a significance level of 6%, falling short of the customary 5% level. It must be remembered, however, that this result is obtained using the Bank's own availability figures. Any residual uncertainty in the degree to which the Bank's hiring criteria have been job-correlated would operate to bring this figure closer to the traditional 5% level. Indeed, the overall Z-score is highly sensitive to small changes *385 in the overall availability figure. Thus, for example, if the court were for some reason to reject the Bank's weighted average availability for, e. g., petroleum engineers, (.0147) on the ground of an insufficient showing of job-relatedness, and to use instead the .388 availability of females in the D/FW SMSA, the overall Z-score would jump from -1.88 to -2.00, and 5% significance would be present. Mindful that use of the 5% significance level is a custom rather than a rigid rule to be mechanically applied without regard to the circumstances, see section IX(A), supra, the court is convinced that the sensitivity of the availability figures to minor flaws in job-correlation renders the result for 1970-74 statistically significant.

H. Summary.

In summary, the court finds that plaintiffs have established a prima facie case of hiring discrimination for black applicants for nonexempt positions during the period from 1969 through 1973 and for female applicants for exempt positions during the entire class period. That prima facie case was rebutted as to female exempt applicants in 1975 and later years. A Phase I finding of liability therefore exists for unsuccessful black applicants for nonexempt positions on or before December 31, 1973, and for unsuccessful female applicants for exempt positions on or before December 31, 1974.

X. Terminations.

The plaintiffs' statistical presentation on the issue of terminations consists of a table presented as part of the report of Dr. David Morgan. That table presents the numbers and proportions of blacks, females, and other groups in the Bank work force, juxtaposed with numbers and proportions of those groups terminated in general and for specific reasons. No calculations were presented to show the statistical significance of these results. Relevant portions of these data appear in the margin.[217]
Dr. Morgan's data suffer from a fundamental flaw. No effort has been made to segregate involuntary terminations from voluntary terminations. As noted in Markey v. Tenneco Oil Co., 439 F.Supp. 219, 243 (E.D.La.1977), an employer "can only be held responsible for the employees that it discharged, not for those who left [the employer] of their own accord." While in an appropriate case a plaintiff might be able to show that the employer's other discriminatory conduct made life so intolerable for the group discriminated against as to amount to constructive discharge, no such showing has been made or attempted here. The standards applicable to proof of constructive discharge are rigorous, see Miller v. Texas State Board of Barber Examiners, 615 F.2d 650, 652 (5th Cir.), cert. denied, ___ U.S. ___, 101 S.Ct. 249, 66 L.Ed.2d 117 (1980); Young v. Southwestern Savings & Loan Association, 509 F.2d 140, 144 (5th Cir. 1975), and neither proof of wage discrimination, Bourque v. Powell Electric Manufacturing Co., 617 F.2d 61 (5th Cir. 1980); Cullari v. East-West Gateway Coordinating *386 Council, 457 F.Supp. 335, 341 (E.D.Mo.1978), nor proof of promotion discrimination, Muller v. United States Steel Corp., 509 F.2d 923, 929 (10th Cir.), cert. denied, 423 U.S. 825, 96 S.Ct. 39, 46 L.Ed.2d 41 (1975), is alone sufficient.
In her post-trial brief, plaintiff Johnson presents a summary of 1969-72 involuntary termination data for black employees. Defendant urges that consideration of this summary by the court would be tantamount to the reception of new evidence after trial. This court considered and rejected a similar contention in L. C. L. Theatres v. Columbia Pictures Industries, Inc., 421 F.Supp. 1090, 1103 & n.9 (N.D.Tex.1976), rev'd in part on other grounds, 566 F.2d 494 (5th Cir. 1978). The summary in question contains nothing more than termination data contained in a computer print-out (Plaintiffs' Exhibit 121) introduced into evidence, together with numbers and percentages of blacks in the Bank's work force, derivable (albeit with minor discrepancies) from the Bank's EEO1 reports (Plaintiffs' Exhibits 511 et seq.). The assembly of the data in summary form is so straightforward a procedure that exposure to the adversary process would have shed little additional light here.
Although properly before the court, this summary does not satisfy the plaintiffs' prima facie burden on the termination issue. As with Dr. Morgan's summary, no calculations of statistical significance are presented.[218] While the Supreme Court in Hazel-wood School District v. United States, 433 U.S. 299, 311 n.17, 97 S.Ct. 2736, 2743 n.17, 53 L.Ed.2d 768 (1977), avoided the suggestion "that precise calculations of statistical significance are necessary in employing statistical proof," the fact remains that a plaintiff must present at least some evidence that statistical disparities are not due to chance. D. Baldus & J. Cole, supra n.55, ง 1.231, at 48, see MacRae v. McCormick, 458 F.Supp. 970, 980-81 (D.D.C.1978) (evidence of disparity without any evidence of statistical significance "of little probative value"). Where the sample sizes are as small as in the plaintiffs' summary (11 to 21 blacks terminated per year out of 72 to 112 total involuntary terminations), the possibility of a chance result cannot easily be discounted absent calculation of Z-scores, randomness ranges, or other appropriate statistical measures of disparity. See D. Baldus & J. Cole, supra n.55, งง 9.1 & 9.12; cf. Mayor of Philadelphia v. Educational Equality League, 415 U.S. 605, 621, 94 S.Ct. 1323, 1333, 39 L.Ed.2d 630 (1974) (concern for smallness of sample presented by 13-member panel "well founded").[219]

XI. Terms and Conditions

In addition to their challenges to the Bank's hiring, initial placement, promotion, compensation, and termination of its employees, the plaintiffs raise a variety of challenges to what may loosely be termed the "terms and conditions" of their employment. These challenges relate to several specific practices not directly related to the employees' economic welfare. Each such challenge will be discussed in the sections which follow.

A. Departmental Segregation.

The plaintiffs assert that the Bank has followed a practice of relegating blacks *387 and females to certain undesirable departments, often inconspicuous to the public. In support of this allegation, they rely on several statistical "snapshots" of the Bank workforce, broken down by race, sex, and department,[220] together with anecdotal evidence of the distribution of blacks and females in various departments as viewed by individual witnesses as they walked through the Bank. The court finds the latter evidence to be too sporadic and incomplete to lend any support to the departmental segregation allegations.
At the outset, it is necessary to distinguish departmental segregation from other forms of alleged discrimination. The initial placement and promotion analyses are all concerned with vertical segregation: the assignment of females or blacks to lower-level jobs in a discriminatory fashion and discriminatory barriers to their upward movement. Departmental (i. e., horizontal) segregation relates to assignment of employees at given (vertical) job levels to departments on the basis of their race or sex. Since disproportionate pay is an indicator of vertical discrimination, the harm caused by departmental segregation is usually noneconomic in nature. See James v. Stockham Valves & Fittings Co., 559 F.2d 310, 333 (5th Cir. 1977), cert. denied, 434 U.S. 1034, 98 S.Ct. 767, 54 L.Ed.2d 781 (1978); Swint v. Pullman-Standard, 539 F.2d 77, 92-93 (5th Cir. 1976).
The various departmental snapshots relied upon by the plaintiffs are inadequate to establish a prima facie case of departmental segregation. These snapshots show disparities between the proportions of blacks and females in the Bank workforce and the corresponding proportions in various departments, but such disparities are not alone sufficient. Swint v. Pullman-Standard, supra, at 94-95 & n.40; Lee v. City of Richmond, 456 F.Supp. 756, 770 (E.D.Va. 1978). No calculations of statistical significance are provided. Thus the court cannot determine whether the disparities shown are due to segregation, to differences in job mixes among the departments, or to chance. Gross examination of departmental racial and sexual proportions reveals no obvious instances of gross disparity. In the most useful of the "snapshot" exhibits (Plaintiffs' Exhibit 638), departmental proportions of females range from 37.5% to 63.5%, compared to an overall proportion of 58.9%.[221] With the exception of two departments, *388 all departments contain blacks, with proportions ranging from 2.0% to 22.6% against an overall proportion of 14.2%. While there is some evidence of a concentration of blacks in the Finance & Administration Department, this is probably the product of a differing mix of skilled and unskilled jobs between that department and other departments, and does not establish that blacks of a given level of qualification are being shunted into that department. Absent a showing of historical segregation or any of the other factors identified in Swint v. Pullman-Standard, supra, as adjuncts to statistical evidence of departmental segregation, plaintiffs have failed to present a prima facie case of horizontal segregation.
The failure of plaintiffs to establish a prima facie case renders detailed discussion of the Bank's rebuttal effort unnecessary. Dr. Stolzenberg performed a regression study to measure the effects of race and sex on the quality of Bank employment as perceived by its employees. Eleven questions were asked of a sample of Bank employees, the responses coded on a scale of 1 to 4, and regressions performed using race, sex, and length of Bank employment[222] as independent variables and individual question response and aggregate satisfaction levels as the dependent variables.[223] Additional *389 regressions were run with controls for the proportion of blacks and females in the respondent's department.[224] While these results show that blacks more often perceive their work to be dull and their tasks to be too easy, they also show that blacks and females are more likely to believe themselves to have enough authority, that blacks are more likely than whites to find travel convenient, and that females are more likely than males to find their responsibilities to be clearly defined. No consistent pattern of racial or sexual dissatisfaction is shown. The overall coefficients for race and sex are positive, and the sex coefficient is statistically significant, indicating greater overall satisfaction (at least according to this measure) for females than for males.[225]

B. Maternity Leave.

The Bank's maternity leave policy has varied widely during the period covered by this lawsuit. Prior to 1971, regular full-time employees with three years' service were eligible (in their supervisors' discretion) for a six-month unpaid leave beginning at the end of the sixth month of pregnancy. Since 1979, all regular full-time employees may receive paid leave, with starting and ending dates flexible depending on the employee's physical condition as evaluated by her doctor. Between these years, a variety of hybrid policies has been in effect.[226]
*390 Although the plaintiffs' claims are not specific, it appears that the plaintiffs attack these policies in five respects:
(1) That maternity leave was without pay until 1979.
(2) That seniority does not accrue during maternity leave.
(3) That the grant of maternity leave was subject to length-of-service requirements until 1974.
(4) That maternity leave was required to begin at a predetermined stage of pregnancy until 1971.
(5) That maternity leave was fixed at a six-month duration until 1971.[227]
These challenges will be discussed seriatim.
The plaintiffs' challenge to the Bank's failure to pay sick pay to employees on maternity leave, while providing sick pay to those with other disabilities, is foreclosed by Nashville Gas Co. v. Satty, 434 U.S. 136, 143-45, 98 S.Ct. 347, 352-353, 54 L.Ed.2d 356 (1977). Relying on General Electric Co. v. Gilbert, 429 U.S. 125, 97 S.Ct. 401, 50 L.Ed.2d 343 (1976), the Court in Satty held that provision of sick pay for most disabilities, but not for pregnancy-related disabilities, did not violate Title VII absent a showing by the plaintiff that such a denial is a "pretex[t] designed to effect an invidious discrimination against the members of one sex." 434 U.S. at 144, 98 S.Ct. at 352, quoting from General Electric Co. v. Gilbert, supra, at 136, 97 S.Ct. at 408. No such showing has been made or attempted here. While Title VII was amended effective October 31, 1978, to effectively overrule Gilbert and Satty, see 42 U.S.C. ง 2000e(k), the Bank has since approximately that time provided sick pay for pregnancy on the same basis as other disabilities.
The Court in Satty distinguished the provision of economic benefits, such as sick pay and insurance coverage, as to which an employer is not required to treat pregnancy in the same manner as other disabilities, from the assessment of penalties or burdens, such as denial of accumulated seniority, as to which an employer must give equal treatment to pregnancy-related disability. Unlike the situation in Satty, however, Bank employees retain accumulated seniority when they take maternity leave. While it is true that additional seniority does not accrue during the actual period of leave, the plaintiffs have failed to show that this policy is different from that applied to males on sick leave (disparate treatment) or that it causes females to lose more seniority due to disability than males (disparate impact). See Harriss v. Pan American World Airways, Inc., 437 F.Supp. 413, 437 (N.D.Cal. 1977).
Until 1971, the Bank limited maternity leave to employees with three years' service. In 1971, this requirement was shortened to six months on advice of counsel. In 1974, the service requirement was eliminated altogether. Although there was no specific policy on non-pregnancy sick leave, beginning in 1971, a "guideline" limited sick leave to those with six months' service.
Denial of leave to those with little seniority, with a corresponding likelihood of termination due to pregnancy disability, is more closely akin to denial of accumulated seniority (a burden) than to denial of pay or insurance coverage (an economic benefit). See Harper v. Thiokol Chemical Corp., 619 F.2d 489, 491 (5th Cir. 1980) (same for unjustified *391 refusal to allow employee on maternity leave to return to work). The EEOC has ruled since 1970 that service requirements applicable to maternity leave but not to other types of leave violate Title VII. EEOC Decision No. 72-562, [1970] EEOC Dec. ถ 6184 (two-year requirement for maternity leave but not for other leave illegal in absence of legitimate business considerations justifying such a requirement); see EEOC Decision No. 72-1919, [1972] EEOC Dec. ถ 6370 (one-year service requirement, not applicable to other leaves, violative of Title VII by virtue of disparate impact); cf. EEOC Decision No. 74-68, [1973] EEOC Dec. ถ 6422 (same where no leaves granted to disabled employees with less than one year's service); 29 C.F.R. ง 1604.10(b). See generally Annot., 27 A.L.R. Fed. 537, ง 3[b] (1976).
The Bank has relied on two opinions of the EEOC which its counsel believed in 1970 authorized differing length-of-service requirements for maternity leave and other types of leave. On October 17, 1966, the General Counsel of the EEOC stated:
In a recent opinion letter regarding pregnancy, we have stated "The Commission's policy in this area does not seek to compare an employer's treatment of illness or injury with his treatment of maternity since maternity is a temporary disability unique to the female sex and more or less to be anticipated during the working life of most women employees."
General Electric Co. v. Gilbert, supra, at 142, 97 S.Ct. at 411. A similar statement in a November 15, 1966, opinion was relied upon by the Bank's counsel in a 1970 opinion letter upholding differential length-of-service requirements. See Plaintiffs' Exhibit 738. It must be noted, however, that both these EEOC opinions arose in the context of salary continuation-an economic benefit. Indeed, the latter opinion states. "[G]enerally speaking, a leave of absence should be granted for pregnancy whether or not it is granted for illness." (Emphasis supplied). While this statement goes beyond the position ultimately adopted by the Supreme Court in Satty, it does serve to distinguish the provision of leave from payment of salary during leave. Nor may the Bank find comfort in a February 20, 1967, opinion also relied upon by its counsel: that opinion holds only that "where it is an employer's policy to deny leave of absence to any employee with less than twelve months' service, it is not a violation of Title VII to deny maternity leave pursuant to this policy." See Plaintiffs' Exhibit 738 (emphasis supplied). While the Bank may have relied in all good faith on the opinion of its counsel (although the policy was changed in any event shortly after that opinion was issued), the opinion cannot insulate the Bank from liability for discrimination later found to have been unlawful.
It follows that Phase I liability has been established for all females denied maternity leave for failure to meet the three-year service requirement.[228] The presumption of discrimination created by this finding can of course be rebutted: in Phase II, the Bank may attempt to show the existence of a de facto service requirement applicable to other types of disability leave during the early years, thereby vitiating liability for differential treatment of maternity leave in individual cases.[229]
While the Bank argues that it had no inflexible requirement that maternity leave begin at the end of the sixth month of pregnancy, this contention is supported by the evidence only for the 1971-74 and later *392 periods. A 1967 personnel manual states that leave "will start" at the end of the sixth month, with no provision for exceptions. Plaintiffs' Exhibit 580, at 16; see Plaintiffs' Exhibit 598. Courts are virtually unanimous that a policy requiring leave to be taken at a specific stage of pregnancy violates Title VII where the employer cannot justify such an absolute standard. Berg v. Richmond Unified School District, 528 F.2d 1208, 1213 (9th Cir. 1975), vacated on other grounds, 434 U.S. 158, 98 S.Ct. 623, 54 L.Ed.2d 375 (1977), vacated on other grounds, 572 F.2d 709 (9th Cir. 1978); Fabian v. Independent School District No. 89, 409 F.Supp. 94, 100 (W.D.Okl.1976); Stansell v. Sherwin-Williams Co., 404 F.Supp. 696, 702 (N.D.Ga.1975); Vineyard v. Hollister Elementary School District, 64 F.R.D. 580, 583 (N.D.Cal.1974); Singer v. Mahoning County Board of Mental Retardation, 379 F.Supp. 986, 988-89 (N.D.Ohio 1974), aff'd, 519 F.2d 748 (6th Cir. 1975); Newmon v. Delta Air Lines, Inc., 374 F.Supp. 238, 244-45 (N.D.Ga.1973); EEOC Decision No. 75-095, [1974] EEOC Dec. ถ 6444; EEOC Decision No. 74-112, [1974] EEOC Dec. ถ 6428; EEOC Decision No. 73-0520, [1973] EEOC Dec. ถ 6389. But see de Laurier v. San Diego Unified School District, 9 Empl. Prac.Dec. ถ 9893 (S.D.Cal.1974) (mandatory eighth-month leave permissible although mandatory fifth month leave would be impermissible), aff'd on this point, 588 F.2d 674 (9th Cir. 1978). See generally Annot., 27 A.L.R.Fed. 537, งง 4[a], [b] & [c] (1976).[230] Hence Phase I liability is established for all females forced to begin leave at an arbitrary stage of pregnancy.[231]
Finally, the plaintiffs complain of the six-month durational limitation in effect until 1971.[232] Such a policy, if not applied to other forms of disability leave, would violate Title VII. Wetzel v. Liberty Mutual Insurance Co., 511 F.2d 199, 207-08 (3rd Cir. 1975), vacated on other grounds, 424 U.S. 737, 96 S.Ct. 1202, 47 L.Ed.2d 435 (1976); Vineyard v. Hollister Elementary School District, supra; EEOC Decision No. 75-095, supra; see 29 C.F.R. ง 1604.10(b). See generally Annot., 27 A.L.R.Fed. 537, ง 4[d] (1976). The plaintiffs have produced no evidence, however, that a different policy applied to other types of sick leave. This being the case, no liability is established as to the durational requirement.

C. Marriage Policy.

The Bank has at various times followed a policy of declining to hire close relatives of existing employees, and of requiring the resignation of one spouse if two existing employees married. If two employees each having three years' seniority married, the Bank reserved a discretionary right to require one spouse to resign, and in any event assigned the spouses to different departments. See Plaintiffs' Exhibit 734 & 735. The plaintiffs have shown no disparate impact of this facially neutral policy.

D. Training.

The Bank provides or has provided a variety of training and educational programs through the years. These include in-house programs, such as a variety of personal development courses, specialized skill courses, and American Institute of Banking courses, and outside programs, *393 such as a tuition assistance program. Detailed discussion of these programs is unnecessary: with two exceptions, these programs were administered in a race- and sex-neutral manner. No disparate impact of any program requirements has been shown.
The court has found that females were discriminated against in exempt hiring through 1974, section IX(H), supra, and that various groups suffered initial placement or promotion discrimination, section VIII(C), supra. To the extent that discrimination prevented individuals from entering the credit training program, the Bank will be liable as part of the hiring and placement/promotion cases.
At various times throughout the class period, the Bank has offered personal development courses entitled "Personality, Appearance, and Personal Attitude" and "Image for Success." The former course, designed for females and attended exclusively by females, emphasized personal appearance and manner.[233] While attendance was strongly recommended for some employees, no employee was ever required to attend. The latter course, which focused on personal image through appearance, mental attitude, and communicative skills,[234] was attended primarily (and exclusively during the earlier years) by males.
While the court must find that the sexual segregation of these courses constituted a violation of Title VII, any economic harm caused thereby must be characterized as de minimis. Each course constituted at best a minor aspect of the Bank's overall training program. Bank officials testified that attendance at the courses was not considered in evaluating any employee for promotion. The courses were short (approximately twelve class hours), and content for the male and female courses was similar. While the "Image for Success" course seems to have included a wider variety of interpersonal skills, there is no indication that females were thereby deprived of substantial employment opportunities.[235] Finally, the two courses were merged in 1973, obviating the need for any prospective injunctive relief.

E. Dress.

The court finds no systematic discrimination in the application or enforcement of any dress requirements.

F. Summary.

In summary, Phase I liability for discriminatory terms and conditions of employment *394 is found only as to certain aspects of the Bank's maternity leave policy. Phase I liability exists as to those female employees denied maternity leave for failure to meet the three-year service requirement, and those female employees forced to begin maternity leave at an arbitrary stage of pregnancy.

CONCLUSION
This opinion has been written and rewritten, or equally accurate, has been calculated and recalculated, over the past year. It is modular in structure but hopefully possessed of sufficient internal consistency to squeeze as a whole into a traditional suit of judicial opinion. Even if that effort was successful, it has to judicial eyes a surrealistic cast, mirroring the techniques used in its trial. Excursions into the new and sometimes arcane corners of different disciplines is a familiar task of American trial lawyers and its generalist judges. But more is afoot here, and this court is uncomfortable with its implications. This concern has grown with the realization that the esoterics of econometrics and statistics which both parties have required this court to judge have a centripetal dynamic of their own. They push from the outside roles of tools for "judicial" decisions toward the core of decision making itself. Stated more concretely: the precision-like mesh of numbers tends to make fits of social problems when I intuitively doubt such fits. I remain wary of the siren call of the numerical display and hope that here the resistance was adequate; that the ultimate findings are the product of judgment, not calculation.
We know the mathematical techniques also may be freighted with collateral consequences, some antithetical to the aims of the legislation being enforced. For example, we blink at reality when we say that ratios and quotas are not being reinforced when liability is heavily hinged upon "disparities" and deviations from numerical exemplars. And the techniques of this suit highlight our present social conundrum of nourishing ethnicity in an effort to starve it.
There is a tendency to turn these suits into morality plays as indeed they can be. Here the story is not of good people versus bad people but of an employer caught by changing social order, ponderous in response, but now running hard to get in line.
One school of jurists would hold that we ought privately to resolve the close questions and write an opinion whose public face belies a nonexistent confidence with a blurring of decisional juncture points. I acknowledge the utility of that approach in given cases, but not here. I write this unusual conclusion to this unusual opinion to make plain that this court did not select a numbers field for this contest but instead has been forced to judge a fight, there fought.
To place the court's findings in perspective, an additional observation is required. Despite their recent recognition, the econometric techniques employed in this case are not discrimination CAT scanners-ready to detect alien discrimination in corporate bodies. It may reveal shadows but its resolution is seldom more precise.
Ultimately the findings of fact here are not numerical products and sums but a human judgment that the facts found are more likely true than not true. With that standard, stripped to essentials, and within the decisional limits placed upon me by higher courts, this is what I think happened, approximately.
NOTES
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[131]  This action has been the subject of three prior published opinions: Vuyanich v. Republic National Bank, 409 F.Supp. 1083 (N.D.Tex.1976); Vuyanich v. Republic National Bank, 78 F.R.D. 352 (N.D.Tex.1978); Vuyanich v. Republic National Bank, 82 F.R.D. 420 (N.D.Tex.1979).
[132]  Another supervisor had previously told Ms. Vuyanich that "the world was not ready for mixed marriages." A third supervisor, however, on being told of Vuyanich's impending marriage, had stated "if that's what you want to do, do it."
[133]  Ms. Johnson was turned down the same day at two other leading Dallas banks.
[134]  February 16, 1969, is the date 180 days prior to the filing of the Vuyanich claim with the EEOC. Employees who left the Bank prior to that date may not take advantage of the Vuyanich filing, and hence are timebarred from asserting their claims. See In re Consolidated Pretrial Proceedings, 582 F.2d 1142, 1149 (7th Cir. 1978), cert. deferred sub nom. Zipes v. Trans-World Airlines, Inc. and Trans-World Airlines, Inc. v. Zipes, 442 U.S. 916, 99 S.Ct. 2834, 61 L.Ed.2d 282 (1979); Laffey v. Northwest Airlines, Inc., 567 F.2d 429, 472-74 (D.C. Cir.1976), cert. denied, 434 U.S. 1086, 98 S.Ct. 1281, 55 L.Ed.2d 792 (1978); Wetzel v. Liberty Mutual Insurance Co., 508 F.2d 239, 246 (3d Cir.), cert. denied, 421 U.S. 1011, 95 S.Ct. 2415, 44 L.Ed.2d 679 (1975).
[135]  A sixth potential subclass, that of unsuccessful female applicants for nonexempt positions, was decertified due to a lack of commonality, in light of the insufficient evidence of discrimination against that group. Vuyanich, supra n.1, 82 F.R.D. at 438.
[136]  A description of the intervenors and their respective claims was contained in the court's earlier opinion:

Marjorie Lee Jackson was employed by the Bank as an exempt employee from September 13, 1971 until October 9, 1973. She worked in Estate Administration as an administrative assistant. She testified that she was the only female holding such a position at the Bank. She further testified that although she had greater responsibility than the men holding the same position, she believes that she was paid less. She stated that although officers came from the ranks of administrative assistants in her department, she was not promoted. Jackson further testified that she requested transfer into a trust new business position but was informed by a male bank officer that he did not expect to have to hire women into such positions for at least five years.
*   *   *   *   *   *
Marisu Fenton worked for the Bank from May 22, 1972 until August 3, 1973. She complains of discrimination in job classification, training, promotion, transfer, compensation, working conditions, and job content. Although she had a degree in business administration and graduated in the top 10% of her class, she was hired as a nonexempt employee. She testified that many other educated and qualified women were in comparable positions. She further testified that even though the evaluations she received from her superiors were consistently favorable, she never received any training which would have enabled her to move into management. Nor did she receive a promotion that resulted in an increase in salary. Although Fenton was a nonexempt employee, she testified that the Bank kept no records of the hours she worked and did not pay her overtime when she worked more than 40 hours a week.
*   *   *   *   *   *
Dorothy Hooks was employed by the Bank from 1963 until 1970. She complains of race discrimination in pay, promotion, training, transfer, job assignment and classification and job content. Specifically, she alleges that she, like Vuyanich, was confined to Money Orders away from the public. During the seven years that she was at the Bank, she worked her way up through various positions in Money Orders, but observed many white employees move at a more rapid pace. Before she became a supervisor, she was required to train several white males for supervisory positions. After Hooks complained, she was promoted to supervisor. She testified that white males holding comparable positions were paid more than she was. When a different white male became the head of Money Orders, Hooks testified that he took away her supervisory authority, and then would not allow her to transfer. She asserts that as a result of these racist practices, she was constructively discharged.
Vuyanich, supra n.1, 82 F.R.D. at 439-40.
[137]  Black females in nonexempt positions are members of both of the indicated subclasses. To the extent these individuals assert sex discrimination, they are treated as members of the female nonexempt employee subclass; to the extent that their claims involve race discrimination, they are members of the black nonexempt employee subclass. Likewise, black females who unsuccessfully applied for nonexempt positions may assert race discrimination claims (but not sex discrimination claims) as members of the fifth subclass. Discrimination on account of an individual's status as a black female constitutes both sex and race discrimination. See Jefferies v. Harris County Community Action Association, 615 F.2d 1025 (5th Cir. 1980).
[138]  The applicant subclasses were originally certified to run from April 16, 1971, 180 days prior to Ellen Johnson's EEOC charge. On September 7, 1979, the court expanded these subclasses, at least for phase one purposes, to run from February 16, 1969, relying on Satterwhite v. City of Greenville, 578 F.2d 987, 994 n.8 (5th Cir. 1978), vacated, 445 U.S. 940, 100 S.Ct. 1334, 63 L.Ed.2d 773 (1980); Sanchez v. Standard Brands, Inc., supra; and Oatis v. Crown Zellerbach Corp., 398 F.2d 496, 499 (5th Cir. 1968), in holding that the applicant subclasses could take advantage of the earlier EEOC charge filed by Joan Vuyanich.
[139]  On October 16, 1979, the court appointed itself a special master to take the testimony of Dr. Francine Blau in Champaign, Illinois. To the extent that this opinion relies on the testimony of Dr. Blau, it constitutes the court's report in its capacity as special master, and is hereby adopted as such. See Fed.R.Civ.P. 53(e)(1) and 53(e)(2).
[140]  Compare Bartelson v. Dean Witter & Co., 86 F.R.D. 657, 23 Empl.Prac.Dec. 30,962 (E.D.Pa. 1980); Quigley v. Braniff Airways, Inc., 85 F.R.D. 74, 78-79, 83 (N.D.Tex.1979); Beasley v. Griffin, 81 F.R.D. 114, 116-17 (D.Mass.1979); Wajda v. Penn Mutual Life Ins. Co., 80 F.R.D. 303, 307-09 (E.D.Pa.1978); Parker v. Bell Helicopter Co., 78 F.R.D. 507, 513-14 (N.D.Tex. 1978); Arnett v. American National Red Cross, 78 F.R.D. 73, 77 n.6 (D.D.C.1978); Adams v. Jefferson Davis Parish School Board, 76 F.R.D. 621, 623 (W.D.La.1977); Dickerson v. United States Steel Corp., 439 F.Supp. 55, 61-62 (E.D. Pa.1977), vacated on other grounds sub nom. Worthy v. United States Steel Corp., 616 F.2d 698 (3d Cir. 1980) ("across-the-board" suits survive Rodriguez) with Local 194, Retail Wholesale and Department Store Union v. Standard Brands, Inc., 85 F.R.D. 599, 605 (N.D. Ill.1979); Hubbard v. Rubbermaid, Inc., 78 F.R.D. 631, 642-43 (D.Md.1978); Carpenter v. Herschede Hall Clock Division, 77 F.R.D. 700, 701 (N.D.Miss.1977); Miller v. Motorola, Inc., 76 F.R.D. 516, 518 (N.D.Ill.1977) (such suits do not survive Rodriguez). See Alexander v. Gino's, Inc., 621 F.2d 71, 75 (3d Cir. 1980) (Rodriguez dictates that there be some class that satisfies strict requirements of Rule 23(a); "across-the-board" concept then permits broadening to consider claims technically beyond scope of class claims); DeGrace v. Rumsfeld, 614 F.2d 796, 811 (1st Cir. 1980) (pretermitting question of whether "across-the-board" actions survive Rodriguez). See generally Note, Antidiscrimination Class Actions Under the Federal Rules of Civil Procedure: The Transformation of Rule 23(b)(2), 88 Yale L.J. 868, 882-83 (1979); Comment, The Proper Scope of Representation in Title VII CLass Actions: A Comment on East Texas Motor Freight Systems, Inc. v. Rodriguez, 13 Harv.C. R.-C.L.L.Rev. 175, 186 (1978).
[141]  The consistency between Rodriguez and the "across-the-board" approach was thus stated in Arnett v. American National Red Cross, supra n.10:

This Court does not believe that the Supreme Court intended [Rodriguez] to preclude an across-the-board attack by one who was discriminated against -whether it be in the hiring, promotion, or discharge aspects of the defendant's practice. What the Supreme Court did intend was to preclude the maintenance of a class action by one who was not discriminated against at all.
78 F.R.D. at 77 n.6 (emphasis in original; citation omitted).
[142]  The Bank points to a series of lower court cases in which classes limited to discrimination in specific employment practices (e. g., hiring, termination, etc.) were certified. It argues that these cases, while consistent with an "across-the-board" approach, recognize a generally applicable principle that after Rodriguez Title VII class actions must be so limited. Otherwise stated, the Bank argues that if "across-the-board" suits have survived Rodriguez, that case at least mandates that the size of the "board" be reduced to specific employment practices. This argument loses sight of the case-by-case approach by which purported "across-the-board" suits must be evaluated. The fact that in other cases plaintiffs have not met their burden of showing systemwide discriminatory intent (or have failed to meet other requirements of Fed.R.Civ.P. 23(a) such as adequacy of representation) does not indicate that an "across-the-board" suit may not be certified where those burdens have been met.
[143]  E. g., use of a single multiple regression for initial placement/promotion and salary discrimination on the part of plaintiffs' experts. See section VIII(A)(1)(i), infra.
[144]  This argument is directed against the subclass of female nonexempt employees. That subclass, represented by Marisu Fenton, is the only subclass not represented in whole or in part by one of the original named plaintiffs, both of whom have exhausted their EEOC remedies.
[145]  The case of Fewlas v. Allyn & Bacon, Inc., 83 F.R.D. 161 (D.Mass.1979), cited by the Bank, is not to the contrary. In that case, after requested certification had been granted in part, five prospective intervenors sought to represent the uncertified portion of the class. The court denied permission to intervene since the intervenors had not filed EEOC complaints, holding that the intervenors were "not in the same class as the plaintiff." 83 F.R.D. at 164. Fewlas was not an "across-the-board" suit: the court specifically stated that there was no basis for concluding that the claims of the certified and uncertified portions of the class were interdependent. Id. Moreover, in the present case, unlike Fewlas, the prospective intervenors were already members of a broadly certified class.
[146]  The case must also fall within one or more subsections of Fed.R.Civ.P. 23(b). Since satisfaction of this requirement turns largely on the nature of the plaintiffs' claims, principally as set forth in the pleadings, a determination that the requirement has been satisfied is not likely to require reevaluation as the case progresses. In the present case, the Bank makes no claim that the case is not covered by Fed.R.Civ.P. 23(b)(2).
[147]  The Bank argues that this severance operates to its prejudice, in that it is deprived of an additional opportunity to demonstrate that class certification was improper. Such an argument rings hollow in light of the fact that the Bank was twice afforded the opportunity of challenging the class certification in formal hearings. While the court has a continuing duty to monitor and modify the class, that duty does not extend to affording a defendant unlimited opportunities to develop evidence in opposition to class certification at the expense of trial efficiency and convenience.
[148]  The Supreme Court's recent decisions in United States Parole Commission v. Geraghty, 445 U.S. 388, 100 S.Ct. 1202, 63 L.Ed.2d 479 (1980), and Deposit Guaranty National Bank v. Roper, 445 U.S. 326, 100 S.Ct. 1166, 63 L.Ed.2d 427 (1980), may arguably be read as casting some doubt on the "continuing nexus" line of cases. In light of those decisions' emphasis on Article III standing considerations, however, and the statement in Geraghty that "[t]he question of who is to represent the class is a separate issue" from the case or controversy question decided there, 445 U.S. at 407, 100 S.Ct. at 1213, it would seem that the Rule 23 "continuing nexus" cases would retain their vitality. But cf. Satterwhite v. City of Greenville, 445 U.S. 940, 100 S.Ct. 1334, 63 L.Ed.2d 773 (1980) (vacating and remanding in light of Geraghty and Roper); Armour v. City of Anniston, 445 U.S. 940, 100 S.Ct. 1334, 63 L.Ed.2d 774 (1980) (same). In any event, the "continuing nexus" test does not operate to destroy class certification here.
[149]  This is of course not meant to imply that any of the named plaintiffs' claims have in fact failed. As previously noted, those claims have been severed for later consideration.
[150]  The Court's reference to decertification presumably refers to decertification on such grounds as lack of numerosity or lack of adequate representation.
[151]  No effort is made in the following narrative to provide a definitive description of the Bank. The objective is instead to provide an overview of its structure and personnel practices. By necessity we paint with a broad brush in this section in an effort to provide a backdrop to the detailed statistical data that follow.
[152]  The Bank also has an "Executive Recruiting Program" where promising students are pointed out to high level bank officials by college faculty members.
[153]  The personnel request is made by use of a requisition form submitted by a line supervisor. The requisition form until 1968 had blocks to allow an expression of sex preference. In 1967, some 67% of the requests made had sex preference checks, and in 64% of all cases the request was met. By 1969, 11% had preference stated and the extent to which the preference was honored is not known. Nor is the relative percentage of requested males and females known. There were occasional explications of the requests such as "cute size preferred" (presumably a request for a female) but there was no pattern to such requests. At the same time there is no evidence that any forms contained references to race. The form was changed in 1968, although many of the old forms continued in actual use on a sporadic basis for some time thereafter.
[154]  As will be seen in the review of the data, the arrival of Pistor and movement in the statistical measures of racial and sexual influences on hiring decisions enjoy a coincidence in time. While an inference of a causal nexus from statistical results is eschewed by the social scientist, the intuitive inference allowed to judges under the law reveals more than coincidence at work. The Bank does not emphasize any increased intensity of effort in the hiring of females and blacks under Pistor, but instead emphasizes claimed increases in availability of "qualified" females and, to a lesser extent, blacks in the labor pool.
[155]                                            Years                                       Record
  Data Base           Contents             Covered    Source           Frequency       Type        Used by
  Defendant:
  A020 tape      Company                   1973-78    existing         semi-monthly    fixed       to prepare
                 Employee number                      record           (a few                      A035 &
                 Cost center                                           tapes                       A500
                 Exempt status                                         lost or                     tapes
                 Salary                                                non-existent)
                 Employee status
                 Haypoints/grade
                 Termination date
                 Date of hire
                 Date last perf. appr.
                 Official title
                 Position title
                 Officer level
                 Sex and race
                 Termination reason
                 Type last salary incr.
                 Amount last salary incr.
                 Rating last salary incr.
                 Date of birth
                 + other information
  A030 tape      Employee name             1973-78    existing         continuous      fixed       to prepare
                 Educational data                     record                                       A035 &
                 + other information                                                               A500 tapes
  A035 tape      Above items               1973-78    A020             continuous      variable    Stoikov
                                                      A030
                                                      + new EEO
                                                      codes
                                                      + 18 employees
  A500 tape      Employee number           1969-72    personnel        snapshot        fixed?      Stoikov
  (also          Name                                 records                                      Stolzenberg/Snyder
  known as       Sex and race
  A040)          Date of hire
                 Cost center at hire
                 Year
                 Cost center
                 Job family code
                 Grade or Haypoints
                 Officer level
                 Date of promotion
                 Job family code before
                   promotion
                 Job family code after
                   promotion
                 Date of termination
                 Cost center at term.
                 Termination reason
  questionnaire  29 questions (see         1979       personnel        one-time        fixed       Stoikov
                 Appendix B of                        jackets +                                    Stolzenberg/Snyder
                 Defendant's Exhibit 32)              responses



  terminations   Involuntary/voluntary     1969-78    A035             one-time        fixed       Stolzenberg/Snyder
                 Sex and race                         A500
                 Years formal schooling               personnel
                 College major
                 College degree                       jackets
                 Typing ability
                 Shorthand ability
                 Number of office machines
                 Position in last job
                 Time in last job
                 Age at application
                 Year of application
  hiring         Job family                1970-78    personnel        one-time        fixed       Stoikov
  data           Position title                       jackets
                 Name
                 Employee number
                 Date of hire
                 Experience
                 Number years
                 Education
                 Study area
                 Major
                 School
                 County/City
  Plaintiffs:
                 Items from A020           1973-78    A020             annual          fixed       Morgan
                 + highest grade compl.               A035                                         Blau/Madden
                 + date of birth
                 Items from A500           1969-72    A500             annual          fixed       Morgan
                 + highest grade compl.               +                                            Blau/Madden
                 + date of birth                      selected
                 + salary                             records
                                                      from personnel
                                                      jackets
A variety of miscellaneous data sources, such as the Bank's EEO1 reports
(Plaintiffs' Exhibits 511 et seq.), were also utilized by the various experts.

[156]  A "record" is a computer data entry relating to a single subject (e. g., a particular employee). A record consists of a number of "fields" (individual items of data; e. g., name, date of birth, etc.).
[157]  The Bank believed "that these errors were caused by a continuation of an historic misinterpretation by Bank personnel of EEOC guidelines regarding code assignment and by deficiencies in the Bank's computer procedures which prohibited code changes following some promotional or transfer movements." Defendant's Post-Trial Brief at 51.
[158]  Most fields were found to be accurate in 97-99% of all cases. See Plaintiffs' Exhibits 508 and 509.
[159]  The omission and error rates for various educational fields were as follows:



Field                        Omissions           Errors
School Type                     27%                 1%
High School Code                29%                 1%
College Code                    28%                 0%
Graduate Code                   25%                 0%
School Unknown                                      3%
High School Dates               32%                 3%
College Dates                   28%                 2%
Graduation Dates                24%                 1%
High School Graduate            29%                 1%
College Graduate                28%                 1%
Graduate School Graduate        24%                 0%
College Degree                  26%                 1%
Graduate Degree                 24%                 0%
College Major                   32%                 1%
Graduate Major                  25%                 1%
Grade Completed                 31%                10%
Average                         27%                 2%

The data in this table were derived from Plaintiffs' Exhibit 509.
[160]  The court does note, however, that two employees asked Dr. Stoikov to read the questionnaire to them during its administration.
[161]                    Salary                 DOB                 Education
              Missing   Wrong      Missing   Wrong        Missing    Wrong
1969           12.5%     6.9%        8.1%     5.0%         12.5%     25.6%
1970            7.5%     7.5%        7.5%     2.5%          9.4%     27.5%
1971            9.4%     8.1%        6.3%     5.6%          9.4%     21.9%
1972           18.8%     2.5%       14.4%     5.6%         20.0%     23.1%
Overall        12.0%     6.3%        9.1%     4.7%         12.8%     24.5%
          The data in this table appear in Defendant's Exhibit 553.

[162]  Indeed, Exhibit D to Defendant's Exhibit 553 suggests that errors and omissions are more or less randomly distributed among races and sexes, at least for 1969:



     EXEMPT
                        Black            White                Total     
                   M      F     T     M      F      T     M     F      T
     % Workforce                      94.8  5.0  99.8     95.0  5.0  100.0
     % Omitted                        98.7  1.3 100.0     98.7  1.3  100.0



     NONEXEMPT
                       Black               White                Total    
                   M      F     T     M      F      T     M     F      T
     % Workforce   1.1  10.6  11.7    18.1  62.8  80.9    20.3  79.7 100.0
     % Omitted     2.2   8.0  10.2    18.6  61.3  79.9    22.1  77.9 100.0

[163]  We refer to both types of evidence collectively as "anecdotal evidence."
[164]  At trial, Mr. Croft testified that his use at deposition of the term "spiritual change" had been inappropriate.
[165]  Cf. Underwood, Law and the Crystal Ball: Predicting Behavior with Statistical Inference and Individualized Judgment, 88 Yale L.J. 1408, 1420-23 (1979) (distinction between "clinical" approach wherein experienced decisionmakers use subjective judgment to evaluate each applicant on an individual basis in light of experience, and "statistical" approach).
[166]  We do not suggest that all employment decisions must be based on productivity. See e. g., Garcia v. Gloor, 618 F.2d 264, 269 (5th Cir. 1980) (e. g., Title VII does not forbid hiring only persons born under a certain sign of the zodiac). However, since productivity differences are the bases offered to justify differentials in wages, initial placement, and promotion, our analysis as to these issues similarly focus on the productivity justification under Title VII.
[167]  Comment, The Business Necessity Defense to Disparate-Impact Liability Under Title VII, 46 U.Chi.L.Rev. 911, 921 (1979) ("Disparate-Impact Liability"); Fiss, A Theory of Fair Employment Laws, 38 U.Chi.L.Rev. 235, 237-38 (1971); Note, Business Necessity Under Title VII of the Civil Rights Act of 1964: A No-Alternative Approach, 84 Yale L.J. 98, 103 (1974) ("No-Alternative Approach").
[168]  This tension between recognizing the right of employers to differentiate on the basis of qualifications and the difficulty of avoiding its potential for abuse was evident in the passage of Title VII. Senator Tower offered an amendment directed at permitting the use of ability tests even when standardized on advantaged groups. No-Alternative Approach, supra n.37, at 104 n.31. His first proposal was rejected as being too loosely drawn, but passed when drawn more narrowly. See 111 Cong.Rec. 13724, 13492 (1964); Griggs v. Duke Power Co., 401 U.S. 424, 434-36, 91 S.Ct. 849, 855-856, 28 L.Ed.2d 158 (1971). Senator Tower's immediate focus was upon a narrow part of the problem of skill measures-the written test-and represented an effort to preserve the right of employers to test for differing skill levels. Even these were so suspect that the amendment, resulting EEOC guidelines, and some judicial decisions construing the amendment have imposed such stringent burdens of validation as to undermine the entire effort. That is, the effect of placing the burden upon employers of proving that various predictors or criteria they used were true productivity factors when direct measures of productivity were not available has been at times to cause Title VII to operate as if Congress had actually subscribed to equal achievement with a resultant focus upon group status, not individual abilities.
[169]  See Disparate-Impact Liability, supra n.37, at 926-28; No-Alternative Approach, supra n.37, at 103 n.29.
[170]  "Congress clearly intended to impose on employers any potential loss of efficiency resulting from the inability to use race as an employment qualification." No-Alternative Approach, supra n.37, at 102-03. The pernicious effects of using race and sex as predictors of an individual's actual characteristics are so deleterious that even where an employer may find it efficient to use race and sex, he may not legally do so. Cf. Fiss, supra n.37, at 241-43 & 257-63; Brilmayer, Hekeler, Laycock, & Sullivan, Sex Discrimination in Employer-Sponsored Insurance Plans: A Legal and Demographic Analysis, 47 U.Chi.L.Rev. 505, 526-29 (1980); Underwood, supra n.35, at 1434-36.

Under most, but not all, circumstances, the prohibition of discrimination is fully consistent with maximization of productivity for the individual businessman and for society. See Fiss, supra n.37, at 257-63. But see Gwartney, Asher, Haworth, & Haworth, Statistics, the Law and Title VII: An Economist's View, 54 Notre Dame L.Rev. 633, 637 n.11 (1979).
[171]  Cf. Smith v. Olin Chemical Corp., 555 F.2d 1283, 1286-88 (5th Cir. 1977) (employer does not have to justify with evidentiary proof of business necessity the exclusion of any person with a bad back as a laborer).
[172]  See, e.g., id.
[173]  Throughout this opinion we employ the term "predictor" broadly; it refers to any factor or attribute used by the employer, consciously or unconsciously, in differentiating among employees or applicants. For instance, a requirement of a college degree is a predictor, just as a variable in a multiple regression model for compensation is one as well.
[174]  This is one aspect of "improper treatment across twins," referred to in section VI(E), infra.
[175]  The following discussion of the mathematics of regressions draws heavily, in terms of both structure and content, on Fisher, supra.
[176]  There is a third, less widespread use of multiple regression which we will not here discuss.
[177]  A variable is something that can take on different values. Since each variable can assume various values, it must be represented by a symbol instead of a specific number. For instance, we might represent price by P and imports by I. When we write P = 3 or I = 18, however, we are "freezing" these variables at specific values (in appropriately chosen units). A. Chiang, Fundamental Methods of Mathematical Economics 9 (2d ed. 1974).

Variables often appear in combination with "constants," such as in the expression 7P or 18.51. A "constant" is some number that does not change, and hence is the opposite of a variable. When a constant is joined to a variable, it is often referred to as the "coefficient" of that variable. We can let a symbol (for instance, "a") stand for a given constant; when symbols are used to designate constants, we often refer to them as "parameters." Id.
[178]  See n.47, supra.
[179]  Dependent variables are also referred to as "endogenous" variables. See G. Maddala, Econometrics 5 (1977).
[180]  Independent variables are also referred to as "exogenous variables" or "explanatory variables." Id.; Editors' Introduction, supra.
[181]  For instance, the classic supply and demand model seeks to explain the sales of a commodity in a particular market in relation to price; it consists of three equations, namely, a demand equation, a supply equation, and a market adjustment equation. These equations will contain other variables in addition to the quantity and price of the commodity in question, such as disposable income in the demand equation and factor prices in the supply equation. The explanation achieved by the model is then conditional on the values of these other variables, variables which are not determined or explained by the model. J. Johnston, Econometric Methods 2 (2d ed. 1972).
[182]  We use the term "linear equation" or "linear function" for any relationship such as Q = cA + fB + gC where Q, A, B, and C are all variables and c, f, and g are all constants. See W. Baumol, Economic Theory and Operations Analysis 14-15 (3d ed. 1972).
[183]  Certainly in a broad sense almost all variables are endogenous and the only exogenous variable one can think of are such things as weather. But in any particular econometric study, this is a matter of approximation. For instance, while studying the demand for gasoline by households, we can treat the quantity demanded as endogenous and income and price as exogenous, arguing that the household does not have control over these. G. Maddala, supra n.49, at 5.
[184]  See G. Maddala, supra n.49, at 127; R. Wonnacott & T. Wonnacott, supra, at 88-90. Cf. Fisher, supra, at 714-15 ("[A] properly done study begins with a decent theoretical idea of what variables are likely to be important .... [A] study that casts about for a good-looking relationship by trying all sorts of possibilities is very likely to come up with relationships where none exist.")
[185]  We will discuss these "structural assumptions" when we discuss the situations in which the least squares estimates may be unreliable. See Fisher, supra, at 708-11; D. Baldus & J. Cole, Statistical Proof of Discrimination งง 8A.2-8A.43, at 276-86 (1980).
[186]  This is not strictly correct, however, the distinction between a "disturbance term" and the "error" or "residual" is unimportant for our purposes. See J. Johnston, supra n.51, at 123.
[187]  "Dummy variables" are sometimes used as explanatory variables. See R. Wonnacott & T. Wonnacott, supra, at 100-103 (use of dummy variable "1" for wartime and "0" for peacetime, in econometric modeling of sales of government bonds).

When some explaining factors assume only discrete values (such as "male" or "female") and others assume continuous values (such as age expressed in years, or, better, in months), a middle course between ANOVA and regression can be effected by the use of multiple regression with "dummy variables." A dummy variable is given the values zero and one to correspond with the presence or absence of a particular attribute. For example, a regression could estimate an employee's wage by using the employee's years of work experience, years of education, and a dummy variable to reflect whether the employee were male. The weighting on the dummy variable would not be as readily interpretable as a weighting on a more usual type of explaining characteristic, but the question of the overall effect of the characteristic represented by the dummy variable could be answered.
Note, Beyond the Prima Facie Case in Employment Discrimination Law: Statistical Proof and Rebuttal, 89 Harv.L.Rev. 387, 399 (1975) (footnote omitted) ("Beyond the Prima Facie Case").
[188]  As this court explained in Cooper v. University of Texas, 482 F.Supp. 187, 194 (N.D. Tex. 1979):

With a test of statistical significance we learn the probability that the observed value could have happened by chance, i.e., the probability that in a random sample of an appropriate test population the variable would exhibit a value as extreme as that observed. A test of statistical significance is thus itself based on the results of a hypothetical experiment. We suppose that an infinite number of samples of the same size are drawn from the test population. The probability of the observed value occurring by chance is equal to the proportion of the samples in which the value is at least extreme as the observed value. It has become a convention in social science to accept as statistically significant values which have a probability of occurring by chance 5% of the time or less. See generally N. Nie, C. Hull, J. Jenkins, K. Steinbrenner, & D. Bent, Statistical Package for the Social Sciences 222 (2d ed. 1975).
[189]  We are discussing the concept of statistical significance, not "practical significance:"

A measure of statistical significance ... provides a basis for inferring whether there is, in fact, some disparity in results for the minority and majority groups. Practical significance, in contrast, refers to the magnitude of the disparity between the results for the two groups. This information is conveyed by the measure of disproportionate impact and provides a basis for assessing the level of injury to the minority group.
D. Baldus & J. Cole, supra n.55, ง 9.41, at 317 (footnote omitted).
In Title VII cases, "[t]he degree of [d]iscrimination practiced by an employer is unimportant ..."; "[d]iscriminations come in all sizes and all such discriminations are prohibited by the Act." Rowe v. General Motors Corp., 457 F.2d 348, 354 (5th Cir. 1972). See Hodgson v. American Bank of Commerce, 447 F.2d 416, 420 (5th Cir. 1971).
[190]  When regression techniques more advanced than ordinary least squares are used, not all such assumptions may need to hold for valid results to be generated by the regression analyses.
[191]  For a simple explanation of bias, see Fisher, supra, at 709.

We do not discuss explicitly the associated concept of "efficiency." Because of the particular unbiased and "efficient" estimates of coefficients possible using the ordinary least squares technique, ordinary least squares is often described as the "best linear unbiased estimator." See R. Wonnacott & T. Wonnacott, supra, at 55-69; J. Johnston, supra n.51, at 126; Kmenta, supra, at 9-14, 161; R. Pindyck & D. Rubinfeld, supra, at 20-24.
[192]  See n.57, supra.
[193]  For a discussion of the normal distribution, see section IX(A), infra.
[194]  When the natural logarithm of Y is used as the dependent variable, it is, roughly speaking, as if the value of Y were being determined by the explanatory variables acting not in an additive fashion, but in a multiplicative fashion. For instance:
      (7a) 1n Y = a + b1x1 + b2x2
is equivalent to
      (7b) Y = eaeb1x1eb2x2
                            , where e = 2.71828.

If Y were salary, x1 experience, and x2 education, then the behavioral assumption underlying these equations would be that schooling "intensifies" experience, and vice versa. See Testimony of Ross Stolzenberg.
[195]  Simply because a variable in an equation has a statistically significant coefficient associated with it does not prove it is job-related. All it may prove is that the employer has used the variable as a predictor. The law requires that predictors with disparate impact be job-related. Contra, Gwartney, Asher, Haworth, & Haworth, supra n.40, at 655-58.
[196]  This does not mean, of course, that "raw differentials" can always establish a prima facie case. Cf. Hazelwood School District v. United States, 433 U.S. 299, 308 n.13, 97 S.Ct. 2736, 2742 n.13, 53 L.Ed.2d 768 (1977). See e.g., Pouncy v. Prudential Insurance Co. of America, 499 F.Supp. 427, 23 Empl.Prac.Dec. 31,114, at 16,751 (S.D.Tex.1980).
[197]  See section VI(B), supra (example of possession of degree being weighted more than is justified). This feature of job-relatedness could be viewed as one aspect of "improper treatment across twins," described infra.

It should be observed that this aspect of the latter concept arises in this litigation only in the hiring case (section IX, infra). It does not arise in the initial placement, promotion, or compensation cases because the Bank's rebuttal models fail for other reasons, and so we do not reach the job-relatedness issue.
Cf. Horner v. Mary Institute, 613 F.2d 706, 715 n.9 (8th Cir. 1980).
[198]  See generally Fisher, supra, at 722-24; Beyond the Prima Facie Case, supra n.57, at 401-04.
[199]  We do not reach the legal implications of an empirical finding of a "mixed case" because in this litigation we do not need to do so. The above discussion of the "mixed case" is only meant to be a background to an understanding of the use of a dummy variable index for group status. Cf. Waintroob, The Developing Law of Equal Employment Opportunity at the White Collar and Professional Level, 21 Wm. & Mary L.Rev. 45, 98-103 (1979).
[200]  See n.67, supra.
[201]  One can also differentiate for certain other reasons. "Like the Equal Pay Act, Title VII exempts from its coverage wage differentials paid pursuant to an incentive system or to a bona fide seniority or merit system." Note, The Bennett Amendment-Title VII and Gender-Based Discrimination, 68 Geo.L.J. 1169, 1173 (1980). The plaintiffs' theory of wage differentials is that productivity and group status explain differentials; the Bank's explanation is based on a differing productivity theory. We thus need only consider how econometrics may reveal non-productivity-based compensation. See n.36, supra.
[202]  As an instance of how one's occupation may affect productivity, a brilliant employee, equally good as an electrical engineer and a messenger, is likely "worth" more to his employer as an engineer than as a messenger.
[203]  Section 703(h) provides in part:

It shall not be an unlawful employment practice under this title for any employer to differentiate upon the basis of sex in determining the amount of the wages or compensation paid or to be paid to employers of such employees if such differentiation is authorized by the provisions of section 6(d) of the Fair Labor Standards Act of 1938, as amended.
42 U.S.C. ง 2002e-2(h).
[204]  Because the number of employees in each job may well be limited, statistical significance may be a near physical impossibility due to sample size problems; the court might look at the overall pattern of pay differentials in the various regressions for each job (or sample of jobs) at the firm, even though pay differentials in individual regressions may not be statistically significant. Cf., e. g., Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. 702, 718 (1980) (value of statistically significant results in multiple regressions); R. Wonnacott & T. Wonnacott, Econometrics 88-90 (2d ed. 1979); D. Baldus & J. Cole, supra n.55, at 316-17.
[205]  The Fifth Circuit stated:

[W]e find that while in some instances the statistical facts spoke for themselves, as in the absence of promotions of black professional workers, in other cases, there was evidence beyond the statistical facts and analysis that would support an inference of discrimination, as in the case of salaries.
Having found that the plaintiffs did establish a prima facie case, either on the basis of statistical data by itself or by the case of statistical data and analyses supported by other evidence, the burden fell to the defendants to rebut the plaintiffs' case.
Wade v. Mississippi Cooperative Extension Service, supra, at 517.
[206]  Of course, the employer cannot rely on the market to pay members of one race less than another.
[207]  That is, there is no reason to suppose that if an employer has 100 jobs, and the same Hay points were assigned to 50 pairs of jobs (one job predominantly white and the other predominantly black), that it is always the "black" job of each pair that is valued lower in the market-place.
[208]  The Bank may of course also show that there are other legitimate reasons for differentials, see n.71, supra, or that such control for work is otherwise inadequate. See generally section VII(C), infra (introduction) (re challenging prima facie case).

We do not mean to suggest that adding a Hay point variable to the pay regression is a perfect control for differentials due to differences in occupations. See, e. g., Testimony of Dr. Ross Stolzenberg (on how Hay points do not solve the problem; for instance, the coefficient of the education variable may be different for two occupations). However, to require perfection in the prima facie case is not justified. Esoteric statistical arguments of the sort cited by Dr. Stolzenberg may, however, be raised on rebuttal.
[209]  There was testimony, for instance, that petroleum engineers had to be paid more than the normal Bank "line."
[210]  We do not deal in this section with the anecdotal evidence or the system for determining compensation used by the Bank. See sections VIII(A), infra, and V(D), supra.
[211]  John Spaulding, D.B.A. (Doctor of Business Administration), is an associate professor at the College of Business of North Texas State University. Dr. Spaulding is a statistician who is a member of the American Statistical Association, the American Production and Inventory Control Society, and the American Institute of Decision Science.
[212]  David Morgan, Ph.D., is an Assistant Professor of Geography and Political Science at the University of Texas at Dallas. His area of expertise is the collection and analysis of social science data.
[213]  Francine Blau, a Ph.D. in economics, is an Associate Professor of Economics and Labor and Industrial Relations at the University of Illinois at Urbana-Champaign. See Testimony of Dr. Francine Blau at 4; Plaintiffs' Preliminary Report Identifying Expert Witnesses (at Exhibit E). She has taught a number of courses in labor economics, including the economics of discrimination, women in the labor market, and the economics of human resources.
[214]  Janice Madden, a Ph.D. in economics, is an Associate Professor of Regional Science at the University of Pennsylvania. She has published a book in the field of labor economics, is on the editorial board of the International Economic Review, and has taught in the areas of economics of race and sex discrimination and statistical methodology. See Testimony of Dr. Janice Madden at 4-7.
[215]  We do not list the regression results of Plaintiffs' Exhibit 1410: the regressions therein do not have any controls for work performed and so are not probative of salary discrimination. See sections VII(A), supra, and VII(C), infra. See also n.117, infra. Similarly, we do not here list the results of Dr. Madden's "cohort" analysis in Plaintiffs' Exhibit 1308 (at Tables 13 and 14). See also section VIII(A)(1)(ii), infra.

We do not discuss the separate regressions for "other minority" females because they are not among the plaintiffs. See, e. g., Plaintiffs' Exhibits 504-AA to 504-FF.
We also do not list those figures listed in some of the plaintiffs' analyses as being differentials associated with black females, as the figures listed therein were obtained by simply adding the coefficient obtained for the race dummy variable to that obtained for the sex dummy variable. Cf. Testimony of Dr. Janice Madden at 129.
[216]  Thus, looking at regression (2) of the 1969, 1970, 1971, and 1972 tables following, "[b]lack males earned 20.1 to 30.9 percent less than white males of equivalent age and education ..." Plaintiffs' Exhibit 501, at 2.
[217]  We do not list R2 in this opinion because of its limited utility. See section VI(D), supra. Similarly, we do not list the results of F tests because of their limited utility. See R. Pindyck & D. Rubinfeld, Econometric Models and Economic Forecasts 60 (1976).

All the tables for plaintiffs following were drawn up on the assumption that where the plaintiffs had presented results in the form of percentages, those percentages were simply the coefficients multiplied by 100. This assumption is not critical to any of our analyses. See n.110, infra. A blank in the "Other" column does not necessarily mean that a dully variable for the "Other" group status was not included in the regression; this regression result was sometimes not presented in the report from which the table was drawn.
[218]  For this and tables for other years see section VIII(A)(1)(i), infra.
[219]  Listed above and in tables for other years only where reported in associated, non-computer-printout, exhibit; where a digit is illegible in such associated reports, a hyphen is placed. Cf. section VII(B)(2) n.91, infra.
[220]  Judith Stoikov, a Ph.D. in economics, is a Senior Consultant with National Economic Research Associates, a firm of consulting economists. At this firm, she has, among other things, assisted clients in developing analyses of their employment practices for submission to the Office of Federal Contract Compliance and developed responses to charges of discrimination brought to court. See Defendant's Exhibit 32 (at Appendix A). Prior to joining this firm, she was an Associate Professor in the Economics Department of the State University of New York at Cortland, where she has taught courses in, among other things, the economics of slavery and discrimination and quantitative methods in economic analysis. See Testimony of Dr. Judith Stoikov at 8.
[221]  Cf. Testimony of Dr. Janice Madden at 35 & 37-42 (re Plaintiffs' Exhibit 1308 regressions using universe rather than sample, examination of possible effects of missing data, and statement that "We have consistently attempted to analyze the full bank force. There is no sampling here.").
[222]  Specifically, the plaintiffs' experts raised the "truncation bias" problems stemming from running separate regressions for professionals and nonprofessionals. Dr. Stoikov stated that if a sample is stratified on the basis of an endogenous variable there might be a possibility of truncation bias but that she stratified on the basis of exogenous variables. She said she "designed the sample to be stratified on the basis of status at hire and this would have been the way of avoiding truncation bias problems." Testimony of Dr. Judith Stoikov at 297. She stated that this a priori expectation was confirmed by a sophisticated statistical test. See id. at 300-301 and 408-11.

See Defendant's Post-Trial Brief at 497 n.204 and Defendant's Exhibit 32 at 6 (re professional/nonprofessional, exempt/nonexempt).
[223]  It is unclear to the court just what "Banking major in Professional Status" designates, but this will turn out to be unimportant to our analyses. See section VII(C), infra. Cf. Testimony of Dr. Judith Stoikov at 138.
[224]  A female plaintiff can, for instance, run individual regressions for each of a sample of jobs at the place of employment. The overall pattern of coefficients of dummy variables in the set of regressions obtained might be used as the basis for determination of wage discrimination, whether or not such coefficients individually viewed were statistically significant. See n. 74, supra.
[225]  Requiring an explanatory variable that controls for job level when considering the issue of wage discrimination is not proscribed by James v. Stockham Valves & Fittings Co., 559 F.2d 310 (5th Cir. 1977), cert. denied, 434 U.S. 1034, 98 S.Ct. 767, 54 L.Ed.2d 781 (1978). James was not a wage discrimination case; instead, it dealt instead with allegations of racially segregated facilities, discrimination in job assignment, promotion, training, and transfer, and discriminatory testing, education, and age requirements. See id. at 313. As to the job allocation claim, the court looked at, among other things: (i) the fact that the average job class (on a hierarchy of job classes) for blacks was lower than the average job class for whites and (ii) disparities in black and white employee wages and gross earnings (the average hourly earnings rate of black employees as of September 1973 being 37ข less per hour than the average earnings of white employees). Id. at 321 & 326-27.

It was proper for that court to reject the earnings regression offered by defendant to explain away the 37ข difference in "job assignment" by including an explanatory variable which itself was dependent on job level (that is, skill level was one of the explanatory variables, and the defendant determined that one had "skill" only if he worked in certain job classes). Otherwise, the very job placement discrimination of the employer would have been used to explain away the 37ข indicator of job placement discrimination.
[226]  Isolated instances of such market reasons, such as the fact testified to by Thomas Barksdale that petroleum engineers, because of market conditions, were paid more than those with similar Hay points, do not rise to the level of a general argument that such abnormal market conditions systematically account for differential payment of blacks in comparable jobs.
[227]  Of course, there may be some rare instances where the data base is so inaccurate that even in the absence of a showing of systematic bias, any models based thereof become non-probative.
[228]  While the employer may wish to use group characteristics in making social science studies of past behavior, it may not use group characteristics, real or imagined, as a factor in employment decisions. Current law has forbidden, under Title VII, an employer's prediction of what an individual is like on the basis of his group status, even if the prediction is well founded statistically. See Los Angeles Department of Water & Power v. Manhart, 435 U.S. 702, 98 S.Ct. 1370, 55 L.Ed.2d 657 (1978). We see no reason why a court should similarly be prevented from statistical ascription to an actual group of individuals (that is, the employees at Republic). Los Angeles v. Manhart, supra, determined that although as a class, women live longer than men, the employer violated Title VII in requiring its female employees to make larger contributions to its pension fund than its male employees. The reasoning unambiguously rejects ascriptions of group characteristics to affected individuals:

This does not ... involve a fictional difference between men and women. It involves a generalization that the parties accept as unquestionably true: Women, as a class, do live longer than men. The Department treated its women employees differently from its men employees because the two classes are in fact different. It is equally true, however, that all individuals in the respective classes do not share the characteristics that differentiates the average class representatives. Many women do not live as long as the average man and many men outlive the average woman. The question, therefore, is whether the existence or nonexistence of "discrimination" is to be determined by comparison of class characteristics or individual characteristics. A "stereotyped" answer to that question may not be the same as the answer that the language and purpose of the statute command.
The statute makes it unlawful "to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual's race, color, religion, sex, or national origin." 42 U.S.C. ง 2000e-2(a)(1) (emphasis added). The statute's focus on the individual is unambiguous. It precludes treatment of individuals as simply components of a racial, religious, sexual, or national class.
.   .   .   .   .
[T]here is no assurance that any individual woman working for the Department will actually fit the generalization on which the Department's policy is based.
.   .   .   .   .
Even if the statutory language were less clear, the basic policy of the statute requires that we focus on fairness to individuals rather than fairness to classes.
435 U.S. at 707-09, 98 S.Ct. at 1374-1379.
[229]  In here proceeding to an analysis of wage disparities after controlling for productivity, we do not mean to ignore Fisher v. Procter & Gamble Mfg. Co., 613 F.2d 527, 544 (5th Cir. 1980). Fisher was not a compensation case, and we decline to extend its embrace of "raw" figures here because of the dangers of running afoul of note 13 of Hazelwood School District v. United States, 433 U.S. 299, 308, 97 S.Ct. 2736, 2741, 53 L.Ed.2d 768 (1977). Moreover, the sort of raw wage differentials here involved do not even come close to the ratio of percentage of blacks in skilled positions to their percentage in the work force in Fisher.
[230]  Assuming arguendo that the use of the dummy variables for officer or other exempt status does not take care of the "2-pay structure problem."
[231]  Dr. Stoikov appears to have testified to the converse: with truncation bias, she believes there would be overestimation of discrimination in the nonprofessional regression and underestimation in the professional regression. See Testimony of Dr. Stoikov at 151-52. If this converse set of propositions is true, the above sorts of arguments as to black nonexempts and black exempts could be switched as well.
[232]  Numbers of black exempts at the Bank appear to be as follows:



     1973:                           5
     1974:                          14
     1975:                          19
     1976:                          16
     1977:                          21
     1978:                          24

Calculated from P. England, Employment of Women and Minorities at Republic National Bank: A Comparison with Availability Estimates (1979) (Table 6) (Plaintiffs' Exhibit 507F).
[233]  Comparing white females with white males is one permissible way of detecting (pure) sex discrimination; comparing black males with white males is one permissible way of detecting (pure) race discrimination. Cf. Plaintiffs' Exhibit 504, at 4 ("controlling for race, women earn 37.50 percent less than men with similar levels of education and experience"; "[w]ithin each sex group, blacks earn 16.77 percent less than whites with comparable education and experience.") & 8 ("The percentage salary differentials between white males and black males, white females, and black females are shown in Table 5 [Plaintiffs' Exhibit 504-E] for all active employees.").
[234]  For instance, Dr. Stoikov testified:

And so what I then did was to rerun the regression again eliminating white females from the regression and then estimating what the effect of race is on the probability of being hired as a professional.
These results are presented in Testimonial Exhibit 24. As expected the regression coefficient when all blacks are compared with white males is not statistically significant at the 5 percent level, regression coefficient is minus 0.0165 and T statistic is minus 0.43.
I was led to the conclusion from this analysis that the estimate that I was observing in Exhibit 22 is due to a comparison of 2 protected groups, blacks and white females, wherein in fact 1 of the protected groups, white females, had been affirmatively placed into the professional pay structure where the other one had been placed without prejudice into the professional pay structure. And that the comparison of the-of the blacks with white females was producing the appearance of discrimination when in fact it was in essence an indication of affirmative action towards white females. I hope that's clear.
Testimony of Dr. Judith Stoikov at 177.
[235]  Concededly, there is some indirect evidence of the actual patterns of employee qualifications, albeit only for small samples of the Bank's employees in one year (1978), and only as to those characteristics gauged in the Bank's pay regression.

For instance, none of the plaintiffs' pay regressions contained a variable (or set of variables) for bank-related major, which assertedly causes "omitted variable" bias. In the Bank's pay regression, after controlling for service at Republic and highest educational level, the Bank added variables for banking major, and the professional male/professional female pay disparity dropped from -17.16% to -15.13% (using stratification by status at hire). See Testimonial Exhibit 14, Defendant's Exhibit 557. This reduction in the disparity might arguably suggest that females at the Bank are less likely to possess a banking major than males.
However, this sort of indirect evidence is not persuasive in this case, through the interaction of four reasons. First, the samples were fairly small in size-there were only 54 females in the professional male/professional female study, 98 females in the nonprofessional male/nonprofessional female analysis, 10 blacks in the professional black/professional white comparison, 46 black females in the black female nonprofessional/white female nonprofessional comparison, and 19 black males in the black male nonprofessional/white male nonprofessional regression. See Testimonial Exhibits 14, 17-19 (Defendant's Exhibit 557).
Second, many of the changes in pay disparity due to the addition of any one set of explanatory variables were either small in magnitude or, on occasion, positive in sign: exclusion of any one set of explanatory variables may not be a serious problem. Indeed, to include these sorts of explanatory variables may cause multicollinearity problems. (One explanatory variable which does seem to consistently decrease disparity by more than a small amount is the set of variables designed to measure career motivation; this obviously may be tainted by any employee perception of Bank discrimination.) From these small magnitudes, it is difficult to infer that the pay differentials in the plaintiffs' models would be reduced to zero if there were no omitted variable bias, especially considering the possibility that other omitted variables, not included in the Bank's pay regression, may bias results in favor of a finding of non-disparity. Third, this indirect evidence relates only to 1978. We hesitate to extrapolate indirect evidence of complex patterns of employee characteristics, especially when those patterns seem to be relatively unpronounced.
Fourth, this indirect evidence does not show the extent to which plaintiffs' modeling results are affected by omitted variables bias.
[**]  Suppose there were 51 high-level executives, 50 males and 1 female, each with identical productivity characteristics (only some of which are included in the regression equation); they score high on each productivity characteristic.

Suppose there were 51 lower-level executives, 50 females and 1 male, each with identical productivity characteristics (only some of which are included in the regression equation); they score low on each productivity characteristic.
If one does the regression controlling for job level, it can be assumed that the coefficient of the dummy variable will likely not suffer from omitted variables bias; after all, the higher-level male is exactly equivalent to the higher level female, and so if any difference in pay exists, it could not be attributed to productivity differences.
Yet, if one were to draw up tables corresponding to Tables 1 and 2, they would show the mean male having higher productivity characteristics than the mean female.
[**]  When we discuss the use of age as proxy, we mean to include any proxies for general experience that do not use actual general experience.
[**]  See n.117, infra (re "mining" for regression results in the mass of numbers offered, and its application to this case).
[**]  See the section VII(B) tables for a description of the sex-experience interaction terms used for the periods 1969-72 and 1973-78.
[**]  Dr. Blau approximated the proportional reduction in salary (1-ex) by the magnitude of the coefficient of the dummy variable (x, where x is the coefficient of a dummy variable). This approximation was obtained by using the first term of the Maclaurin series expansion
1 - ex = -x + (1/2)x2 - (1/6)x3 + (1/24)x4 - (1/120)x5 + ... .

This approximation is valid for small values of x, but breaks down as x increases.
[**]  We deal separately with allegations of departmental segregation in section XI(A), infra. Various comparisons between segments of the Bank's workforce and (1) the general labor force in the Dallas/Forth Worth SMSA, City of Dallas, and overall Bank workforce; (2) the corresponding segments of the work forces of all banks filing EEOI reports; and (3) the corresponding segments of the work forces of banks in a study done by the Council on Economic Priorities, performed by Dr. Paula England, may arguably be construed as supporting the plaintiffs' case on the question of initial placement and promotion. We reject those studies here for the same reasons they are rejected in connection with the hiring case. See sections IX(B), and IX(D), infra.

We do not here describe the new hires analysis of Dr. Stoikov; discussion of that analysis is left to the hiring case (section IX, infra). That analysis focuses not on whether those who have been hired have received proper initial placement in relation to their individual qualifications, but on whether the right numbers of females (or blacks) are being hired. The fact that some females (or blacks) are being hired into higher-level job categories might indicate that there is no blanket exclusion of females (or blacks) from those jobs, but that fact says little about whether those people who were hired were properly placed in light of their individual qualifications. Alternately viewed, Dr. Stoikov's new hires analysis does not separate the effects of possible hiring discrimination from those of possible initial placement discrimination.
[**]  See column of figures entitled "differential due to differences in occupational distributions" in the tables in section VII(B)(1), supra.
[*]  It is important here to note that even were one to assume arguendo that only Equal Pay Act standards apply to female compensation discrimination, nothing would indicate that women ought be treated differently from men when it comes to Title VII occupational distribution discrimination. Thus any figure used to indicate such discrimination which could be used for blacks could also be used for females-absent independent nonstatutory reasons-even though that figure is based in part on a discrimination measure (that for unequal pay for comparable work) not legally cognizable for females.
[**]  The court is uncertain as to the role of the sex/experience term in column 3 for black males. However, it is not necessary to define the distinction between columns 2 and 3 for black males considering the similarity in pattern and size. Cf. Testimony of Dr. Janice Madden at 50.
[**]  Of course the model may just be malfunctioning, as by inaccuracy in controlling for productivity. However, as Dr. Madden pointed out with respect to the productivity error, "it has to be that ... [she is] not measuring productivity confined [sic] with time." Testimony of Dr. Janice Madden at 52.
[**]  While the plaintiffs also offer gross pay differentials (that is, pay differentials not adjusted for qualifications), we focus on the two more probative pay differentials offered. We do not decide, however, whether with this sort of cohort analysis gross pay differentials are probative.
[**]  This interpretation of the regression results presented in Plaintiffs' Exhibit 1309 is confirmed by comparing the 1970 column (2) figure for blacks and females with the two corresponding numbers on one of the computer printouts (Plaintiffs' Exhibit 130), at page 6, in column "B" of the table, entitled "Variables in the Equation." None of the symbols for variables listed thereunder appears to designate a sex experience interaction term. On the same page, under a table designated "Variables Not in the Equation," is a variable with the symbol "SEXEXO," which likely designates the sex experience interaction term. (Compare also regressions 9 and 11 for 1969-1972 and regressions 11 and 13 for 1973-78 in the tables in section VII(B), supra.)

It appears that Dr. Madden did perform one set of regressions which included sex/experience interaction terms as well as Hay points. See, e. g., Plaintiffs' Exhibit 127 (Computer Printout, p. 9). However, such set is not legally cognizable for a number of reasons:
(i) Dr. Madden neither testified about this set of regressions, nor mentioned it in the corresponding report (Plaintiffs' Exhibit 1309). Given the complexity of this case, it should not be the function of the court to mine for gems in the set upon set of regressions. We would not have encountered this particular set of regressions but for our decision to look at computer printouts in connection with understanding Plaintiffs' Exhibit 1309. And while it may be a good assumption that the variable designated "SEXEX" on the computer printout in the set is intended to designate a sex/experience interaction term in this set of regressions as well, we have no evidence of this. Moreover, the fact that Dr. Madden did not refer at all this set of regressions which included sex/experience interaction terms, but did so elsewhere in answer to the "age as proxy" problem, is itself warning to the court that perhaps this set is unreliable. Finally, because this set of regressions was not raised in any of the reports relied on by the plaintiffs at trial, or testified to at trial, this set of regressions was not the focal point for adversary contest. This particular set of regressions may have had serious problems peculiar to it. (For instance, in the computer printout designated by Plaintiffs' Exhibit 126, dealing apparently with '69 hires, to include both "SEXEX" and "YRSBANXZ" terms, was impossible due to "F-Level or tolerance-level" problems. See Plaintiffs' Exhibit 126, at 71.) The adversary contest, with experts on both sides, would have been helpful here. See generally section V(C), supra.
(ii) Assuming arguendo that we correctly interpret the "B" column of the table "Variables in the Equation" as the regression coefficients, we would still be left with the task of somehow utilizing the other numbers on that table in the computer printout-and perhaps numbers elsewhere on the printout as well-to determine the statistical significance of the group status coefficients. This we cannot do in the absence of explicit directions for doing so, for reasons of institutional competence, discussed in section V(C), supra, which principle we also invoke in refusing to rework the Bank's overly disaggregated analysis of sections VIII(A)(2)(i) and VIII(A)(2)(iii), infra.
[**]  It appears that these are percentage approximations. See n. 110, supra.
[**]  These results are presented in section VII(B)(1), supra (tables) as well, albeit outlined in different form.
[**]  Dr. Madden also offers an additional econometric reason for downward bias. See Testimony of Dr. Janice Madden at 83.
[**]  Concededly, Dr. Stoikov appears to have testified to the general effects of truncation bias as being the converse of what plaintiffs' experts believe. See Testimony of Dr. Judith Stoikov at 151-52. But she did not testify specifically as to the effect of truncation bias on this particular set of regressions. See n. 101, supra.
[**]  In 1969-72, there was a 17-grade system. See Testimony of Dr. Janice Madden at 81.
[**]  In the period covered by this table, there was a 10-grade system. See Testimony of Dr. Janice Madden at 81.
[**]  Patrick Odell, a Ph.D. in mathematical statistics, is a Professor of Mathematic Sciences and Environmental Sciences at the University of Texas at Dallas who has published over 30 papers in peer review journals. He is a member of a number of technical and scientific societies, including the American Statistical Association and the Texas Academy of Science.
[**]  See section IX(E), infra.
[*]  See section IX(E), infra.
[**]  See section IX(A), infra.
[*]  See section IX(E), infra.
[**]  Because the coefficients from probit regressions are not as easily interpretable as those from ordinary least squares regressions, the table is not reproduced here. See Testimony of Dr. Judith Stoikov at 179.
[*]  Dr. Stoikov also used the probit estimations to derive comparative probabilities of being hired into the professional pay structure for males and females, blacks and nonblacks, assuming various employee characteristics. See Testimony of Dr. Judith Stoikov at 180-82.
[**]  Dr. Odell did not testify at any length as to the nature of his analyses in Defendant's Exhibits 24-27.
[**]  See Defendant's Exhibits 25 and 27.
[**]  See, e. g. section IX(G), infra.
[**]  See, e. g., n.210, infra.
[**]  This was done by attributing all variance in grade or Hay points which could be statistically attributed to either sex or experience to experience.
[**]  Dr. Madden recognized that grades were an ordinal index; someone at a grade level of 10 is not doing a job twice as difficult as someone at grade 5. See Testimony of Dr. Janice Madden at 84. She also recognized that:

[T]he dependent variable in this situation in the variable situation in the 1969 to '72 cases can really only have a unit value of 1 to 17. In other words, 1, 2, 3, 4, 5, 6, on to 17, but it cannot be a value of 10.5, it cannot be a value of 19. It could not be a value of negative .3. The linear regression technique can technically on individual give us predictions which are outside the range of possibility. So the way to correct for this is to use a multinomial logit ....
Id. at 88.
Dr. Madden herself first raised the issue of ordinality. In nevertheless relying on these studies as evidence for placement discrimination, it was implicit that she did not believe that this was an error, or, if an error, one so serious as to affect the direction or significance of her dummy coefficients. As to the other possible problem, Dr. Madden believed that while it would have been preferable to use a logit analysis, it would have been expensive, and appeared to say that the problem would not affect her results when one is studying here, not damages, but the "general effects of race and sex." See Testimony of Dr. Janice Madden at 88-89.
[**]  This is not meant in any way to criticize counsel because the transposition is easily made.
[**]  Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. 702, 708 (1980).
[**]  Dr. Spaulding does appear to have testified that the dependent variable must be distributed normally. See Testimony of Dr. John Spaulding, Jr., at 1062. Even if he is correct, the Bank has not shown how much this violation of the assumptions of ordinary least squares affects the plaintiffs' modeling results.
[**]  The analysis in section VIII(A)(1)(vi), supra, only peripherally deals with discrimination in initial placement/promotion against exempt females: the discrimination experienced by a female holding a nonexempt job in not being initially placed into an exempt position, and discrimination experienced by a female holding a nonexempt job who should have been promoted into an exempt job, are both instances of discrimination against nonexempt females, not against exempt females. Moreover, as will be discussed, infra in the context of the Bank's analyses, the two questions studies at section VIII(A)(1)(vi), supra, are very narrow ones. Finally, the analysis in section VIII(A)(1)(vi), supra, only deals with one year.
[**]  We make the assumption, favorable to the plaintiffs, that the percentage of females hired as exempts in 1969 is equal to the percentage of females hired as exempts in 1978 (approximately 8.8%). See Testimonial Exhibit 10 (Defendant's Exhibit 557).
[**]  Plaintiffs have not argued the applicability of the Fisher v. Procter & Gamble Manufacturing Co., supra n.99, "compelling level" standard; to have done so would have been unjustified. As to female exempts, plaintiffs have not shown, nor attempted to argue, that there was a "compelling level" of underrepresentation in the higher echelons of the female exempt hierarchy. Cf. id. at 543-44. Moreover, the plaintiffs have not attempted to show that Republic is also an employer who has adopted a policy and practice of hiring in at lower level (exempt) jobs and promoting to upper level (exempt) jobs based upon training received and skills developed at the employer itself, other than in isolated programs such as the credit analyst training program. Cf. id. at 544. That is, Procter & Gamble created a pool for promotion trained by itself for the skill level requisite for promotion.
[**]  We do not reach the question of whether the Bank's technique of determining skill-refined availabilities is proper.
[**]  See, e. g., section VIII(A)(1)(vi), supra. We also do not reach the issue of whether the Bank has justified (in legal terms) the use of predictors included in the defendant's regressions. See section VI(E), supra.
[**]  We do not reach the question of whether the Bank's technique for determining skill-refined availabilities is proper.
[**]  The general combinatorial formula for the number of sequences of r events or a particular type which may occur in a group of n total events is:

[**]               Number of                            Number of
              Nonblacks          Positions         Sequences        Probability
                           A     B     C     D
                  4       .75   .75   .75   .75         1               .3164
                  3       .75   .75   .75   .25         4               .4219
                  2       .75   .75   .25   .25         6               .2109
                  1       .75   .25   .25   .25         4               .0469
                  0       .25   .25   .25   .25         1               .0039
                                                                       ______
                                                                       1.0000

[**]  This is known as a "one-tailed" test. Another approach, designated a "two-tailed" test, would be to examine the probability that zero, one, two, eight, nine, or ten females would be hired by chance. This approach would take into account the possibility that extreme results in favor of females may sometimes occur by chance just as extreme results disfavoring females may occur by chance. There is no clear convention in favor of the use of one- or two-tailed tests in employment discrimination litigation. D. Baldus & J. Cole, supra n.55, ง 9.221, at 307-308. See section VI(D)(2), supra.
[**]  See n.58, supra.
[**]  This proposition is stated by Baldus and Cole as follows:

The statistical test can say nothing about causation. The test does, however, provide relevant evidence on which an inference about causation can properly be based. For, if the observed result would occur by chance only 1 time out of 100 in a random system, the occurrence of the result observed in the case before the court would have to be considered a rare event (unless the system were not random), thereby suggesting chance was not the cause. And if bias is the only other plausible explanation for the result, one's belief that the defendant intentionally discriminated will be strengthened.
Second, even if such a result would happen only 1 out of 100 times in a purely random system, a statistical test cannot measure the likelihood that a particular outcome was or was not a chance result. The test of significance states the likelihood of seeing evidence of the type observed over the long run, if the system were effectively blind to color or sex. It simply does not follow that the chances are 99 out of 100 that the result in the instant case was not a chance result, let alone intentionally caused.
D. Baldus & J. Cole, supra n.55, ง 9.42, at 320-21.
[**]  .01 is another commonly used level of significance.
[**]  Technically, the most precise measure of the probability of extreme results is given by a distribution known as the hypergeometric distribution. This distribution takes into account the fact that as a black or female is hired, the availability pool for blacks or females becomes smaller for future hires. Where, as here, the availability pool is much larger than the total number of hires, the binomial distribution provides an excellent approximation for the hypergeometric distribution. See generally G. Wadsworth & J. Bryan, supra, at 55-59.
[**]  Consider, for example, the calculations necessary to determine whether, given a female availability of 50%, 250 female hires out of a total of 1,000 hires is statistically significant. For each number between zero and 250 (and for each number between 750 and 1,000 if a two-tailed test is used), a binomial coefficient must be computed using the formula given in n.146, supra. Each such calculation will require 1,998 multiplications and one division. The binomial coefficients thus obtained must then be multiplied by .5 raised to the 1000th power, in a manner analogous to that used in n.147, supra, to obtain the probability of a given number of female hires. The 251 probabilities thus obtained (or 502 if a two-tailed test is used) must then be summed to obtain the statistical significance level. Thus the calculation of a single significance level, involving sample sizes not unlike those which are often encountered in Title VII class actions, requires no less than 1,004,501 arithmetic steps.
[**]  The normal distribution is the "bell curve" familiar in a variety of contexts.
[**]  The standard deviation is a mathematical measure of the degree to which deviations from the expected value are likely to occur. See Hazelwood School District v. United States, supra n.99, 433 U.S. at 308-09 n.14, 97 S.Ct. at 2742 n.14; Castaneda v. Partida, supra, 430 U.S. at 496-97 n.17, 97 S.Ct. at 1281 n.17. For the binomial distribution and its normal approximation, the standard deviation is given by the formula SD =  NPQ, where N = the number of hires, P = the availability proportion, and Q = 1 - P. See EEOC v. United Virginia Bank, 615 F.2d 147, 151 (4th Cir. 1980); Quigley v. Braniff Airways, Inc., supra n.10, at 81 n.7; G Wadsworth & J. Bryan, supra, at 132; R. Burington, supra, at 256; Comment, Statistics and Title VII Proof: Prima Facie Case and Rebuttal, 15 Hous.L.Rev. 1030, 1051 n.192 (1978).
[**]  A formula for the calculation of Z-scores is given in n.162, infra.
[**]  A negative sign indicates fewer hires from a given racial or sexual group than expected in the absence of discrimination. A positive sign indicates more hires than expected.
[**]  For example, the .01 level of significance corresponds to a Z-score of ฑ 2.57. R. Burington, supra, at 259.
[**]  The use of Z-scores is, however, not without pitfalls. The relationship between Z-scores and statistical significance levels is nonlinear; that is, while a Z-score of -1.96 corresponds to the .05 level of significance, a Z-score twice as large does not correspond to the .025 level. Nor does a Z-score twice as large indicate a result "twice as significant" in a more subjective sense. The Z-score cannot be used meaningfully without reference to the Z-scores corresponding to standard levels of significance.
[**]  Since we are using a two-tailed test, the actual significance level is the probability of zero female hires (.3164) plus the probability of four female hires (.0039). See n.147, supra.
[**]  Dr. England is a post-doctoral research fellow at Duke University, on leave from her position as Assistant Professor of Sociology and Political Economy at the University of Texas at Dallas. She holds a Ph.D. in sociology from the University of Chicago. Among her research interests are women in the labor force and society, the sociology of leisure, and the application of statistics to substantive problems.
[**]  A comparison of the proportion of blacks in the Bank work force with availability of blacks in the Dallas/Fort Worth SMSA and City of Dallas labor forces yielded the following results:



                     Blacks            
            Work        Z          Z
Year        Force     (SMSA)     (City)
1969        .076      -6.30     -14.45
1970        .070      -7.25     -15.48
1972        .089      -4.95     -13.88
1973        .112      -2.07     -12.15
1974        .143      +2.26     - 8.94
1975        .143      +2.36     - 9.33
1976        .138      +1.60     - 9.80
1977        .139      +1.73     - 9.57
1978        .162      +5.00     - 6.86

A similar study for females was not performed. A similar comparison of "officials and managers" produced the following results:


                    Blacks                              Females                    
         Work        Z           Z         Z      Work      Z          Z        Z
Year     Force     (SMSA)      (City)    (RNB)    Force   (SMSA)    (City)    (RNB)
1969     .003      -6.40      - 9.05     -4.73    .003    -13.57    -14.57    -20.12
1970     .003      -6.71      - 9.50     -4.73    .028    -13.32    -14.38    -18.44
1972     .002      -7.73      -10.90     -6.29    .085    -12.80    -14.05    -20.31
1973     .002      -8.09      -11.40     -7.51    .080    -13.62    -14.92    -21.57
1974     .006      -8.25      -11.79     -8.88    .118    -12.57    -13.97    -21.78
1975     .011      -8.33      -12.14     -9.02    .128    -12.76    -14.24    -22.24
1976     .014      -8.23      -12.14     -8.72    .151    -11.79    -13.31    -21.47
1977     .017      -7.95      -11.87     -8.49    .176    -10.47    -11.99    -20.48
1978     .000      -4.20      - 5.88     -4.84    .008    - 8.58    - 9.22    -13.04

A similar comparison of "professional" personnel produced the following results:


                    Blacks                              Females                    
         Work        Z          Z          Z      Work      Z        Z          Z
Year     Force     (SMSA)     (City)     (RNB)    Force   (SMSA)   (City)     (RNB)
1969     .000      -0.85      - 1.19     -0.64    .200    -0.86     -1.01     - 1.73
1970     .000      -1.01      - 1.41     -0.73    .143    -1.33     -1.49     - 2.10
1972     .015      -3.89      - 5.77     -3.01    .201    -4.44     -5.18     - 8.71
1973     .005      -5.10      - 7.27     -4.73    .320    -1.94     -2.88     - 7.18
1974     .019      -4.68      - 7.04     -5.11    .317    -2.10     -3.07     - 8.01
1975     .035      -4.17      - 6.79     -4.66    .355    -1.02     -2.05     - 7.08
1976     .052      -3.29      - 5.97     -3.64    .385    -0.09     -1.09     - 5.99
1977     .039      -3.80      - 6.34     -4.16    .362    -0.77     -1.75     - 6.80
1978     .033      -7.44      -11.98     -8.07    .305    -4.49     -6.24     -15.32

A similar comparison of exempt employees produced the following results:


                    Blacks                              Females                    
         Work        Z           Z         Z      Work      Z        Z          Z
Year     Force     (SMSA)     (City)     (RNB)    Force   (SMSA)   (City)     (RNB)
1973     .007      -9.71      -13.94     -10.14   .149    -13.22    -14.89    -24.03
1974     .017      -9.51      -14.19     -10.46   .180    -12.28    -14.10    -24.12
1975     .022      -9.25      -14.11     -10.24   .205    -11.01    -12.89    -22.68
1976     .020      -9.04      -13.68     - 9.99   .235    - 8.84    -10.66    -20.84
1977     .026      -8.65      -13.46     -10.43   .265    - 7.21    - 9.08    -20.19
1978     .029      -8.47      -13.36     -10.75   .271    - 6.91    - 8.80    -20.27

The data in these tables were derived from the data given in Tables 1-7 of Dr. England's report (Plaintiffs' Exhibit 507). Z-scores in these and


succeeding tables were calculated using the formula , where:
P = the proportion of blacks or females in the availability group under
    comparison;
Q = 1-P;
p = the proportion of blacks or females in the Bank group under comparison;
    and
N = the number of employees in the Bank group under comparison.
  See pp. 5-6 of Dr. England's report. This formula is equivalent to that noted by this court in
  Quigley v. Braniff Airways, Inc., supra n.10, at 81 n.7. See n.155, supra.

[**]  A comparison of the proportions of black or female "officials and managers" and "professionals" at the Bank with similar proportions in all banks filing EEO1 reports is shown below:



                                    Blacks                               
            Officials & Managers                   Professionals         
               EEO1                                    EEO1
Year   RNB    Banks    Difference               RNB   Banks    Difference
1969   .003   .009     -.006                    .000  .019     -.019
1973   .002   .022     -.020*                   .005  .039     -.034*
1975   .011   .029     -.018*                   .035  .052     -.017
* denotes statistically significant result at the .05 level.
                                           Note 163 continued on next page



                                 Females                                           
            Officials & Managers              Professionals        
                    EEO1                              EEO1
  Year   RNB   Banks   Difference          RNB   Banks   Difference
  1969   .003  .121    -.118*             .200   .210    -.010
  1973   .080  .191    -.111*             .320   .262    +.058
  1975   .128  .247    -.119*             .355   .308    +.047
* denotes statistically significant result at the .05 level.
      The data in these tables appear in Tables 11 and 12 of Dr. England's report.

A similar comparison between the proportions of minority and female "officials and managers" and "professional, technical, and sales" personnel at the Bank and at banks in the CEP study is shown below:


                                   Minorities                                        
             Officials & Managers            Prof., Tech., & Sales   
                      CEP                              CEP
  Year     RNB   Banks   Difference         RNB   Banks   Difference
  1971-72  .007  .048    -.041*            .039  .093      -.054*
  1975     .028  .099    -.071*            .083  .168      -.085*
                                     Females                                          
  1971-72  .085  .160    -.075*            .217  .223      -.006
  1975     .128  .262    -.134*            .294  .348      -.105*
* denotes statistically significant result at the .05 level.

The data in these tables appear in Table 9 of Dr. England's report. 1971 72 figures represent 1971 data for the CEP study, 1972 data for the Bank.
[**]  This comparison relates females as a proportion of those hired into exempt positions by the Bank to proportions in the SMSA and City labor forces, to the proportion of females among all Bank employees, and the proportion of females receiving bachelors' degrees:



                        Z         Z         Z             Z
                                                      (Degrees
  Year      Hires     (SMSA)    (City)    (RNB)      Conferred)
  1971      .141      -4.86     -5.46     - 7.64     -5.71
  1972      .157      -5.34     -6.05     - 9.48     -6.34
  1973      .171      -5.06     -5.77     - 9.28     -6.11
  1974      .166      -5.49     -6.24     -10.38     -6.69
  1975      .190      -3.72     -4.31     - 7.37     -4.84
  1976      .253      -2.70     -3.34     - 6.61     -3.95
  1977      .317      -1.60     -2.33     - 6.18     -3.08
  1978      .344      -0.71     -1.23     - 3.92     -2.05

The data in this table were derived from the data given in Table 8 of Dr. England's report, using the formula given in n. 162, supra.
[**]  This comparison relates minorities as a proportion of those interviewed and minorities as a proportion of those hired to minorities as a proportion of all Bank applicants:



                                                   Z                    Z
       Year     Applicants     Interviews     Interviews    Hires     Hires
       1974     .389           .296           - 5.19        .314     -1.90
       1975     .386           .287           -12.09        .247     -6.52
       1976     .453           .320           -14.54        .271     -8.55
       1977     .404           .221           -20.39        .325     -4.12

The data in this table were derived from the data given in Table 13 of Dr. England's report, using the formula given in n.162, supra.
[*]  This comparison technically measures an aggregate of hiring discrimination and initial placement discrimination. That is, an unexplained disparity between the racial or sexual makeup of those in (or hired into) job X and those qualified for job X may be due to one or both of two causes:

(1) Qualified applicants for job X are being turned away due to their race or sex; or
(2) Qualified applicants for job X are being placed in job Y (for which they are overqualified) due to their race or sex.
Case (1) represents hiring discrimination; case (2) represents initial placement discrimination. While with this comparison it is impossible for any given job or group of jobs to disentangle the effects of these two forms of discrimination, an aggregate disparity over all jobs must reflect hiring discrimination, since initial placement discrimination merely shifts new hires from a proper to an improper placement. When separate hiring analyses are performed for exempts and for nonexempts, some vestige of the entanglement remains in the exempt hiring analyses, since initial placement discrimination may shift some new hires out of the exempt category into the nonexempts. When viewed against the totality of hiring in the exempt category, however, this effect is likely to be small.
[**]  In Fisher v. Procter & Gamble Manufacturing Co., supra, n.99, at 543-44, the Fifth Circuit held that in some cases a "compelling level of racial underrepresentation in a sizeable workforce" may establish a prima facie case without adjustment for qualifications. In so holding, the Court relied on several pre-Hazelwood cases. See Jones v. Tri-County Electric Cooperative, Inc., 512 F.2d 1, 2 (5th Cir. 1975); United States v. Hayes International Corp., 456 F.2d 112, 120 (5th Cir. 1972); Parham v. Southwestern Bell Telephone Co., 433 F.2d 421, 426 (8th Cir. 1970). See also Louis v. Pennsylvania Industrial Development Authority, 371 F.Supp. 877, 884-85 (E.D.Pa.), aff'd mem., 505 F.2d 730 (3d Cir. 1974), cert. denied, 420 U.S. 993, 95 S.Ct. 1430, 43 L.Ed.2d 674 (1975). The court emphasized the defendant's "entry level hiring policy" as a "distinctive circumstance" justifying the use of raw population statistics:

When a company adopts a policy and practice of hiring in at low level unskilled jobs and promoting to upper-level positions based upon training received and skills developed at the plant itself, it cannot convincingly challenge the prima facie showing under the Hazelwood "qualifications" dicta.
613 F.2d at 544. The circumstances which led the court in Fisher to accept general workforce comparisons are absent here. With the exception of the credit analyst program, the Bank has no formal training programs. For the credit analyst program, the Bank insists only on those credentials it believes are necessary to perform at the entry level position of credit analyst and to satisfactorily complete the training program; it does not require that entering credit analysts possess the skills required of a commercial loan officer.
[**]  More precisely, the test should be whether required skills which cannot be easily acquired are evenly distributed across racial or sexual lines in the proffered availability pool. If, for example, a given skill is possessed by sexual or racial groups in the same proportion that those groups represent in the availability pool, use of proportions in the pool will provide an accurate proxy for proportions of those qualified. Thus if half of all speakers of French are female, general population statistics (in which roughly half of all members are female) may be used in a case involving the hiring of French translators, notwithstanding the fact that only a small proportion of the population can speak French and the fact that knowledge of French is not easily acquired. Of course, proof that a given skill is evenly distributed may be more difficult to come by than proof of the proportions of sexes or races which possess that skill. This may explain the reliance of the courts on a requirement that the skill be easily acquired or commonly possessed in the population before general population figures may be used.
[**]  Such treatment is appropriate due to the differing legal effect of presuming a job requirement valid or invalid at the prima facie stage. If the plaintiff's statistics are required to overreflect job qualifications, it is possible that her prima facie case will thereby fail and hence that discrimination will go undetected. Undercorrection for job qualifications, on the other hand, merely casts upon the defendant the burden of rebutting the plaintiff's statistics by showing that an apparently discriminatory result is in reality explainable through the use of job-related qualification levels. See Shoben, supra, at 15-16.
[**]  Throughout this opinion, the term "prima facie case" has been used to denote the showing a plaintiff must make in order to survive a motion for judgment under Fed.R.Civ.P. 41(b) and to require the defendant to present defensive evidence. See Cooper v. University of Texas, supra n.58, at 197. Other courts have given the term a more expansive definition, referring to the quantum of proof, both at the close of the plaintiff's case and afterwards, necessary to shift to the defendant the burden of demonstrating that observed disparities are the product of the application of job-related criteria. E. g., EEOC v. Datapoint Corp., 570 F.2d 1264, 1269-70 (5th Cir. 1978); Croker v. Boeing Co., 437 F.Supp. 1138, 1183 (E.D.Pa. 1977). Whether rebuttal taking the form of an attack on the validity of the plaintiff's statistical approach is characterized as a defense to an already established prima facie case, or as a factor preventing a prima facie case from ever coming to existence, is a matter of semantics. Regardless of how such proof is characterized, "[t]he plaintiff's statistics need only be shown incompetent and unreliable to defeat them." Dickerson v. United States Steel Corp., 472 F.Supp. 1304, 1308 (E.D.Pa.1978), vacated on other grounds sub nom. Worthy v. United States Steel Corp., 616 F.2d 698 (3rd Cir. 1980). Accord, Cooper, supra n.58, at 197. If this is done, the burden of showing job-relatedness never arises.
[**]  For example, computer technicians accounted for only 2% of the Bank's nonexempt hires in 1970-78. See n.187, infra.
[**]  66% of 1970-78 nonexempt hires were in the lower two levels of "general clerk" positions, and an additional 6% were in the lower two levels of "clerk/typist" positions. See n.187, infra.
[**]  This analysis is fully consistent with Hester v. Southern Railway Co., supra. In Hester, the only jobs under consideration required a specific level of typing ability, and hence availability figures reflecting that qualification were required. While the Bank's nonexempt jobs include jobs requiring equal or greater typing or other skill than those in Hester, the skills typically required for nonexempt jobs at the Bank are not as specific as those required in Hester.
[**]  The UVB approach also would frustrate suits in which, as here, hiring practices involving a variety of different jobs are simultaneously challenged. The decision is thus more in tune with the Fourth Circuit's restrictive approach to "across-the-board" suits than with the Fifth Circuit's liberal attitude. See section II(A), supra.
[**]  E. g., officials and managers, accountants, securities analysts and economists, and petroleum engineers. This is not to say, of course, that a college degree is necessarily required for these positions, but merely that the range of skills typically required for these positions is roughly comparable to that possessed by those in the labor force with a college degree or greater education.
[**]  Nor have the plaintiffs justified the use of the Bank's overall work force as an availability pool from which exempt employees may be drawn.
[**]  No data were presented for female non-exempt hiring inasmuch as the subclass of female nonexempt applicants was decertified. See section I, supra.
[**]  The following are results of a comparison of the proportions of blacks among the Bank's nonexempt work force with proportions of blacks in the Dallas/Fort Worth SMSA and City of Dallas:



        Total       Black            Difference     Z    Difference     Z
Year  Nonexempts   Nonexempts  Ratio  from SMSA   (SMSA)  from City   (City)
1973     1381        231      .167     +.040     + 4.49    -.055     -4.89
1974     1382        303      .219     +.092     +10.30    -.003     -0.25
1975     1547        324      .209     +.082     + 9.74    -.013     -1.19
1976     1559        309      .198     +.071     + 8.44    -.024     -2.26
1977     1482        299      .202     +.075     + 8.64    -.020     -1.88
1978     1431        341      .238     +.111     +12.61    +.016     +1.48

A comparison with the overall Bank work force, cf. nn. 162, 164, supra, is not shown, since the assumption on which such a comparison would be based (i. e., that the overall work force is a potential availability pool) is meaningless in the case of nonexempt hiring.
[**]  The Z-score for 1976 (-2.26), corresponding to a significance level of .024, R. Burington, supra, at 259, is hard to interpret. Falling as it does, however, among nonsignificant figures and near a year (1978) in which overutilization was present, it is not likely that that figure represents an isolated year in which discrimination was present.
[**]  Aggregation of these two groups by the court is not possible due to the fact that Dr. England computed significance levels for this study using the binomial distribution rather than the more easily computed Z-scores based on the normal distribution. Nor would an aggregation of "officials and managers" and "Professionals" correspond precisely to the Bank's exempt workforce.
[**]  EEO1 reports must be filed by all employers, including banks, having 100 or more employees. 29 C.F.R. ง 1602.7. Hence Republic, which had between 1600 and 2400 employees during the relevant period and is one of the nation's largest banks, is substantially larger than most banks filing EEO1 reports, the EEO1 data will include data from banks engaged primarily in retail lending and from banks in branch banking states. Each of these factors means that the skill mix at EEO1 banks may be substantially different from that required by the job mix at Republic.
[**]  While the plaintiffs argue that the relevant comparison is with city figures, which show more minorities than the CEP SMSAs, the CEP itself regarded SMSA data as "the most accurate available base for labor force data" for the banks it studied. T. Simcich, supra, at 169. This conclusion coincides with testimony that Republic recruits its exempt work force from a labor market as broad or broader than the Dallas/Fort Worth SMSA.
[**]  Defendant also relies on statistical studies performed by Dr. Odell. Although Dr. Odell's methodology differs from Dr. Stoikov's in several important respects, his results are largely duplicative of Dr. Stoikov's. For this reason, and because Dr. Stoikov's analysis is in certain respects more comprehensive than Dr. Odell's, Dr. Stoikov's results will be referred to exclusively throughout this and succeeding sections. Dr. Odell's results have been briefly discussed in connection with the placement/promotion analysis. See section VIII(A)(2)(iii), supra.
[**]  Those hired within one year of receiving a degree were presumed to have been hired on the basis of that degree, while others were presumed to have been hired on the basis of their prior work experience.
[**]  The job families as grouped by Dr. Stoikov are as follows:



Group      Description
EXEMPT:
Officials and Managers:
A         Executives; Department Managers; Staff Division Managers
B         Accounting Managers
C         Securities Managers; Marketing Managers
D         Commercial Loan Managers
E         Data Processing Managers
F         Trust Managers
G         Internal Services Managers-Legal
H         Operating Services Managers
Professionals-Accountants:
I         Accountant-Junior
J         Accounting Analyst
K         Accountant-Senior; Accounting Supervisor
Professionals-Securities Analysts and Economists:
L         Stock Analyst; Bond Analyst; Economic Analyst; International Trader
Professionals-Marketing:
M         Marketing Analyst
N         Technical Representative
O         Retail Representative-Junior
P         Retail/Corporate Representative-Senior
Professionals-Commercial Loan Analysts:
Q         Statement Analyst
R         Credit Analyst
S         Unit Administrator
T         Analytical Section Administrator
Professionals-Commercial Loan Representatives:
U         Loan Representative; Loan Officer-B
V         Loan Officer-A; Loan Officer (Assistant Vice President)
W         Loan Vice President-Officer B; Loan Vice President-Officer A/Manager B
Professionals-Data Processing:
X         Computer Analyst; Programmer
Y         Programmer-Analyst; Programmer-Analyst-Senior
Z         Lead Programmer-Analyst; Senior Project Manager; Manager-System
               Section
Professionals-Trust Representatives:
AA        Trust Administrator-Assistant; Trust Administrator
BB        Trust Administrator-Senior; Trust Administrator-Supervisor
Professionals-Consumer Loan Representatives:
CC        Loan Administrator
Professionals-Internal Services:
DD        Personnel
EE        Security
FF        Corporate Planning
GG        Facilities
HH        Public Relations
II        Systems & Methods Planner
Professionals-Operative Services:
JJ        Operating Services Administrator
KK        Operating Services Administrator-Senior
LL        Operating Services Administrator-Supervisor
                                           Note 185 continued on next page



Professionals-Engineers:
MM        Petroleum Engineer
NONEXEMPT:
Computer Technicians:
NN        Junior Computer Operator
OO        Computer Operator; Senior Computer Operator; Lead Computer
               Operator
PP        Production Coordinator
Office and Clerical-Teller:
QQ        Checking/Savings Teller; Note Teller; Check Order/Hold Debit
               Teller; Collateral Teller; Currency Control Teller; Foreign
               Exchange Teller
RR        Lead Teller; Senior Teller; Lead Receiving Teller; New Accounts
               Representative; Senior New Accounts Representative; Oil Loan
               Teller; Senior Oil Loan Teller; Security Teller; Tax-RR-Customer
               Note Teller
Office and Clerical-Accounting Clerk:
SS        Accounting Clerk-Junior; Accounting Clerk-Senior; Lead Accounting
               Clerk
TT        Accounting Clerk-Supervisor
Office and Clerical-General Clerk:
UU        General Clerk-Junior; General Clerk-Senior
VV        Lead General Clerk; General Clerk-Supervisor
Office and Clerical-Clerk/Typist:
WW        Clerk/Typist-Junior
XX        Clerk/Typist-Senior
YY        Lead Clerk/Typist; Clerk/Typist-Supervisor
Office and Clerical-Secretary:
ZZ        Secretary-Junior; Secretary-Senior; Administrative Secretary;
               Executive Secretary
Office and Clerical-Business Machine Operator:
AAA       Keypunch Operator-Junior; Supervisor-Keypunch Operator
Service Workers-Security Guards:
BBB       Security Guard; Console Operator; Senior Security Guard; Supervisor-Security
               Guards
This table is derived from Defendant's Exhibit 1.

[**]  The randomness range was computed using the binomial distribution except where the number of hires equalled or exceeded thirty and the availability equalled or exceeded 5%. In those cases, the normal distribution was used.
[**]  This table shows the availability, randomness range, and number of hires for 1970-74:



Job                            Blacks                           Females         
Family    Total              Randomness   No.                 Randomness   No.
Group     Hires      Avail.  Range        Hires      Avail.   Range        Hires
exempt:
  A         1        .0098      0-1         0        .1724      0-1         0
  B         4        .0116      0-1         0        .2526      0-4         0
  C         4        .0098      0-1         0        .1994      0-3         0
  D        14        .0131      0-2         0        .2299      0-7         0
  E         2        .0285      0-2         0        .0736      0-2         0
  F         2        .0131      0-1         0        .2299      0-2         0
  G         1        .0234      0-1         0        .0537      0-1         1
  H         3        .0171      0-2         0        .1924      0-3         0
  I        59        .0405      0-11        1        .1979      6-18       14
  J        18        .0128      0-2         0        .2360      0-9         5
  K         8        .0099      0-2         0        .2565      0-5         1
  L        44        .0205      0-4         0        .1510      2-11        4
  M         3        .0367      0-2         0        .1522      0-3         1
  N        17        .0492      0-4         0        .2892      1-10       15#
  O         4        .0267      0-2         0        .3378      0-4         1
  P        21        .0176      0-3         0        .3122      2-12        3
  Q         3        .0304      0-2         0        .1844      0-3         2
  R       107        .0405      0-13        3        .1416      8-22        8*
  S         3        .0301      0-2         0        .1477      0-3         0
  T         1        .0033      0-1         0        .2006      0-1         1
  U         7        .0175      0-2         0        .1985      0-4         2
  V         9        .0056      0-2         0        .1843      0-5         0
  W         5        .0098      0-1         0        .2305      0-4         0
  X        15        .0016      0-1         0        .1565      0-6         3
  Y        63        .0078      0-5         1        .1437      4-15        8
  Z         3        .0278      0-2         0        .0649      0-2         0
  AA       37        .0352      0-4         0        .1701      2-11        7
  BB        6        .0019      0-1         0        .1767      0-4         0
  CC        5        .0262      0-2         0        .1860      0-4         0
  DD        6        .0190      0-2         0        .1983      0-4         2
  EE        3        .0244      0-2         0        .0100      0-1         0
                                             Note 187 continued on next page



  FF       0
  GG       1         .0210      0-1        0         .2052       0-1         0
  HH       2         .0072      0-1        0         .3396       0-2         0
  II       6         .0248      0-2        0         .0686       0-3         0
  JJ      44         .0322      0-5        3         .2490       5-17        5*
  KK       6         .0260      0-2        0         .2210       0-4         0
  LL       2         .0044      0-1        0         .1898       0-2         0
  MM       3         .0000      0-3        0         .0147       0-1         0
nonexempt:
  NN       2         .0093      0-2        1         .6768       0-2         0
  OO      38         .0180      0-3        3         .2263       4-14        5
  PP       2         .0111      0-1        1         .1562       0-2         0
  QQ      65         .0380      0-10       2         .9518      58-65       53*
  RR      11         .0380      0-3        1         .9518       8-11       10
  SS     215         .0285      1-11      24#        .8530     173-194     202#
  TT       2         .0279      0-2        0         .8463       0-2         0
  UU    1880         .1065    174-226    417#        .7074    1291-1369   1487#
  VV      31         .0469      0-5        0         .8526      23-30        3*
  WW     121         .0762      4-15      27#        .9660     113-121     120
  XX      42         .0765      0-8        3         .9637      38-43       41
  YY       1         .0740      0-1        0         .9481       0-1         1
  ZZ     261         .0317      3-14       4         .9817     252-260     261#
  AAA    104         .0564      1-10      31#        .9216      90-101     101
  BBB     35         .0322      0-4        8#        .0071       0-2         4#
* denotes underutilization statistically significant at the .05 level
# denotes overutilization statistically significant at the .05 level

The data in this table were taken from Tables 2 and 3 of Dr. Stoikov's "testimonial exhibits" (Defendant's Exhibit 557).


                  The following table shows similar results for 1975-78:
Job                            Blacks                           Females
Family    Total              Randomness   No.                 Randomness   No.
Group     Hires      Avail.  Range        Hires      Avail.   Range        Hires
exempt:
  A        3         .0219      0-2        0         .2369       0-3         0
  B        5         .0196      0-2        0         .3338       0-4         0
  C        2         .0202      0-1        0         .3332       0-2         0
  D        4         .0269      0-2        0         .3110       0-4         0
  E        2         .0194      0-1        0         .1043       0-2         0
  F        0
  G        1         .0145      0-1        0         .0746       0-1         0
  H        1         .0274      0-1        0         .2640       0-1         0
  I       51         .0390      0-6        5         .2677       7-20       15
  J       24         .0115      0-2        0         .3111       3-12        9
  K       12         .0080      0-2        0         .3298       0-8         3
  L       19         .0239      0-3        0         .2523       1-9         3
  M        8         .0310      0-2        0         .3110       0-6         4
  N       21         .0355      0-3        0         .3433       3-12       14#
                                                Note 187 continued on next page



  O        8         .0252      0-2        0         .4113       0-7         3
  P       14         .0181      0-2        0         .3787       1-10        2
  Q        2         .0324      0-2        1         .2938       0-2         0
  R       74         .0404      0-11       7         .2017       8-22       25#
  S        6         .0211      0-2        0         .2087       0-4         0
  T        0
  U        7         .0191      0-2        0         .2639       0-5         3
  V       11         .0100      0-2        0         .3318       0-8         2
  W        8         .0202      0-2        0         .3130       0-6         0
  X       10         .0013      0-1        1         .2521       0-6         3
  Y       44         .0027      0-4        0         .2040       4-14        9
  Z       11         .0212      0-2        0         .0790       0-4         0
  AA      29         .0356      0-4        2         .2320       2-12       13#
  BB       5         .0040      0-1        0         .2405       0-4         0
  CC       2         .0322      0-2        0         .2562       0-2         0
  DD      10         .0149      0-2        1         .2748       0-6         6
  EE       0
  FF       2         .0280      0-2        0         .2170       0-2         1
  GG       1         .0126      0-1        0         .2750       0-1         1
  HH       2         .0072      0-1        0         .2191       0-2         1
  II       5         .0181      0-2        1         .0890       0-3         1
  JJ      41         .0298      0-4        2         .3233       7-19       17
  KK       6         .0208      0-2        0         .2873       0-5         2
  LL       2         .0090      0-1        0         .3428       0-2         0
  MM       3         .0067      0-1        0         .0147       0-1         0
nonexempt:
  NN       9         .1165      0-4        0         .7195       3-9         0*
  OO      42         .0111      0-6        2         .2791       6-17        1*
  PP       7         .0180      0-2        1         .1954       0-4         1
  QQ      65         .0695      0-10       4         .9325      57-65       59
  RR       8         .0695      0-3        1         .9325       5-8         8
  SS      76         .0344      0-10       9         .9294      66-75       66*
  TT       1         .0337      0-1        0         .9220       0-1         0
  UU    1413         .1291    158-207    468#        .7515    1030-1094   1081
  VV      19         .0567      0-4        2         .9288      14-19        6*
  WW      95         .0920      3-14      31#        .9675      89-95       93
  XX      39         .0924      0-8       12#        .9652      35-40       39
  YY       1         .0893      0-1        0         .9496       0-1         1
  ZZ     281         .0382      4-17      12         .9832     272-280     281#
  AAA     61         .0681      0-9       22#        .9231      52-60       59
  BBB     42         .0332      0-5       11#        .0104       0-6         3
* denotes underutilization statistically significant at the .05 level
# denotes overutilization statistically significant at the .05 level

The data in this table were taken from Tables 4 and 5 of the testimonial exhibits (Defendant's Exhibit 557).
[**]  Job families were aggregated into the following groups:



     Group       Job Families
     A-H         Officials and Managers
     I-K         Accountants
     L-P         Securities Analysts and Economists; Marketing
     Q-W         Commercial Loan Analysts; Commercial Loan Representatives
     X-Z         Data Processing
     AA-CC       Trust Representatives; Consumer Loan Representatives
     DD-II       Internal Services
     JJ-MM       Operating Services; Petroleum Engineers
     NN-PP       Computer Technicians
     QQ-TT       Teller; Accounting Clerk
     UU-VV       General Clerk
     WW-AAA      Clerk/Typist; Secretary; Business Machine Operator
     BBB         Security Guards

[**]  This table shows the availability, randomness range, and number of hires for 1970 74:



Job                               Blacks                         Females         
Family        Total            Randomness    No.              Randomness   No.
Group         Hires    Avail.     Range      Hires     Avail.    Range     Hires
exempt:
A-H            31      .0141      0- 3         0       .2076      2-11       1*
I-K            85      .0318      0-10         1       .2115     11-25      20
L-P            89      .0261      0-10         0       .2239     12-28      24
Q-W           135      .0351      0-14         3       .1522     12-29      13
X-Z            81      .0074      0- 6         1       .1432      5-18      11
AA-CC          48      .0301      0- 5         0       .1726      3-13       7
DD-II          18      .0206      0- 3         0       .1398      0- 6       2
JJ-MM          55      .0288      0- 9         3       .2310      7-19       5*
nonexempt:
NN-PP          42      .0215      0- 4         5#      .2444      5-16       5
QQ-TT         293      .0310      3-15        27#      .8786    246-268    265
UU-VV        1911      .1055    175-228      417#      .7098   1318-1395  1490#
WW-AAA        529      .0504     17-37        65#      .9648    502-519    524#
BBB            35      .0322      0- 4         8#      .0071      0- 2       4#
The following table shows similar results for 1975-78:
exempt:
A-H            18      .0218     0- 3         0       .2687      1- 9         0*
I-K            87      .0271     0-10         5       .2882     17-33        27
L-P            70      .0272     0- 9         0       .3298     15-31        26
Q-W           108      .0332     0-12         8       .2293     16-33        30
X-Z            65      .0056     0- 5         1       .1902      6-19        12
AA-CC          36      .0310     0- 4         2       .2345      3-13        13
DD-II          20      .0161     0- 2         2       .2170      0- 9        10#
JJ-MM          52      .0266     0- 9         2       .3021      9-22        19
nonexempt:
NN-PP          58      .0283     0- 9         3       .3373     13-27         2*
QQ-TT         150      .0515     2-13        14#      .9309    134-146      133*
UU-VV        1432      .1281   159-208      470#      .7539   1048-1112    1087
WW-AAA        477      .0573    17-37        77#      .9708    456-470      473#
BBB            42      .0332     0- 5        11#      .0104      0- 6         3
* denotes underutilization statistically significant at the .05 level
# denotes overutilization statistically significant at the .05 level
  The data in these tables were taken from Tables 6 9 of the testimonial exhibits (Defendant's
  Exhibit 557).

[**]  The question of what the Bank must show to establish job-relatedness (e. g., "business purpose," "business necessity," or other standards) is treated in sections VI(B) and (E), supra, and will not be discussed again here. It is instead the purpose of the present section to discuss the type of evidence which may be used to meet the appropriate standard of proof. Likewise, no mention is made in this section of proof of the presence or absence of alternative selection procedures with lesser disparate impact, since that topic relates primarily to the question of what must be shown rather than how it must be shown.
[**]  See n. 190, supra.
[**]  In addition to this difficulty, the fact that such employees are often engaged in team efforts, wherein the marginal productivity of any one individual is difficult to measure, and the fact that such employees are often hired not only for their ability to perform in entry-level positions but also for their potential to advance to top-level positions, also render traditional validation difficult. Gwartney, Asher, Haworth, & Haworth, supra n.40, at 642-43 & n.20.
[**]  "[A] content strategy is not appropriate for demonstrating the validity of selection procedures which purport to measure traits or constructs, such as intelligence, aptitude, personality, common sense, judgment, leadership, and spatial ability." 29 C.F.R. ง 1607.14(C)(1), quoted in Hunt & Pazuniak, supra, at 128.
[**]  "There are circumstances in which a user (of a selection procedure) cannot or need not utilize the validation techniques contemplated by these guidelines. In such circumstances, the user should utilize selection procedures which are as job related as possible and which will minimize or eliminate adverse impact, as set forth below." 29 C.F.R. ง 1607.6(B). "When an informal or unscored selection procedure which has an adverse impact is utilized, the user should eliminate the adverse impact, or modify the procedure to one which is a formal, scored or quantified measure or combination of measures and then validate the procedure in accord with these guidelines, or otherwise justify continued use of the procedure in accord with Federal law." 29 C.F.R. ง 1607.6(B) (emphasis supplied). The Department of Justice has reaffirmed the availability of alternate procedures for showing job-relatedness: "12. Q. How can users justify continued use in accord with federal law of a procedure which has an adverse impact and which is not feasible or appropriate to validate? A. That subject is one to which the Guidelines are not addressed. In Griggs v. Duke Power Co., 401 U.S. 424, 91 S.Ct. 849, 28 L.Ed.2d 158, the Supreme Court indicated that the burden on the user was a heavy one, but that the selection procedure could be used if there was a "business necessity" for its continued use. The federal agencies will consider evidence which shows "business necessity" to justify continued use of a selection procedure." United States Department of Justice, Questions and Answers on the Federal Executive Agency Guidelines on Employee Selection Procedures, 8 Lab.Rel.Rep. 771, 774-5 (1977), quoted in Hunt & Pazuniak, supra, at 127 n.43.
[**]  Indeed, a few lower courts have upheld degree requirements despite little or no job-relatedness evidence. See EEOC v. Georgia-Pacific Corp., 456 F.Supp. 1227, 1237 (N.D.Miss. 1977) (foresters "of necessity" must have forestry degrees); Keyes v. Lenoir Rhyne College, 15 Fair Empl.Prac.Cas. 914, 923 (W.D.N.C. 1976), aff'd, 552 F.2d 579 (4th Cir.), cert. denied, 434 U.S. 904, 98 S.Ct. 300, 54 L.Ed.2d 190 (1977) (Ph.D. "presumably evidence" of job-related skills and knowledge); Arnold v. Ballard, 390 F.Supp. 723, 738 (N.D.Ohio 1975). See also Lerner, Washington v. Davis: Quantity, Quality and Equality in Employment Testing, 1976 Sup.Ct.Rev. 263, 306 (1977) (advocating that validation not be required for educational requirements for professional jobs).
[**]  The Tenth Circuit has advanced a further reason for relaxing proof requirements in this context, suggesting that the employer's burden of showing job-relatedness should diminish as the degree of skill and the economic and human risks of hiring an unqualified applicant increase. Spurlock v. United Airlines, Inc., supra, at 219.
[**]  It is this give-and-take which mandates the rejection of Dr. Madden's argument that availability should be measured by choosing the one credential used by the Bank for each job family which gives the highest female or black availability. The fact that one math major, who may have been first in her class, taken several business-related courses, and had other outstanding credentials, was hired as a credit analyst, see n.212, infra, does not mean that all math majors are qualified for that position.
[**]  A more general case of hiring decision-making would occur if the employer sought to justify a hiring "equation"; that is, if the employer used โ consciously or unconsciously โ a formula whereby attainment of, e. g., 100 points were necessary for a positive hiring decision. He might assign 30 points to possession of a particular degree, 50 points to graduation in a particular field of study, 50 points to experience of a certain sort, and 90 points for having good sales ability. In such a case, as to the predictors with disparate impact, the employer would have to show that a race- and sex-blind employer would weigh those predictors similarly. See section VI(E), supra.
[*]  This concept has been discussed in the contexts of econometric modeling and job-relating a predictor to the extent an employer weights that predictor, as "proper treatment across twins." See sections VI(B) and VI(E), supra.
[**]  For example, a law firm could justify an absolute requirement that hires for attorney positions possess law degrees, since all attorneys must have such a degree in order to be considered qualified under state law.
[*]  For example, a university might justify a practice of hiring 60% physicists and 40% chemists into positions as physical chemistry professors by showing that the pool of qualified physical chemistry professors is made up 60% of physicists and 40% of chemists.
[**]  For example, a professional football team could justify a practice of occasionally hiring a college baseball player by showing that a few qualified football players have backgrounds in college baseball rather than football.
[*]  The theoretical underpinnings of the use of the employer's hiring history in determining weights for a weighted average availability are rather complex. The only assumption required for such use, however, is that within groups possessing a given credential used for hiring, members of different sexes or races are equally likely to be qualified. Consider, for example, the example of physical chemistry professors given in n.201, supra. The pool of all physicists and all chemists may be diagrammed as follows:

We are given the availability of females among physicists, ap = FP/P, and among chemists, ac = FC/C, and want to know the availability of females among those qualified, that is, QF/Q. The employer must produce job-relatedness evidence which shows that the proportion of the employer's hires which are physicists (symbolized hp, 60% in our example) and the proportion which are chemists (symbolized hc, 40% in the example) reflect the respective proportions which qualified physicists and chemists represent among all those qualified. In mathematical terms, he must show that hp = QP/Q and hc = QC/Q.
Applying some elementary algebra:
QF/Q = QFP/Q + QFC/Q (qualified
       females are either physicists
       or chemists)
     = (QFP/QP) (QP/Q) + (QFC/QC)
       (QC/Q) (algebraic manipulation)
     = (QFP/QP) hp + (QFC/QC) hc
       (from the job-relatedness
       requirement above)
We make the assumption that a female physicist (or chemist) is as likely to be qualified as a male physicist (or chemist). I. e., QFP/QP = FP/P, QFC/QC = FC/C (females are qualified among physicists or chemists in proportion to their numbers).
QF/Q = (QFP/QP) hp + (QFC/QC) hc
       (as above)
     = (FP/P) hp + (FC/C) hc (based
       on the above assumption)
     = aphp + achc (from the
       definitions of ap and ac)
Thus we see that the availability of female physical chemistry professors (QF/Q) equals the availability of female physicists (ap) weighted by the proportion of hires which were physicists (hp, or 60%) plus the availability of female chemists (ac) weighted by the proportion of hires which were chemists (hc, or 40%). The same procedure can be used to obtain weighted average black availabilities.
[**]  See, e. g., Movement v. GMC, supra (weighted average of SMSA counties, based on employees); United States v. Ironworkers Local 86, 443 F.2d 544, 551 & n.19 (9th Cir.), cert. denied, 404 U.S. 984, 92 S.Ct. 447, 30 L.Ed.2d 367 (1971) (city data); Gay v. Waiters' and Dairy Lunchmen's Union, 489 F.Supp. 282, 302-03 (N.D.Cal.1980) (weighted average, based on applications, between central city and balance of SMSA); Greenspan v. Automobile Club of Michigan, 495 F.Supp. 1021, 22 Empl. Prac.Dec. ถ 30,812, at 15,166 (E.D.Mich.1980) (statewide and national data); Baker v. City of Detroit, 483 F.Supp. 930, 959 (E.D.Mich.1979) (greater weight given to city over SMSA based on applications); Drayton v. City of St. Petersburg, 477 F.Supp. 846, 857-58 (M.D.Fla.1979) (SMSA data, based on employees and applications); Louisville Black Police Officers Organization, Inc. v. City of Louisville, 21 Empl.Prac. Dec. ถ 30,330 (W.D.Ky. 1979) (weighted average, based on applications, between city, county, and SMSA, adjusted for various factors); EEOC v. E. I. du Pont de Nemours & Co., supra, at 236-38 (SMSA data; use of weighted average of concentric circles around defendant's plant rejected for failure to consider patterns of social and economic integration); Smith v. Union Oil Co., 17 Empl.Prac.Dec. ถ 8,411 (N.D.Cal.1977) (SMSA, with higher weight for central city); Domingo v. New England Fish Co., 445 F.Supp. 421, 431-33 (W.D. Wash.1977) (weighted three-state average based on hiring rejected in seasonal industry with isolated plants; industry-wide data used); League of United Latin American Citizens v. City of Santa Ana, supra, at 896-97 (city data used despite contrary hiring pattern). See generally Gastwirth & Haber, Defining the Labor Market for Equal Employment Standards, 99 Monthly Lab.Rev. 32 (1976) (advocating weighted average approach using residences of current employees).
[**]  Since the geographical regions which naturally suggest themselves for availability analysis, and for which data tend to be available, are not mutually exclusive and often tend to be concentric, considerable caution must be employed in choosing appropriate regions. For example, an employer who hires exclusively from the central city can with equal correctness state (1) that his entire work force comes from the city; (2) that it comes from the county; (3) that it comes from the SMSA; (4) the state; (5) the region; and so forth. These markets may have widely differing availabilities of blacks and females. The employer must be required to use the region which most accurately describes the relevant geographical market (or some sort of weighted average, see the cases cited in n.204, supra) and for which adequate statistical data are available.
[**]       Job Family       Occupational or educational qualification and weight
        Group          (with geographical weights)                              
      NN           93.3%     wide aggregation of clerical job codes, including
                               inexperienced unemployed (พ DFW, ผ Texas)
                    6.7%     Business-related bachelor's degree
      OO           97.4%     Computer and peripheral equipment operators
                               (75% DFW, 15% Texas, 10% US)
                    1.3%     Business-related bachelor's degree
                    1.3%     Mathematics bachelor's degree
       PP          59.3%     Computer specialists and computer and peripheral
                               equipment operators (85% DFW, 15% US)
                   29.6%     Managers and administrators (85% DFW, 15% US)
                   11.1%     Bachelor's degree (any major)
      QQ          100.0%     Bank tellers (พ DFW, ผ Texas)
      RR          100.0%     Bank tellers (พ DFW, ผ Texas)
      SS          100.0%     Bookkeepers (DFW Market)
      TT          100.0%     Bookkeepers and clerical supervisors (DFW market)
      UU          100.0%     wide aggregation of clerical job codes, including
                             inexperienced unemployed (DFW market)
      VV          100.0%     aggregation of clerical job codes (DFW market)
      WW          100.0%     Receptionists and typists (DFW market)
      XX          100.0%     Typists (DFW market)
      YY          100.0%     Clerical supervisors and typists (DFW market)
      ZZ          100.0%     Secretaries and stenographers (DFW market)
      AAA         100.0%     Key punch operators (DFW market)
      BBB         100.0%     Protective service workers (DFW market)

"DFW" refers to the Dallas/Fort Worth SMSA. Occupational categories refer to Census Bureau occupational classifications. See Bureau of the Census, U.S. Department of Commerce, Classified Index of Industries and Occupations X-XIV (1971) (Defendant's Exhibit 558). The weights in this table are taken from Appendix B to Dr. Stoikov's report (Defendant's Exhibit 33).
[**]  97.4% of the Bank's hires into family OO held that credential. See n.206, supra.
[**]  It should be noted that, while occupational and educational weights for exempt and technical job families were based on the actual qualifications possessed by those hired, the 100% weights in clerical and service job families were based on an unverified assumption that only tellers can be tellers, typists can be typists, and so forth. Acceptance of comparisons based on this assumption would allow a defendant to discriminate so long as its discrimination did not exceed that engaged in by society as a whole.
[**]  Of course, the Bank is entitled to exclude from the availability pool those who, while qualified for a position, are not likely to apply for that position due to their overqualification. Thus while lawyers presumably possess the skills necessary to be janitors, the availability pool for janitorial jobs need not include those presently employed as lawyers. This factor could not, however, justify the restriction of the availability pool to only those now employed in janitorial positions, since some people presently otherwise employed might well be interested in such positions. While Dr. Stoikov states (at page 5 of her report, Defendant's Exhibit 33) that "occupational qualifications were interpreted broadly to encompass all employed and unemployed persons having comparable levels of skill," an examination of the narrowness of fit between census occupational classifications and most job families belies this assertion.
[**]  That the Z-score (or, equivalently, the randomness range) is sensitive to fragmentation of the population can be seen from the presence of N in the denominator of the fraction given in n.162, supra. For example, when a population of size N is broken into two populations of size N/2, the Z-score is reduced by 29%, without any corresponding change in the underlying disparities. Thus if an employer hires 16 females out of 100 hires, where female availability is 25%, statistical significance present when the group is taken as a whole (Z = -2.08) disappears when two subgroups of 50 are considered (Z = -1.47). See generally D. Baldus & J. Cole, supra n.55, ง 9.221, at 309-10.
[**]                                        1970-74   1975-78
     weighted average availability       .1845     .2601
     total hires                           542       456
     total female hires                     83       137
     females as proportion of hires      .1531     .3004
     standard deviation                  9.030     9.368
     Z                                 -1.8817   +1.9626
     significance level                  .0600     .0497

[**]  The weights used by Dr. Stoikov for exempt availabilities are as follows:



Job
Family      Occupational or educational qualification and weight
Group       (with geographical weights)                                 
  A          75.0%    Bank officers & financial managers (US market)
             25.0%    Petroleum engineers (US market)
  B         100.0%    Accountants (1/2 DFW, 1/2 US)
  C         100.0%    Bank officers & financial managers (1/4 Texas,
                        3/4 US)
  D         100.0%    Bank officers & financial managers (US market)
  E         100.0%    Computer systems analysts & computer specialists
                        (60% DFW, 20% Texas, 20% US)
  F         100.0%    Bank officers & financial managers (US market)
  G         100.0%    Lawyers (US market)
  H          75.0%    Bank officers & financial managers (US market)
             25.0%    MBA degree
  I          42.1%    Accountants (90% DFW, 10% Texas)
             49.8%    Business-related bachelor's degree
              8.1%    MBA degree
  J          75.0%    Accountants (80% DFW, 10% Texas, 10% US)
             10.7%    Bank officers & financial managers (80% DFW,
                        10% Texas, 10% US)
              2.4%    Business-related bachelor's degree
              8.3%    MBA degree
              3.6%    Law degree
  K          95.0%    Accountants (80% DFW, 10% Texas, 10% US)
              5.0%    MBA degree
  L          52.1%    Bank officers & financial managers (1/3 DFW,
                        1/3 Texas, 1/3 US)
             15.6%    Economists & stock and bond salesmen (3/4 DFW,
                        1/4 US)
             19.0%    Business-related bachelor's degree
              1.6%    Mathematics bachelor's degree
             12.7%    MBA degree
  M          63.6%    Public relations men and publicity writers &
                        advertising agents and salesmen (1/2 DFW,
                        1/4 Texas, 1/4 US)
             27.3%    Business-related bachelor's degree
              9.1%    MBA degree
  N          19.7%    Bank officers & financial managers (90% DFW,
                        10% Texas)
             39.4%    Salesmen and sales clerks (90% DFW, 10% Texas)
             20.5%    Business-related bachelor's degree
              3.0%    Mathematics bachelor's degree
             14.4%    Bachelor's degree (any major)
              3.0%    Master's degree (other than MBA or computer
                        science)
  O          16.7%    Salesmen of services and construction (5/6 DFW,
                        1/6 US)
             25.0%    Bank officers & financial managers (5/6 DFW,
                        1/6 US)
             50.0%    Professional, Technical, and Kindred workers
                        generally (5/6 DFW, 1/6 US)
              8.3%    Clerical and kindred workers generally (5/6 DFW,
                        1/6 US)
                                          Note 212 continued on next page



  P          33.3%    Salesmen of services and construction (70% DFW,
                        13.3% Texas, 16.7% US)
             23.3%    Bank officers & financial managers (70% DFW,
                        13.3% Texas, 16.7% US)
             43.3%    Professional, Technical, and Kindred workers
                        generally (70% DFW, 13.3% Texas, 16.7% US)
  Q          60.0%    Bank officers & financial managers (1/2 DFW,
                        1/4 Texas, 1/4 US)
             40.0%    Business-related bachelor's degree
  R           9.2%    Bank officers & financial managers (1/2 DFW,
                        1/4 Texas, 1/4 US)
              9.2%    Professional, technical, and kindred workers
                        generally (1/2 DFW, 1/4 Texas, 1/4 US)
             29.0%    Business-related bachelor's degree
              0.6%    Mathematics bachelor's degree
             51.3%    MBA degree
              0.6%    Law degree
  S          35.6%    Assessors, controllers, and treasurers; local
                        public administration (80% DFW, 20% US)
             35.6%    Inspectors, except construction, public
                      administration (80% DFW, 20% US)
             17.7%    Real estate agents and brokers (80% DFW, 20% US)
             11.1%    Business-related bachelor's degree
  T         100.0%    Bank officers & financial managers (1/2 DFW,
                        1/4 Texas, 1/4 US)
  U          58.3%    Bank officers & financial managers (80% DFW, 20% US)
              8.4%    Bachelor's degree (any major)
             33.3%    MBA degree
  V          95.0%    Bank officers & financial managers (1/3 DFW,
                        1/3 Texas, 1/3 US)
              5.0%    MBA degree
  W         100.0%    Bank officers & financial managers (1/4 DFW,
                        3/4 US)
  X          96.0%    Computer programmers (2/3 DFW, 1/3 Texas)
              4.0%    Professional, technical, and kindred workers
                        generally (2/3 DFW, 1/3 Texas)
  Y          82.2%    Computer programmers (60% DFW, 20% Texas, 20% US)
             15.9%    Computer specialists (60% DFW, 20% Texas, 20% US)
              1.9%    Master's degree (computer science)
  Z          92.9%    Computer systems analysts & computer specialists
                        (60% DFW, 20% Texas, 20% US)
              7.1%    MBA degree
 AA           9.8%    Lawyers (75% DFW, 10% Texas, 15% US)
             14.8%    Bank officers & financial managers (75% DFW,
                        10% Texas, 15% US)
             14.8%    Accountants (75% DFW, 10% Texas, 15% US)
              2.5%    Farm managers (75% DFW, 10% Texas, 15% US)
              4.8%    Managers and administrators (75% DFW, 10% Texas,
                        15% US)
              2.5%    Real estate agents and brokers (75% DFW, 10%
                        Texas, 15% US)
             24.3%    Business-related bachelor's degree
              1.5%    Mathematics bachelor's degree
              3.7%    Bachelor's degree (any major)
              6.1%    MBA degree
             15.2%    Law degree
                                            Note 212 continued on next page



  BB         50.0%    Bank officers & financial managers (60% DFW,
                        30% Texas, 10% US)
             50.0%    Purchasing agents & buyers (60% DFW, 30% Texas,
                        10% US)
  CC         57.1%    Bank officers & financial managers (40% DFW,
                        60% US)
             14.3%    Sales managers, except retail trade (40% DFW,
                        60% US)
             28.6%    Business-related bachelor's degree
  DD         87.5%    Personnel and labor relations workers (90% DFW,
                        10% Texas)
             12.5%    Business-related bachelor's degree
  EE        100.0%    Policemen, detectives, guards, and watchmen
                        (DFW market)
  FF         50.0%    Bank officers & financial managers (US market)
             50.0%    MBA degree
  GG         50.0%    Draftsmen (DFW market)
             50.0%    Designers (DFW market)
  HH         75.0%    Editors and reporters (1/3 DFW, 2/3 Texas)
             25.0%    MBA degree
  II         81.8%    Computer systems analysts, computer specialists,
                        & operations and systems researchers and
                        analysts (DFW market)
              9.1%    Computer science bachelor's degree
              9.1%    MBA degree
  JJ         25.9%    Bank officers & financial managers (75.9% DFW,
                        7.1% Texas, 17% US)
              4.7%    Accountants (75.9% DFW, 7.1% Texas, 17% US)
              3.5%    Credit men (75.9% DFW, 7.1% Texas, 17% US)
             31.8%    Professional, technical, and kindred workers
                        generally (75.9% DFW, 7.1% Texas, 15% US)
              7.6%    MBA degree
            19.45%    Business-related bachelor's degree
              4.7%    Bachelor's degree (any major)
              2.4%    Master's degree (other the MBA or computer science)
  KK         58.3%    Bank officers & financial managers (36.4% DFW,
                        9.1% Texas, 54.6% US)
             16.7%    Computer specialists (36.4% DFW, 9.1% Texas,
                        54.6% US)
             16.7%    Professional, technical, and kindred workers
                        generally (36.4% DFW, 9.1% Texas, 54.6% US)
              8.3%    Business-related bachelor's degree
  LL        100.0%    Bank officers & financial managers (1/3 DFW,
                        1/3 Texas, 1/3 US)
  MM        100.0%    Petroleum engineers (US market)
  This table is derived from Appendix B1 of the testimonial exhibits (Defendant's Exhibit 557).

[**]  Dr. Hofstedt is Chairman of the Accounting Department at the graduate business school at Southern Methodist University, and previously taught at Stanford and Cornell Universities. Dr. Hofstedt's teaching emphasizes the banking-related disciplines of accounting, corporate reporting, organizational control, and financial accounting. He spends 30 to 40 days a year as a consultant to the banking industry, specializing in the design and teaching of management training programs.
[**]  Dr. Hofstedt presented the following chart showing the education/experience mix typically required in the Bank's more common job family groups:

[**]  Margaret Bentsen, a commercial loan officer of the Bank and graduate of its credit training program, also testified that graduate level courses in banking and finance were very important for success in the program.
[**]  Dr. Madden argues that "flow" figures should be used for occupational and educational availabilities, that is, the proportions of females and blacks among those newly entering a profession. Dr. Stoikov's educational availabilities, which are based on the numbers of degree recipients in given years rather than the numbers of degree holders in those years, are flow statistics. Use of flow figures for occupational availabilities, however, would be inconsistent with the Bank's approach: the separation of those hired for their educational credentials from those hired for their experience is itself an effort to determine the proportion of Bank employees hired from the "flow" of new graduates from that hired from the "stock" of those already in the workforce. Those newly entering an occupation are accounted for under the Bank's procedure, by classification based either on the degree they have just received or the occupation they have just left.
[**]                                     Black                     Female
                Total            Proportions                Proportions        
      Year   Terminations   Workforce   Terminations   Workforce   Terminations
      1973       574          .139           .150        .634           .805
      1974       584          .158           .216        .631           .795
      1975       500          .149           .168        .619           .774
      1976       488          .153           .186        .623           .738
      1977       561          .168           .203        .639           .756
      1978       302          .178           .245        .632           .772
  This table is derived from Table 3.1 of Dr. Morgan's report (Plaintiffs' Exhibit 922).

[**]  Indeed, it is arguable that calculations of statistical significance in such a summary would constitute the introduction of new evidence, since no testimony at trial established the appropriate statistical techniques to be used in evaluating terminations. See section V(C), supra.
[**]  Because the court has found that plaintiffs failed to establish a prima facie case of termination discrimination, it is unnecessary to discuss in detail the Bank's rebuttal effort on the termination question. In sum, that effort consisted primarily of a report (Defendant's Exhibit 31) by Drs. David Snyder and Ross Stolzenberg and testimony by Dr. Stolzenberg. Those experts performed a multiple regression using as the dependent variable the probability of being involuntarily terminated and a variety of education, experience, and skill measures as independent variables. Separate regressions were run for all nonexempt employees, for nonexempt employees outside the general clerical job families, and for nonexempt general clerical employees. Although the dummy variable for race was statistically significantly positive before explanatory variables were added to the regression, no statistically significant coefficients for race or sex were obtained when controls were added for such variables.
[**]  The following Plaintiffs' Exhibits show the racial and sexual makeup of the Bank's departments:



Exhibit              Date                  Description
631               12/8/72 & 12/5/73        "minorities" by division & section
632               12/8/72 & 12/5/73        "minorities" by division & section
634               7/15/74                  by division
635               7/15/74                  exempt by department or function
638               4/30/75                  by department
7451-7491         4/30/75                  cumulative of #638
7421-7431         7/16/76                  race by some departments

[**]  The following table shows the relevant proportions as of April 30, 1975:



                              Total           %          %
        Department          Employees       Female     Black
    Credit Administration      92            43.5        7.6
    Correspondent              64            59.4        6.3
    Executives                 43             0.0        0.0
    Finance and
      Administration         1215            63.5       22.6
                                            Note 221 continued on next page



Funds Management            45         51.1        2.2
International               96         52.1        0.0
Metropolitan               231         58.4        8.7
Petroleum                   32         37.5        3.1
Real Estate                 49         53.1        2.0
Republic of Texas Corp.    114         57.0       13.1
Staff                       31         58.1        3.2
Trust                      337         61.7        5.3
United States               59         54.0        0.0
Total                     2408         58.9       14.2
  The data in this table were taken from Plaintiffs' Exhibit 638. The group designated
  "Executives" does not comprise a separate department, but includes top-level management in a
  variety of departments.

[**]  The theory behind inclusion of this variable was that dissatisfied employees would be apt to leave the Bank more rapidly than satisfied employees.
[**]  The results of these studies were as follows:



                                             Race        Sex
            Question                     Coefficient  Coefficient
The hours are good.                         .033          .129
Travel is convenient.                       .443*         .147
The physical surroundings are pleasant.     .159         -.129
I can forget my personal problems.         -.141          .112
The work is interesting.                   -.243*         .035
I can see the results of my work.           .163          .013
The problems I can expected to
  solve are hard enough.                   -.418*        -.036
I have enough information to get the
  job done.                                 .095          .035
I receive enough help and equipment.       -.016          .154
I have enough authority.                    .374*         .186*
My responsibilities are clearly
  defined.                                  .212          .273*
* Statistically significant at the .05 level.

These questions were taken from the University of Michigan/Department of Labor Quality of Employment Survey. See R. Quinn, T. Mangione, & S. Seashore, 1972 Quality of Employment Survey (1975).
The data in this table were taken from Table 3 of Dr. Stolzenberg's report (Defendants Exhibit 30).
[**]  Coefficients for race and sex are 1.162 and 1.379, respectively, with the sex coefficient statistically significant at the .05 level. See Table 4 of Dr. Stolzenberg's report (Defendant's Exhibit 30).
[**]  The Bank also relies on a series of workforce utilization studies by Dr. Odell, wherein numbers of blacks and females in a variety of departments and a variety of job families were compared with their respective availabilities. These results are not relied upon for a variety of reasons discussed in greater detail elsewhere:

(1) Zero is part of the randomness range in virtually all cases. See sections VIII(A)(2)(i) and (iii) and IX(G), supra.
(2) The extreme disaggregation destroys statistical significance. See sections VIII(A)(2)(i) and (iii) and IX(G) at n.210, supra.
(3) The geographical markets used have not been fully justified. See sections VIII(A)(2)(iii) and IX(F)(2), supra.
(4) The 100% occupational weighting has not been shown to be job-related. See sections IX(F)(1) and IX(G), supra.
[**]  The Bank's maternity leave policies are summarized in the following table:



         Service       Supvr.       Departure                                                  See
Years     Req't       Approval        Date         Duration       Benefits    Pay     Exhibits
pre-1971  3 years       yes         end of 6th     6 months        full?       no     PX580 (p.16)
                                    month                                                PX598
1971-74   6 months      yes         "normally"     flexible; max.  full        no     PX579
                                    6 months;      shorter of 6                       DX465 (pp.54-58)
                                    based on       months or 3                             PX736-40
                                    attending      months after
                                    physician's    delivery
                                    statement      (Notes 1,2,3)
1974-79     none        no?         based on           same        full        no     DX359 (pp. 54-58)
                                    attending      (Notes 1,2,3)                            PX1300
                                    physician's
                                    statement
1979 to     none        no          determined          same       full?       yes    DX345
 present                            by attending    (Note 1?)
                                    physician       (Notes 2,4)
Note 1: Unless physician's statement indicating that return is not advisable
        is submitted to the Bank.
                                           Note 226 continued on next page



Note 2: Must present physician's statement that there is no danger involved
        in returning to work.
Note 3: Must give reasonable notice ("probably 30 days").
Note 4: Must give three weeks' notice.
See Testimony of Thomas Barksdale.

[**]  No challenge is made to the requirement of the supervisor's approval, effective until 1974. Indeed, no such challenge could successfully be made, since the supervisor's approval was required for all other forms of disability leave as well.
[**]  The post-1971 policy does not violate Title VII, since the six-month requirement corresponds to a similar guideline for non-pregnancy sick leave. There has been no showing that a six-month service requirement for disability leaves has any disparate impact on any part of the class.
[**]  For example, if all disability leaves were subject to a three-year requirement, females with less than three years' service have not been discriminated against. Of course, the burden of proving this circumstance will be on the Bank.
[**]  The Bank appears to have relied on a pre-1970 EEOC ruling that "[a]n employer is justified in requiring all pregnant female employees to take a maternity leave of absence 90 days prior to delivery." See Plaintiffs' Exhibit 736. While it is unfortunate that this early EEOC opinion failed to accurately predict the development of the law, the Bank cannot rely on the erroneous opinion as a defense to liability for conduct which the courts (and, for that matter, the EEOC) have since held to be unlawful.
[**]  After 1971, the Bank permitted a flexible commencement date based on a doctor's recommendation. Such a policy, fairly administered, would not violate Title VII. Maclennan v. American Airlines, Inc., 440 F.Supp. 466 (E.D.Va.1977). Moreover, it corresponds roughly to the discretionary policy applied to other types of disability leave. No showing has been made that the flexible maternity leave commencement policy has been operated in a discriminatory manner.
[*]  Since 1971, maternity leave has been extendable where the employee's physician certifies that return is not advisable.
[**]  The Bank described the course as follows: PURPOSE: To demonstrate the effects of personality traits, personal appearance and attitudes on individual behavior, and the correlation between improvements in these factors and improved relations with employees and customers. CONTENT: Twelve hours of presentations, demonstrations, and classroom participation. One half of the program is devoted to personal appearance. The second half emphasizes the inner self and shows participants how to combine outward appearance with an attitude of concern for others, resulting in fruitful personal reltations [sic].

Plaintiffs' Exhibit 791.
[**]  The Bank described the course as follows: PURPOSE: Based on the Bank's philosophy that its success is built on individual successes at every level of capability and experience, this program provides ideas and hard facts which affect your image with others and, therefore, your potential for personal effectiveness or success. CONTENT: Specific subjects covered are: Success and Image.

Shows how your own image with others affects your changes for success. The Way You Communicate. Successful correspondence, conversation, and conference conducting, depend upon the same basic techniques. The Way You Sound. What you say and how you say it reflects your total personality. The Way You Feel. Fitness, health and hygiene combine to optimize your effectiveness. The Way You Are. Your attitudes toward yourself, your job, and your fellow employees are reflected in your image. The Way You Look. The correct wardrobe and styling for you improves your overall appearance. Plaintiffs' Exhibit 791.
[**]  The plaintiffs also complain that the "Personality, Appearance, and Personal Attitude" course was geared toward the personal appearance of white females, constituting racial discrimination in violation of Title VII. While this fact is likewise unfortunate, the harm caused thereby is, like that caused by the sexual segregation of the courses, too speculative to merit relief.
