738 F.2d 574
18 Ed. Law Rep. 823
The PARENT ASSOCIATION OF ANDREW JACKSON HIGH SCHOOL, anunincorporated association, Fred Perez, a minor by hisfather and next friend Bienvenido Perez, Brian and RolandFelder, minors by their parents and next friends Beverly andLeroy Felder, Robin Brown, a minor by her father and nextfriend David Brown, Joan McFarland, a minor by her fatherand next friend Jerome McFarland, on behalf of themselvesand on behalf of all persons similarly situated, Plaintiffs-Appellees,v.Gordon AMBACH, as Commissioner of Education of the States ofNew York, Irving Anker, Chancellor of the City SchoolDistrict of the City of New York, Samuel Plotnick, asDirector of the Division of High Schools of the City SchoolDistrict of the City of New York, Abraham Wilner, asSuperintendent of the Queens Division of High Schools of theCity School District of the City of New York, James Regan,Isaiah Robinson, Stephen Aiello, Amelia Ashe, Joseph Barkin,Robert Christen, Joseph Monserrat, as members of the Boardof Education of the City School District of the City of NewYork, Defendants-Appellants.
Nos. 553, 677, Dockets 83-7670, 83-7704.
United States Court of Appeals,Second Circuit.
Argued Dec. 1, 1983.Decided July 16, 1984.

James I. Meyerson, New York City (Thomas I. Atkins, General Counsel, NAACP, Brooklyn, N.Y., of counsel), for plaintiffs-appellees.
Robert D. Stone, Albany, N.Y., New York State Education Dept., Albany, N.Y., for defendant-appellant Gordon Ambach.
Dody Schorr, New York City (Frederick A.O. Schwarz, Jr., Corporation Counsel for the City of New York, New York City, Ronald E. Sternberg, Robert M. Katz, New York City, of counsel), for defendant-appellant Board of Education.
Before LUMBARD, WINTER and PRATT, Circuit Judges.
LUMBARD, Circuit Judge:


1
Defendants New York City Board of Education and New York State Commissioner of Education appeal from the decision in the Eastern District of New York, Constantino, J., invalidating the Board's plan for reassigning students of Andrew Jackson High School to other City high schools, on the ground that defendants failed to justify as necessary at least one of the racially restrictive provisions of the plan.  We reverse and remand for further proceedings.

I.

2
This case originated in June, 1976, as a class action filed on behalf of students and parents in the Andrew Jackson High School District in Queens, New York, challenging the constitutionality of a proposed school desegregation plan voluntarily adopted by defendant New York City Board of Education, and approved by defendant State Commissioner of Education Gordon Ambach.  The genesis of the plan, known as the "1976 Controlled Rate of Change Plan," is explained in detail in two opinions in an earlier stage of this litigation, Parent Ass'n of Andrew Jackson High School v. Ambach, 598 F.2d 705, 710-711 (2d Cir.1979), reversing in part, 451 F.Supp. 1056, 1063-69 (E.D.N.Y.1978).


3
In brief, the minority enrollment at Andrew Jackson High School, which had been 18% in 1957, steadily increased over the next two decades until, by 1976, it exceeded 99%.  Beginning in 1963, the New York City Board of Education ("the Board") adopted a succession of plans to stem Jackson's acceleration toward an exclusively minority student population by transferring white students in from other zones.  As the statistics show, none of these plans was successful.  In 1973, the Board finally abandoned the hope of integrating Jackson.1   Instead, it developed a "Choice of Admissions" scheme, under which minority students in the Jackson zone had the choice of attending any of a number of receiving schools at which the minority population was lower than at Jackson, and the influx of additional minority students would not destroy whatever racial balance currently existed.


4
With revisions in 1975 and 1976, the scheme became the "1976 Controlled Rate of Change Plan" ("the Plan") at issue here.  The Plan was intended to balance the goal of placing Jackson's minority students in integrated schools of their choice, against the perceived reality that if the minority enrollment in an individual receiving school increased too rapidly or reached a critical absolute level (the so-called "tipping point"), white students would leave the receiving school at an increasing rate.  By controlling both the rate and extent of change in racial composition in the receiving schools, the Board hoped gradually to transfer most or all of the student population out of segregated Jackson, without at the same time triggering resegregation of currently integrated receiving schools, thereby giving the largest number of children an integrated education over the longest period of time.


5
The Plan provided that Black and Hispanic students residing within the Jackson school zone could elect to attend any New York City high school not already overenrolled, in which (1) the percentage of white students in the school exceeds either 50% of the school population or the borough-wide average in the borough in which the receiving school is located (whichever is higher);  provided further that (2) the admission of such students, coupled with the admission of minority students from other integration programs and/or through demographic changes in the attendance area servicing the school, will not (a) decrease the receiving facility's white/minority ethnic balance by 4% or more in any one school year, or (b) produce a rate of change in any one year that exceeds one-fourth of the difference between the school's current white enrollment and a 50% white enrollment, whichever is less.  At the same time, it provided that white students in the Jackson zone could elect to attend any City high school in which the percentage of white students is lower than either the borough's white percentage or 50% of the school population, whichever is less.


6
In June 1976, students and parents at Jackson filed this class action against the Board of Education and the State Commissioner of Education, charging (in relevant portion here) that the Plan's imposition of a racial quota against the admission of minority students to potential receiving schools violated the equal protection clause of the Fourteenth Amendment.  The district court, agreeing with plaintiffs, invalidated the Plan as unconstitutional.  Although acknowledging that the Board's goals in adopting the Plan were benign, the court held that by "limit[ing] minority pupils' access to schools because of their minority status in order to provide integrated schooling for as long as possible to a progressively limited number of minority and other students," the Plan impermissibly recreated "a dual school system" of integrated and segregated schools.  451 F.Supp. at 1080-81.


7
On appeal, we reversed, holding that the Plan's aim "to promote a more lasting integration is a sufficiently compelling purpose to justify as a matter of law excluding some minority students from schools of their choice under the obviously race-conscious Rate of Change Plan."    598 F.2d at 719.  However, we remanded for several factual determinations concerning specific provisions of the Plan, including the two relevant here:  (1) whether the Board had shown a factual justification for its choice of a 50% tipping point;  and (2) whether demographic projections supported the Board's determination that the maximum allowable rate of change in racial composition of a school should be set at the lesser of 4% or  1/4 of the difference between the current white enrollment and 50% white enrollment.


8
On remand before Judge Dooling in February 1980, the Board provided extensive statistical data to support both determinations.  In respect to the 50% tipping point, the Board submitted two exhibits to substantiate its claim that white students left a school at an increasing rate when the percentage of white students dropped below 50%.  The first, Exhibit 3C, compared the rates of change for each City high school when and if it was between two 30-point intervals of ethnic composition--80%-50% white, and 50%-20% white--during the years 1957-1978.  As explained by Philip tama, the Associate Director of Statistics and Demographics in the Board's Office of Zoning and Integration, the Exhibit revealed that most individual schools experienced a sharp increase in the annual rate of change when they moved from the 80%-50% range to the 50%-20% range.  The city-wide interval average (equal to the mean of the schools' averages within each interval) showed that the percentage of white pupils per year declined at an average rate of 2.94% per year during the period that schools were in the interval from 80%-50% white;  and declined at an accelerated average rate of 5.54% during the period they were in the interval from 50%-20% white.  The second, Exhibit 3E, broke down the two 30-percentage point intervals in Exhibit 3C into six 10-percentage point intervals.  The city-wide average figures revealed a steady increase in the decline in the percentage of white pupils in the first three intervals (3.67% from 80%-70%;  4.42% from 70%-60%;  and 4.80% from 60%-50%), and a marked increase in such decline in the fourth interval (6.06% from 50%-40%).2


9
In respect to the 4% maximum allowable annual rate of change, Joseph Elias, Director of the Board's Office of Zoning and Integration, testified that the intent of the Board in limiting the rate of change was to approximate the 3% natural rate of change (through attrition) that the schools were experiencing at the time.  He noted further that the 4% figure referred to the school as a whole (i.e., all four classes), not simply the entering class that contained the new minority transfer students.  As a result, the entering class could actually absorb up to a 16% decrease in white enrollment annually without exceeding the 4% ceiling.


10
In addition, in support of both the 50% tipping point and the 4% rate of change, the Board submitted Exhibit 4, comparing the actual experience in Queens receiving schools since the Controlled Rate of Change Plan went into effect in 1976,3 with projections of what would have happened had all minority students been granted their first choice assignments without regard to extent or rate of change.  (Computations were done only for Queens high schools because virtually all the students who otherwise would have gone to Jackson designated another Queens high school as their first choice.)    As explained by Elias, whose office prepared the Exhibit, the limitations on extent and rate of change had two desirable effects on integration.  They greatly slowed the rate of decrease in white enrollment for those receiving schools that were below the 70% point prior to 1976, in several cases avoiding a projected precipitous drop to well below the 50% mark;  and they nearly equalized the racial balance in all receiving schools, by decreasing more sharply the white enrollment in the few schools that had been well above 70% white in 1975.4   As a result, it was anticipated that by 1980, all Queens receiving high schools would have between 52% and 63% white enrollment if the Plan remained in effect, compared to a projected spread of 39.5% to 75.7% white enrollment if Jackson students were given their first preferences without limitation.


11
Finally, Tama testified that, even with the Plan in effect, in 1980 92.5% of the applicants in the Queens choice of admissions area were assigned to one of their top three, and 98% to one of their top four, choices of schools.


12
Plaintiffs produced no witnesses of their own on remand, and introduced no evidence directly contesting the Board's maximum allowable rate of change figures.  However, following the February 1980 hearing, they submitted a report prepared by Dr. Steven Cohen, an Associate Professor of Sociology at Queens College, taking issue with the statistical methodology employed by defendants in the two exhibits offered to substantiate the 50% tipping point.  When Dr. Cohen recomputed the average city-wide rates of change by his preferred methodology, his figures showed that the greatest decline occurred at the 60%, not the 50%, level.


13
A second hearing was held before Judge Dooling in October 1980, at which Elias and Tama were requestioned, but no new witnesses or evidence were introduced.  Before he could render a decision, Judge Dooling passed away.  The case was reassigned to Judge Costantino on January 22, 1981.  With the agreement of all the parties, Judge Constantino indicated he would decide the matter on the basis of the record before Judge Dooling and any written evidence submitted by the parties subsequent to the first hearing.


14
On July 5, 1983, Judge Constantino issued his decision, finding that the Board had failed to provide factual justification for its choice of a 50% tipping point.  As that finding alone was grounds for declaring the Plan invalid under the terms of this court's remand, Judge Constantino never reached the maximum rate of change issue.  Defendants appeal from that decision.

II.

15
In an earlier stage in these proceedings, we held that the Board's goal of ensuring the continuation of relatively integrated schools for the maximum number of students, even at the cost of limiting freedom of choice for some minority students, survived strict scrutiny as a matter of law.  598 F.2d at 717-720.  However, in remanding to the district court for further factual determinations, we stated that "the Board and the Commissioner must bear the burden of proving that the Plan, in each of its components, is necessary to achieve the goal set forth."    Id. at 721.  The question before us now is whether defendants have met that heavy burden of proof as to the two elements of the Plan still in dispute:  the choice of a 50% tipping point;  and the choice of (a) 4% or (b)  1/4 of the difference between the current white enrollment and 50%, whichever is less, as the maximum allowable rate of change per year.5


16
We begin with the 50% tipping point, the only issue addressed by the district court.  As a matter of normal appellate procedure, this court may overturn the district court's finding that defendants failed to meet their burden of proof with respect to the 50% tipping point only if it was clearly erroneous.  See Fed.R.Civ.P. 52(a).  We conclude that it was, at least on the grounds asserted by the court.


17
The district court based its finding that defendants had failed to provide factual justification for the 50% tipping point on two observations the court drew from its own analysis of defendants' school-by-school raw statistical data:  (1) that three City high schools (Erasmus Hall, Springfield Gardens, and Tilden), two of which defendants' expert had cited as typical illustrations of the 50% tipping point phenomenon, in fact showed almost as great a drop in white enrollment at the 60% or 70% level as they did at the 50% level;  and (2) that of nine public high schools in Brooklyn and another nine in Queens, only two experienced their greatest decline in white enrollment at the 50% level.6   The court made no reference to the two exhibits submitted by defendants showing the average interval drops on a city-wide basis, or to Dr. Cohen's critique of those exhibits.


18
Appellants argue that both observations on which the court relied are based on an erroneous or misleading reading of their statistical data.  We agree.


19
On the court's first point, in defense of its conclusion that "a similar rate of decline occurred [for each of the three schools] at figures where white student population was well above the 50% mark," the court noted that Erasmus Hall, which had a 10.1% decline at the 50% point, had a 8.5% decline at approximately the 60% point;  Springfield Gardens, which had a 10.1% decline at the 50% point, had a 7.5% decline at the 60% point;  and Tilden, which had a 9.7% decline at the 50% point, had a 10.1% decline at approximately the 70% mark.


20
As to Tilden, the district court's statistical inference is incorrect.  The 9.7% decline at the 50% level took place over one year;  the 10.1% decline at the 70% level over two years.  An accurate comparison of the average annual declines at the two levels would thus require halving the 10.1%, with resulting figures of 9.7% at the 50% level and 5.05% at the 70% level--figures that amply support defendants' choice of 50% as the critical tipping point.


21
As to Erasmus Hall and Springfield Gardens, we are unsure what the district court's observation proves.  Defendants contend not that the 50% level triggers the only significant drop in white enrollment, but that it triggers (on average) the most significant drop.  The experience of these two schools bears that observation out.  Finally, although defendants cited to these two schools as illustrative of their argument, their case for the 50% tipping point rests not on the individual experiences of the two schools, but on the average experience of schools on a city-wide basis.


22
On the court's second point, of the eighteen schools the court cited, during the relevant period eleven never went below 50% white enrollment, one more went only barely below 50%, and one was never above 50%.7   As a result, for thirteen of the eighteen schools, there were no, or virtually no, statistics available at the 50% level.  The remaining five schools did have experience on both sides of the 50% level.  But two of the five (Erasmus Hall and Newtown) had their greatest decline at around 50%, thus bearing out defendants' theory.  The other three showed higher declines at levels other than 50%.  But two of these three (Bay Ridge and Hillcrest) had only one year's experience (their first year of operation) substantially above the 50% level, casting doubt on the meaningfulness of their statistics for that year, leaving only one (Bowne) as a reliable example of a tipping point other than 50%.


23
Thus, of the eighteen schools cited by the district court, only one--Bowne--unequivocally diverges from defendants' predictions of a 50% tipping point.  Bowne may not be the exception that proves defendants' rule, but surely it is also not the exception that disproves it.


24
We therefore cannot agree with the district court in holding, on the basis of the evidence on which it purported to rely, that defendants failed to justify the choice of a 50% tipping figure.


25
That leaves two questions:  whether a proper analysis of the evidence submitted on the issue of a tipping point would show that defendants justified their choice of the 50% figure;  and whether the evidence submitted but not considered by the district court on the rate of change issue justifies as necessary defendants' selection of a 4% ceiling.  In the hope that both questions could be resolved by us on the basis of statistical evidence submitted below, see Shackelton v. J. Kaufman Iron Works, Inc., 689 F.2d 334, 337 (2d Cir.1982);  Philip v. Mayer, Rothkopf Industries, Inc., 635 F.2d 1056, 1061 (2d Cir.1980), we have reviewed that evidence with some care.  Reluctantly--given the already lengthy history of this litigation--we conclude that we have no choice but to remand both questions to the district court to be resolved as expeditiously as possible.


26
50% ceiling on minority enrollment.    In support of their claim of a 50% tipping point, defendants rely chiefly on Exhibits 3C and 3E, showing the average annual percentage decline in white enrollment in the City's high schools from 1957-1978.  Although we have doubts how much can be concluded from Exhibit 3C,8 which groups the data into two 30-percentage point intervals, Exhibit 3E, which breaks the data down into six 10-percentage point intervals, appears on its face to support defendants' contention that a significant increase in the rate of departure of white students occurs between the 60%-50% and 50%-40% intervals.  See n. 3 supra.    However, several questions concerning the significance of that statistical finding prevent us from concluding outright, on the strength of that alone, that defendants have met their burden of proof concerning the 50% tipping point.9


27
The first question, raised below by plaintiffs but never addressed by the district court, concerns whether, in computing the city-wide mean averages in Exhibit 3E, defendants properly classified and weighted each school's experience within a given 10-percentage point interval.  Dr. Steven Cohen, plaintiffs' expert, raised two objections to defendants' methodology:  (1) that defendants determined the interval within which a school fell for any given year based on its average white enrollment for the year, rather than on its white enrollment at the beginning of the year;  and (2) that defendants weighted each school's statistical experience within a ten-point interval by school rather than by school year.10   Reanalyzing the Board's raw data under his preferred methodology, Dr. Cohen came up with city-wide averages that show a sharper drop at both the 60% and 30% levels than at the 50% level.  As those figures, if the more accurate, would seriously undercut defendants' case for a 50% tipping point, an informed choice between the two would seem essential.  Lacking adequate information to make that choice ourselves,11 we leave it to the district court to resolve on remand.


28
Second, our own independent review of the data presented in Exhibit 3E raises an additional question not dealt with in the district court.  The city-wide figure in Exhibit 3E for each ten-point interval is equal to the mean average of the individual rates of change for each of the City's 62 nonspecialized high schools with any experience within that interval.  However, during the 1957-1968 period covered in Exhibit 3E, only 25 of the 62 schools had any significant experience on both sides of 50%.12   The other 37 schools either remained above 50% white throughout that entire period, or had dropped below 50% white before it began.  If the time period is further limited to 1968-1978--which the Board elsewhere claims to be the only period on which it is relying to prove the 50% tipping point--the number of schools with experience on both sides of 50% drops down to 8.13


29
What this means is that the statistical experience of the City's school system on the two sides of 50% is based on largely, or almost entirely, nonoverlapping samples.  Whether it is nonetheless legitimate to infer that the same statistical disparity observed between those two samples would hold true for the average individual school were it to cross the 50% point seems to us a crucial methodological question we cannot resolve.  We therefore leave it for the district court to resolve on remand, with the aid of any additional data accumulated by the City since 1978.14   We note for the record, however, that the experiences of those few schools that did straddle 50%, while too small a sample to be conclusive, do not allay our concerns.  Of the 8 such schools for the period 1968-1978, only 4--Tilden, Erasmus, Springfield Gardens and Newtown--showed the highest percentage decline in white enrollment at the 50% level.  Of the additional 17 schools for the period 1957-1978, only 5--Eastern District, Bushwick, Seward Park, Washington and Hillcrest--showed their highest percentage decline at 50%, and 3 of those 5--Eastern District, Washington and Hillcrest--were never above 62% white, casting some doubt on the significance of their comparative experience.


30
Finally, although we conclude that close judicial scrutiny of the 50% ceiling is required because of the unequal burdens that ceiling imposes on Jackson's minority students, see n. 9 supra, we do not mean thereby to trigger a rigid and mechanical application of a strict scrutiny standard.  The extent of proof required of defendants should take some realistic account of the extent of the burdens imposed.  To that end, the parties should provide on remand a more precise and updated picture of the burdens actually imposed on Jackson's minority students by the Plan, particularly the numbers of plaintiffs who have been and will be forced to remain in a largely or exclusively minority school (presumably Jackson) because of a shortage of potential receiving schools that meet the Plan's restrictions.


31
4% ceiling on annual rate of change.    As this issue was never reached below, and as we are remanding in any case for reconsideration of the 50% ceiling on minority enrollment, we leave it for factual resolution by the district court.  However, we note that some limitation on the rate of change seems to us a necessary corollary to a limitation on extent of change.  That is, if the Board is allowed to put a 50% ceiling on minority enrollment but not allowed to control the rate of influx of additional minority students, the result will simply be that first-choice schools will reach the 50% ceiling more rapidly--at which point no additional minority students at all will be allowed in.  As we fail to see how that result would in any way lessen the burden on minority students, we are inclined to think that the case for imposing some limitation on rate of change must stand or fall with the case for a 50% ceiling on minority enrollment.  However, whether the particular figure chosen by defendants as the maximum permissible rate of change has a factual justification is a matter we leave to the district court.


32
Reversed and remanded for further proceedings not inconsistent with this opinion.  We retain jurisdiction.

WINTER, Circuit Judge, dissenting:

33
I respectfully dissent.


34
In 1979, another panel remanded this case to take further evidence in the hope that the present appeal would return "with reasonable speed."    598 F.2d at 722.  Five years later, we remand again in order to have the district court repeat the precise exercise ordered by the earlier panel.


35
In my view, we should reverse and enter judgment for the Board of Education.  The prior opinion held that the de facto segregation at Andrew Jackson High resulted solely from residential demographics and was thus not unconstitutional.  Therefore, the Board was not then, and is not now, under any federal legal obligation to do anything to alter the racial composition of Andrew Jackson High.


36
Notwithstanding the lack of any legal obligation, the Board has sought to create an opportunity for students at Andrew Jackson to pursue their education in integrated schools through a transfer plan that utilizes racial quotas.  In the prior opinion, we held that such quotas are justified by a compelling governmental interest where they result in greater integration than would exist under a racially neutral (and constitutionally valid) system of attendance zones based on residence.  We remanded, however, because of the lack of evidence in the record as to why the various formulae used by the plan to fashion the quotas were chosen.  Given the nature of the underlying legal issue, there is no federal constitutional imperative to require more than a showing that the factual basis for the quotas' formulae is sufficient to ensure that the plan actually decreases segregation.1


37
Although the majority appears to assume the present plan does reduce segregation, it creates a legal rule that such a plan be clearly demonstrated to be the best available to reduce segregation.  Since the Board need not offer any plan, much less one judged by us to be the most desirable, the constitutional source of this requirement is not evident to me.  It is, moreover, a most unwise ruling, for two reasons.


38
First, demographic variables are too numerous and too imponderable to locate so-called "tipping points" with any precision.  They will vary from school to school, neighborhood to neighborhood, and ethnic group to ethnic group.  Moreover, unpredictable events occurring elsewhere which affect the degree of racial tension in the society can quickly change behavior.  One cannot determine a "tipping point" predicting human behavior as scientists predict the movement of planetary objects.  I fear the majority's quest will result only in this case returning to us in 198? with more recent but equally equivocal statistics.


39
Second, the majority's ruling will have the unintended consequence of increasing racial imbalance in education.  While the plaintiffs here may be confident of obtaining a better plan through political efforts if the present one is invalidated, the undeniable precedential effect of this ruling is to throw substantial roadblocks onto the paths of school boards which voluntarily undertake plans to reduce racial imbalance.  Lawyers who counsel such boards will surely advise them that today's decision guarantees protracted and unpredictable litigation arising out of these plans and that such a voluntary effort is simply not worth the cost.  Given the controversy inevitably generated by these plans, this additional obstacle can only decrease voluntary efforts.


40
For the reasons stated, I respectfully dissent.



1
 Among other problems facing the Board, by 1975 the public high school student population for the entire borough of Queens was over 50% minority.  As a result, as the City argued and the State Commissioner of Education agreed, in the unlikely event that a new compulsory scheme for reassigning white students to Jackson succeeded in integrating Jackson, it would do so only at the cost of "impair[ing] racial integration of the high schools of the borough as a whole."    Matter of Parent Ass'n of Andrew Jackson, 15 Educ.Dept.Rep. 483, 485 (1976)


2
 The complete city-wide figures given in Exhibit 3E are as follows:
                Citywide
                Interval
Interval        Average
80-70% White .... 3.67
70-60% White .... 4.42
60-50% White .... 4.80
50-40% White .... 6.06
40-30% White .... 6.01
30-20% White .... 7.45


3
 The exhibit was prepared in February 1980, before data for school year 1979 was complete and before data for school year 1980 was available at all.  As a result, the "actual experience" for 1979 and 1980 was in fact projected on the basis of actual experience in 1976-1978 and the parameters of the school desegregation plan in effect at the time.  The "projected experience" for 1979-1980, in the event the Plan did not continue in effect, was based on the assumption that Jackson students would continue to request as a first choice those high schools that they had requested in 1976-1978


4
 The exhibit bears out the Board's conclusions.  Bayside, Francis Lewis, Van Buren, Forest Hills, Cardozo and Queens Vocational, all of which were below 70% white in 1975, were projected to have more modest declines in white enrollment with the Plan's restrictions than without them, in three cases (Van Buren, Forest Hills and Queens Vocational) avoiding otherwise precipitous drops.  John Adams, Cleveland, and Long Island City, all of which were at or well above 70% white in 1980, were projected to end up with a white enrollment more in line with the borough-wide average for other receiving schools.  Of the remaining two, Richmond Hill was projected to stay above 50% and in line with the borough-wide spread in either case, and only Edison Vocational showed less favorable results with the Plan than without it


5
 For simplicity's sake, the maximum allowable rate of change will hereafter be referred to as 4%.  We do not mean thereby to ignore the Board's actual, and more complicated, formula


6
 It is unclear what reason, if any, the district court had for selecting those 18 schools out of the total of 41 nonspecialized high schools in Brooklyn and Queens, or for that matter, out of the total of 62 in the City as a whole


7
 The eleven that never went below 50% white enrollment are:  Canarsie, Lincoln, Murrow, Madison, Dewey, Cleveland, Long Island City, Van Buren, Richmond, Adams, and Franklin Roosevelt.  The one that went barely below 50% was Bryant (at 48.3%).  The one that was never above 50% was Wingate


8
 By aggregating the data into two groups divided at the 50% point, the Board guarantees that 50% will appear to be the critical line of demarcation, irrespective of where the largest changes in rates of change actually occurred.  Thus, the most one can tell from the comparative city-wide figures presented in Exhibit 3C is that there is on average a higher percentage of annual white flight in the intervals below 50% than in those above 50%.  It is impossible to tell whether that disparity is caused by accelerated rates of decline at 50%, 40%, 30%, or any combination thereof;  or indeed whether the highest single rate of decline may not in fact have occurred above 50%


9
 The dissent urges that because the Plan was voluntarily adopted by the Board, and thus we lack authority to order a new plan in its place, we should require no more than a showing the Plan results in less segregation than no plan at all.  That argument was expressly rejected by this court on prior appeal, see 598 F.2d at 717, and ignores the rationale for applying a strict scrutiny test in the first place:  that a plan utilizing racial ceilings based on the supposed existence of a tipping point, however well-intentioned, imposes on minority students two burdens that are absent from traditional desegregation plans designed to achieve a uniform system-wide white/minority student ratio.  First, where, as here, white enrollment in the system as a whole is inadequate to keep all schools above the perceived tipping point, one group of minority students can be kept in stably integrated schools only at the cost of requiring another group of minority students to remain in largely or exclusively all-minority schools.  598 F.2d at 717-19.  Second, however pragmatic it may be, official sanction of the stigmatizing notion of "white flight" which underlies use of a tipping point places a psychological burden on the entire class of minority students
We held on prior appeal that it was constitutionally permissible to impose those unequal burdens on minority students, but only if it could be shown that they were necessary to inhibit resegregation of the system as a whole.  598 F.2d at 718-19, 721.  We do not construe that to mean, as the dissent suggests, that to survive our scrutiny, a voluntary plan must "be clearly demonstrated to be the best available to reduce segregation."    Rather, we construe it to mean that if particular elements of a voluntary plan create equal protection problems of their own, the Board must justify those elements as necessary to achieve a compelling governmental interest.  In other words, the Board may take constitutionally suspect measures to counteract the perceived problem of accelerated white flight at 50% tipping point, but it must at least show that a problem exists.
If the Board cannot make that showing with respect to a 50% or any other "tipping point," it is not without options to place students in Andrew Jackson in nonsegregated schools.  They can allow a free transfer program within the constraints of space limitations, or a transfer program designed to achieve a uniform white/minority student ratio throughout the borough or city.  As neither alternative would raise equal protection problems of the sort presented here, neither would be subject to the strict judicial scrutiny we apply to this plan.


10
 That is, if School A took 5 years and School B took 2 years to move from 60% to 50% white enrollment, under either methodology the average annual decline within that interval would be 2% for School A (10% divided by 5 years) and 5% for School B (10% divided by 2 years).  But in calculating the system-wide average for that interval, defendants would count School A's and School B's average figures once each, while plaintiffs would count School A's 5 times and School B's 2 times to reflect the number of years' experience each school had within the interval


11
 In regard to locating schools in given intervals, if in fact most or all students make the decision whether to withdraw at the beginning of the year, without any information concerning other students' simultaneous decisions, then plaintiffs' choice of the beginning-of-the-year figure would more accurately reflect whatever effect percentage of white enrollment has on rate of decline.  But if students depart throughout the year, and make the decision whether to do so with some awareness of other students' impending parallel decisions, then defendants' choice of the average percentage of the year would appear to be preferable
In regard to weighting individual rates of change by school versus school year, both parties have offered only conclusory statements in support of their respective positions.  It may be that the choice between the two positions ultimately is arbitrary.  But before so concluding, we think it advisable to give each party on remand an opportunity to present a more persuasive argument on its behalf.


12
 The 25 schools are:  Tilden, Jefferson, Prospect Heights, John Jay, Erasmus, Wingate, Eastern District, Bushwick, Bay Ridge, Seward Park, Washington, Taft, Monroe, Evander Childs, Walton, Franklin D. Roosevelt, Clinton, Andrew Jackson, Springfield Gardens, Bowne, Newtown, Flushing, Far Rockaway, Jamaica, and Hillcrest
An additional 3 schools--Bryant, Washington Irving, and Julia Richman--crossed the 50% point, but by an amount so slight as not to justify inclusion.


13
 The 8 are Tilden, Erasmus, Springfield Gardens, Bowne, Newtown, Flushing, Far Rockaway, and Jamaica


14
 We do not mean thereby to impose on defendants a burden that, by the nature of the proof available, it is impossible for them to meet.  We recognize that no amount of evidence in this area will ever prove that a drop below 50% white enrollment causes an increase in the rate of white departure.  But it can at least show that it correlates statistically with such an increase.  Given the equal protection issues raised by a plan designed to counteract the perceived problem of a tipping point, see n. 9, supra, we think that is the least the City should be required to do.  Furthermore, if it turns out that the statistical experience of New York City schools is inadequate to show conclusively such a correlation, defendants may supplement that data with the experience of other cities or expert testimony


1
 The prior panel, in a footnote, indicated, without further discussion or elaboration, that each component of the plan had to be shown to be "necessary."    Since, as I discuss infra, I believe there is no single best formula, I would not read "necessary" to mean "essential" but rather to mean "appropriate."    See McCulloch v. Maryland, 17 U.S.  (4 Wheat.) 316, 413-14, 4 L.Ed. 579 (1819) (Marshall, C.J.)


