
641 F.3d 706 (2011)
UNITED STATES of America, Plaintiff-Appellee,
v.
Harold JONES, Defendant-Appellant.
No. 09-3664.
United States Court of Appeals, Sixth Circuit.
Argued: March 10, 2011.
Decided and Filed: April 12, 2011.
*708 ARGUED: Brian P. Downey, Schwartz Downey & Co., L.P.A., Cleveland, Ohio, for Appellant. Laura McMullen Ford, Assistant United States Attorney, Cleveland, Ohio, for Appellee. ON BRIEF: Brian P. Downey, Schwartz Downey & Co., L.P.A., Cleveland, Ohio, for Appellant. Laura McMullen Ford, Assistant United States Attorney, Cleveland, Ohio, for Appellee.
Before: KENNEDY and MARTIN, Circuit Judges; MURPHY, District Judge.[*]

OPINION
BOYCE F. MARTIN, JR., Circuit Judge.
Defendant-appellant Harold Jones appeals the sufficiency of the evidence supporting his convictions and his sentence. For the following reasons, we AFFIRM Jones's convictions, but VACATE his sentence and REMAND for resentencing.

I. BACKGROUND
Harold Jones is a podiatrist who served a large, mostly elderly client base in Cleveland, Ohio. On April 9, 2008, a grand jury in United States District Court for the *709 Northern District of Ohio indicted Jones with twenty-seven counts of mail fraud in violation of 18 U.S.C. § 1341, twenty-three counts of healthcare fraud in violation of 18 U.S.C. § 1347, and four counts of aggravated identity theft in violation of 18 U.S.C. § 1028(a). Upon motion by the United States, the court dismissed two counts of healthcare fraud and all of the identity theft counts. Jones proceeded to trial by jury on the remaining charges. The jury convicted him of two counts of mail fraud and one count of healthcare fraud and acquitted him of the other counts.
The substance underlying Jones's convictions involved charging Medicare for services that he did not actually perform for two patients, Mary Carter and James Tubbs. Jones billed Medicare for debriding six or more toenails for Carter on October 14, 2004, but Carter testified that Jones had not treated her since 2000 or 2001. A note in Jones's file for Carter showed that she was last treated on December 13, 2000. The 2004 bill resulted in Medicare paying Jones $29.70. Jones also billed Medicare claiming that his former employee, Cheryl Stewart, treated Tubbs on July 19, 2005. However, this date came five days after Jones fired Stewart. This bill resulted in Medicare paying Jones $91.06. Jones had no notes in his files to corroborate the bills for Carter and Tubbs, which he attributed to sloppy bookkeeping.
The jury acquitted Jones of all the other charges against him. These charges alleged that Jones often performed non-reimbursable services such as nail-trimming and corn- and callus-shaving, but then billed Medicare and Medicaid for those services using codes for reimbursable services, otherwise known as "up-coding." Jones initially billed for these services under codes 11055 and 11056, which cover the paring or cutting of corns and calluses. These codes are only reimbursable if the patient has certain systemic conditions, not if the service is routine. The Healthcare Finance Administration sent Jones a letter explaining that his use of codes 11055 and 11056 appeared to be for routine services and instructing him to cease billing Medicare and Medicaid under these codes for routine services. After receiving this letter, the United States alleged that Jones stopped using those codes and instead used code 11421, the code for excision of a benign lesion, when he cut and pared calluses and corns.
The district court sentenced Jones to one and a half years of imprisonment followed by three years of supervised release. Although the loss from Jones's convicted counts totaled only $120.76, the court ordered Jones to pay a $300 special assessment and $224,133 in restitution for the conduct associated with his acquitted counts. In determining both the imprisonment and restitution components of Jones's sentence, the district court relied on the relevant conduct underlying Jones's acquitted charges. More specifically, it relied upon the amount of loss to Medicare and Medicaid caused by Jones's acquitted conduct.
To establish the amount of loss to Medicare and Medicaid caused by Jones's upcoding, a statistician for the United States identified a representative sample of Jones's bills from 2001 to 2004 that charged code 11421. This sample consisted of 357 bills spread across 264 patients. The United States located Jones's patient files for only 210 out of the 264 patients in the sample size. Taking those 210 files, the United States's podiatry expert, John Stephens, testified that Jones had up-coded the bills in each of them. In forming his conclusion, Stephens noted that certain procedures were missing entirely from the bills that would have accompanied a code *710 11421 if an excision had actually been performed, such as post-operative procedures, anesthesia, and lab reports. The United States alleged that this statistical extrapolation method established that one hundred percent of Jones's bills for code 11421 from 2001 to mid-2005 were fraudulent, resulting in a loss of $224,133.
Additionally, Stewart testified that while she worked for Jones, he instructed her to bill code 11421 when she shaved calluses. Furthermore, an investigator from the Department of Health and Human Services, Special Agent Fairbanks, testified that Jones's bills for code 11421 cost Medicare and Medicaid more than $200,000, but he did not supply an explanation for this figure. Jones based his incorrect billing history on a misunderstanding of the billing codes.
The district court accepted the United States's statistical evidence and determined by a preponderance of the evidence that one hundred percent of Jones's bills for code 11421 from 2001 to mid-2005 were fraudulent. It found that Jones's acquitted conduct resulted in a loss of $224,133.

II. DISCUSSION

A. The Sufficiency of the Evidence Supporting Jones's Convictions
Jones claims that there was insufficient evidence to support his healthcare fraud and mail fraud convictions. When reviewing convictions for sufficiency of evidence, we ask "whether, after viewing the evidence in the light most favorable to the prosecution, and after giving the government the benefit of all inferences that could reasonably be drawn from the testimony, any rational trier of fact could find the elements of the crime beyond a reasonable doubt." United States v. Ross, 502 F.3d 521, 529 (6th Cir.2007) (quoting United States v. M/G Trans. Servs., Inc., 173 F.3d 584, 589 (6th Cir.1999)) (internal quotation marks omitted). This is a "very heavy burden." Id. (quoting United States v. Davis, 397 F.3d 340, 344 (6th Cir.2005)) (internal quotation marks omitted).
There was sufficient evidence to support Jones's healthcare fraud conviction. A defendant violates section 1347, the healthcare fraud statute, if he "(1) knowingly devised a scheme or artifice to defraud a health care benefit program in connection with the delivery of or payment for health care benefits, items, or services; (2) executed or attempted to execute this scheme or artifice to defraud; and (3) acted with intent to defraud." United States v. Martinez, 588 F.3d 301, 314 (6th Cir. 2009) (internal quotation marks and citation omitted). Here, there was evidence that Jones submitted a bill to Medicare for debriding Carter's toenails on October 14, 2004. However, Carter testified that she only saw Jones once in either 2000 or 2001. Jones's medical files for Carter corroborate that she saw him on December 13, 2000, but contain no record of a 2004 visit. Additionally, there is evidence that Jones billed Medicare under Stewart's billing identification number for services provided to Tubbs on July 19, 2005 even though Jones had fired Stewart five days earlier. And, as with Carter, no part of Jones's medical file for Tubbs contained a record of this service. Therefore, when viewing the evidence in the light most favorable to the prosecution, and giving the government the benefit of all reasonable inferences, a rational trier of fact could find that Jones intentionally devised a scheme to defraud Medicare when he submitted the bills for Carter and Tubbs.
Similarly, there was sufficient evidence to support Jones's mail fraud conviction. A defendant violates section 1341, the mail fraud statute, if he had "(1) a scheme to defraud, and (2) [caused] the *711 mailing of a letter, etc., for the purpose of executing the scheme." Id. at 316 (internal quotation marks and citation omitted). Because there was a scheme to defraud regarding Jones's section 1347 violation, it is an easy step to satisfy the remaining requirement of section 1341using the mail to effect the scheme. Jones received payment from Medicare and Medicaid for his fraudulently billed services via the mail. Therefore, when viewing the evidence in the light most favorable to the prosecution, and giving the government the benefit of all reasonable inferences, a rational trier of fact could find that Jones intended to use the mails to commit fraud when he submitted the bills for Carter and Tubbs. Accordingly, sufficient evidence supported Jones's convictions.

B. Whether Jones's Prison Sentence Is Reasonable
The district court calculated Jones's base offense level at seven based on United States Sentencing Guidelines § 2B1.1. According to section 2B 1.1(b)(1)(G) and section 1B1.3, the court then added twelve levels after finding that Jones had caused more than $200,000 of loss to Medicare and Medicaid through the conduct underlying his acquitted charges. Additionally, the court added two levels because Jones had abused a position of trust to commit his offenses. See U.S.S.G. § 3B1.3. These additions created a total offense level of twenty-one. Jones's criminal history category was I, so his corresponding sentencing range was three years and one month to three years and ten months of imprisonment.
However, the district court was not quite comfortable with the sentencing range because it was so largely a product of Jones's acquitted conduct. The court sentenced Jones to one and a half years of imprisonment. This prison term equated to the term that would have been applicable if the twelve-level enhancement had been cut in half.[1]
Jones claims that his sentence is procedurally unreasonable because the court calculated his total offense level incorrectly by applying the twelve-level enhancement for the amount of loss associated with his acquitted conduct. He also claims that the enhancement made his sentence substantively unreasonable because his acquitted conduct had an inordinately high effect on his sentence. He argues that statistical extrapolation should not have been used because there was not enough evidence presented regarding the method, the sample size was too small compared to the large number of patients Jones had, and the jury had acquitted him of the conduct that caused this loss. Essentially, these claims boil down to whether the court could properly calculate the amount of loss using the statistical extrapolation presented by the United States.
This Court reviews sentences for both procedural and substantive reasonableness under the abuse of discretion standard. Gall v. United States, 552 U.S. 38, 51, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007). Sentences are procedurally unreasonable if the district court does not calculate the Guidelines range or calculates it improperly, treats the Guidelines as mandatory, fails to consider the factors in 18 U.S.C. § 3553(a), selects a sentence based on clearly erroneous facts, or gives an inadequate explanation for the sentence. *712 United States v. Peebles, 624 F.3d 344, 347 (6th Cir.2010).
When applying section 2B1.1(b)(1) to determine the amount of loss, the district court "need only make a reasonable estimate" of the amount. U.S.S.G. § 2B1.1 cmt. n. 3(C). The United States's burden is to prove the amount of loss by a preponderance of the evidence. See United States v. Zimmer, 14 F.3d 286, 290 (6th Cir.1994) (holding that the United States must prove the amount of drugs by a preponderance of the evidence when applying a sentencing enhancement for relevant drug activity). When reviewing a district court's application of section 2B1.1(b)(1), we review the district court's factual finding as to amount of loss for clear error. United States v. Warshak, 631 F.3d 266, 328 (6th Cir.2010) (citations omitted). "An error with respect to the loss calculation is a procedural infirmity that typically requires remand." Id. (citation omitted).
The court found that Jones had caused a loss of approximately $224,133,[2] and did so by adopting the statistical extrapolation evidence presented by the United States at trial. The district court provided no other explanation for how it reached that figure. Furthermore, the exhibits used by the United States at trial to explain how that figure was calculated were not introduced into the trial record or explained at the sentencing hearing. As a result, the only evidence relied upon by the district court in making its amount of loss determination at sentencing are the scant details regarding the statistical extrapolation.
A statistical estimate may provide a sufficient basis for calculating the amount of loss caused by a defendant, e.g., United States v. Kohlbach, 38 F.3d 832, 841 (6th Cir.1994); Zimmer, 14 F.3d at 290, but here, the United States's statistical analysis was flawed. The United States's statistician identified a representative statistical sample encompassing 357 bills contained throughout 264 patient files, but the United States found only 210 of those files. The United States did not present evidence that the 210 patient files still formed a representative sample of bills without the missing fifty-four files. Furthermore, it does not appear that the district court even realized that the fifty-four files were missing and it definitely did not make a finding as to whether they were fraudulent. Therefore, the accuracy of the extrapolation method is called into question.
If the United States had presented evidence that the 210 patient files contained a representative sample of bills, then the analysis could have reliably established that one hundred percent of Jones's claims billing code 11421 were fraudulent. However, the United States did not present any such evidence. The only statistically reliable sample group was the 357 bills. For the district court to determine the amount of loss was $224,133, it would have had to rely upon other evidence in addition to the statistical analysis, but it did not as far as we can tell from the sentencing record.
There is no rule that a district court must rely upon statistical analysis in a situation such as this to determine the amount of loss pursuant to section 2B1.1. However, here, the district court relied solely upon a statistical analysis. Without a sound representative sample, that analysis was flawed. As a result, the amount of *713 loss calculation was clearly erroneous. We remand this case so that the correct amount of loss may be established by a preponderance of the evidence. Because the sentence is procedurally unreasonable, we do not reach the question of substantive reasonableness. See Warshak, 631 F.3d at 330 (citing Gall, 552 U.S. at 51, 128 S.Ct. 586).

C. Jones's Fifth Amendment Due Process Rights and Sixth Amendment Right to Trial by Jury
Jones argues that being sentenced for his acquitted counts without having been convicted of them violates his Fifth Amendment due process rights and his Sixth Amendment right to a trial by jury. This Court reviews de novo constitutional challenges to a sentence. United States v. Graham, 622 F.3d 445, 452 (6th Cir.2010).
This question is controlled by our en banc opinion in United States v. White, 551 F.3d 381 (6th Cir.2008) (en banc). There, considering a Sixth Amendment challenge, we held that a "district court's consideration of acquitted conduct in sentencing passes constitutional muster ... insofar as enhancements based on acquitted conduct do not increase a sentence beyond the maximum penalty provided by the United States Code." Id. at 386. Furthermore, the Guidelines permit a sentencing court to "consider[] conduct underlying ... acquitted charge[s], so long as that conduct has been proved by a preponderance of the evidence." United States v. Watts, 519 U.S. 148, 157, 117 S.Ct. 633, 136 L.Ed.2d 554 (1997) (per curiam); accord White, 551 F.3d at 383 (citing Watts, 519 U.S. at 157, 117 S.Ct. 633).
Following White, the district court could properly consider Jones's acquitted conduct in determining his prison sentence. However, because we have decided here that the United States did not establish by a preponderance of the evidence that Jones caused more than $200,000 in loss through his acquitted conduct, Jones's constitutional claims are not completely precluded by White. We, nonetheless, need not address these claims in light of our decision to remand for resentencing. Should the district court again make findings on remand that are not supported by a preponderance of the evidence, Jones may reassert these claims in a later appeal.

D. Whether Restitution Was Permissible
Jones contests the district court's order that he pay $224,133 in restitution, raising the same objections he stated with respect to his prison sentence: that the amount is premised on acquitted conduct and was calculated using a faulty statistical extrapolation method. The Presentence Report recommended, and the district court ordered, that Jones pay restitution to Medicare and Medicaid pursuant to the Mandatory Restitution Act of 1996, 18 U.S.C. § 3663A, for both the fraudulent acts of which he was convicted and the ones of which he was acquitted. The amount of restitution was $224,133. We have already concluded that the district court employed a faulty statistical analysis, albeit for reasons different from Jones's, and that the case must be remanded so that the district court may recalculate the amount of loss caused by Jones's acquitted conduct. Thus, we need not reach his claim inasmuch as it attacks the amount of restitution ordered. That leaves intact, however, the question of whether the district court may order restitution at all for Jones's acquitted charges. "We review de novo the question of whether restitution is permitted under the law. ..." United States v. Elson, 577 F.3d 713, 725 (6th Cir.2009) (citing United States v. Boring, *714 557 F.3d 707, 713 (6th Cir.2009)) (internal quotation marks omitted).
Section 3663A(a)(1) states:
Notwithstanding any other provision of law, when sentencing a defendant convicted of an offense described in subsection (c), the court shall order, in addition to, or in the case of a misdemeanor, in addition to or in lieu of, any other penalty authorized by law, that the defendant make restitution to the victim of the offense. ...
Section 3663A(a)(2) further defines the term "victim" as follows:
For the purposes of this section, the term "victim" means a person directly and proximately harmed as a result of the commission of an offense for which restitution may be ordered including, in the case of an offense that involves as an element a scheme, conspiracy, or pattern of criminal activity, any person directly harmed by the defendant's criminal conduct in the course of the scheme, conspiracy, or pattern.
In the context of mail fraud convictions, we have read this statutory definition of "victim" to allow for restitution for the loss attributable to all the victims of a defendant's scheme to defraud, even when the defendant was not indicted or convicted of fraud with respect to each victim. United States v. Davis, 170 F.3d 617, 627 (6th Cir.1999); see also United States v. Jewett, 978 F.2d 248, 253 (6th Cir.1992). However, this precedent applies only when the loss is attributable to the precise scheme that was an element of the defendant's convicted offense. The Supreme Court's decision in Hughey v. United States, 495 U.S. 411, 110 S.Ct. 1979, 109 L.Ed.2d 408 (1990), which was partially superseded when Congress expanded the meaning of "victim" in § 3663A(a)(2) to its present definition, "continues to `require the court to exclude injuries caused by offenses that are not part of the [scheme] of which the defendant has been convicted.'" Elson, 577 F.3d at 723 (quoting United States v. George, 403 F.3d 470, 474 (7th Cir.2005)).
Jones's healthcare fraud and mail fraud convictions contain as elements schemes to defraud. The district court may, therefore, order restitution for any loss suffered by a victim of Jones's scheme. The next issue, then, is defining the scope of the scheme. We have encountered this issue before, but in the context of a conviction following a plea agreement. See id. (holding that the scope of the scheme is defined by "the plea agreement, the plea colloquy, and other statements made by the parties" when a conviction follows a plea agreement rather than a jury trial). When a defendant is convicted by a jury, however, the scope of the scheme is defined by the indictment for purposes of restitution. United States v. Adams, 363 F.3d 363, 366 (5th Cir.2004); United States v. Ramirez, 196 F.3d 895, 900 (8th Cir.1999); United States v. Henoud, 81 F.3d 484, 489 (4th Cir.1996) (quoting United States v. Brothers, 955 F.2d 493, 497 (7th Cir.1992)).
Jones's indictment defined his scheme as a broad, over-arching plan "to defraud and to obtain money by means of false and fraudulent pretenses, representations and promises." The indictment listed as components the more specific acts falling under that plan such as up-coding, billing for services not rendered to Carter and Tubbs, billing under Stewart's Medicare Provider Number, and others. Therefore, the grand jury defined Jones's scheme as a general plan to defraud Medicare and Medicaid by committing many different distinct acts. The scheme included not only the acts of which he was convicted, but also the ones of which he was acquitted.
*715 Accordingly, once the district court correctly determines the amount of loss caused by Jones on remand, it may properly order him to pay restitution for both his convicted and acquitted conduct.

III. CONCLUSION
Sufficient evidence supported Jones's convictions, but his sentencing went awry. The United States presented flawed statistical evidence to prove what amount of loss Jones had caused through his acquitted conduct. Accordingly, Jones's convictions are AFFIRMED, but his sentence is VACATED. We REMAND the case for resentencing.
NOTES
[*]  The Honorable Stephen J. Murphy, III, United States District Judge for the Eastern District of Michigan, sitting by designation.
[1]  It is unclear from the record whether the district court applied an offense level reduction or varied downward from the sentencing range. See United States v. Douglas, 634 F.3d 852, 864 (6th Cir.2011) (explaining the difference between an improper offense level reduction and a proper downward variance).
[2]  Although the district court did not state this exact figure at the sentencing hearing, it did state that the amount was approximately $224,000, and the amount ordered for restitution in the district court's judgment is $224,133.
