ARNOLD, Circuit Judge.

Valmont Industries, Inc., appeals from a judgment that it violated the Age Discrimination in Employment Act (ADEA), 29 U.S. C. §§ 621 et seq., by terminating Raymond MacDissi on account of his age. MacDissi cross-appeals from the same judgment, which held that Valmont did not terminate him on account of his ethnic origin in violation of Title VII, 42 U.S.C. § 2000e, and dismissed his claim based on 42 U.S.C. § 1981. Both parties find fault with the scope of the District Court’s 1 remedial order. We see no clear error in the trial court’s factual findings on the merits of MacDissi’s claims. Nor do we find any abuse of discretion in its determination of the remedy. We therefore affirm.

The facts, summarized in the light most favorable to the party prevailing before the trial court on each claim, are as follows. MacDissi, a 51-year-old man of Lebanese descent, lost his job as a rate analyst in Valmont’s Traffic Department as part of a company-wide reduction in force in April 1982. At the time of his termination, Mac-Dissi had seven and a half years of experience in the traffic department, more than any other nonmanagerial employee. His annual performance ratings were consistently excellent, and Valmont concedes that MacDissi was qualified to perform his job.

At the time of the layoff, Valmont’s management directed Phil Williams, the head of the Traffic Department, to select two of the department’s nine non-managerial employees for termination. MacDissi, fifty-one, was the oldest of this group, followed by forty-eight-year-old Alberta Stewart. With the exception of a third employee aged forty-five, none of the remaining employees of the department was older than thirty-three at the time of the 1982 layoff. MacDissi and Stewart were the two employees selected for termination.

At trial, Valmont presented two reasons for MacDissi’s termination. First, it claimed that the adoption of a computerized system for analyzing transportation *1057 costs made MacDissi’s special function— manual auditing and analysis of freight costs — obsolete. The evidence showed that some degree of manual auditing of rates was still required after implementation of the computerized system, with other Traffic Department employees assuming responsibility for up to 85% of MacDissi’s former duties. Second, Valmont urged that deregulation of the trucking industry enormously simplified the rate structure by permitting Valmont to shift to a mileage-based tariff system. According to Val-mont, this change eliminated the need for MaeDissi’s special expertise in analyzing freight charges. The evidence showed that this change to a mileage-based rate structure did not simplify charges for less than truckload (LTL) shipments, and that these LTL shipments had become MacDissi’s major area of responsibility by 1982.

MacDissi was the only employee in the Traffic Department who was not born in the U.S. While supervisors and co-workers occasionally joked with MacDissi about Israeli victories in the Middle East, MacDissi conceded at trial that these comments were made in jest.

The District Court granted judgment for Valmont on MacDissi’s Title VII claim, holding that he had not established a prima facie case of discrimination because of national origin. MacDissi’s § 1981 claim had been dismissed before trial on the ground that his Lebanese origin did not constitute a “race” within the meaning of § 1981. The District Court did find that MacDissi had established a prima facie case of age discrimination, and held that Valmont’s proffered reasons for terminating him were not believable. The court awarded MacDissi $104,285.22 in back pay, $3,195.94 in health insurance and medical expenses, and $46,851.75 in prospective wages in lieu of reinstatement. The Court denied liquidated damages under § 626(b) of the Act and prejudgment interest. On Valmont’s motion, the Court struck its original order to provide MacDissi with future pension benefits, after Valmont showed that its plan had already been terminated and Mac-Dissi had already received a lump-sum settlement.

I. The ADEA Claim

On appeal, Valmont urges us to reverse the District Court’s findings on MacDissi’s ADEA claim as clear error. Valmont begins by attacking the sufficiency of Mac-Dissi’s prima facie case of age discrimination, on the ground that a nine-employee Traffic Department is too small a statistical universe for MacDissi’s termination to create a reliable inference of discrimination. Valmont reasons that, because Mac-Dissi failed to satisfy his initial burden under the three-stage sequence of proof described in McDonnell Douglas Corp. v. Green, 411 U.S. 792, 93 S.Ct. 1817, 36 L.Ed.2d 668 (1973), the District Court clearly erred by proceeding to examine Val-mont’s reasons for terminating him under the subsequent stages of the McDonnell Douglas sequence.

The problem with Valmont’s focus on the sufficiency of MacDissi’s prima facie case is that it requires us to review the District Court’s judgment in the same piecemeal way rejected in U.S. Postal Service Board of Governors v. Aikens, 460 U.S. 711, 103 S.Ct. 1478, 75 L.Ed.2d 403 (1983). Once the District Court has decided the ultimate question of discrimination vel non, the pri- or question of whether the plaintiff has made out a prima facie case loses its significance. Our task as an appellate court is not to reach separate conclusions on the adequacy of the evidence adduced at each of the three McDonnell Douglas stages of proof — it is simply to decide whether the record supports the ultimate finding of discrimination. See Aikens, 460 U.S. at 714-16, 103 S.Ct. at 1481-82; Barber v. American Airlines, 791 F.2d 658, 659-60 (8th Cir.), cert. denied, 479 U.S. 885, 107 S.Ct. 278, 93 L.Ed.2d 254 (1986).

Taken as one component of MacDissi’s case, a comparison of the ages of those laid off to the ages of those retained does provide at least some support for MacDissi’s claim. Both parties have complicated the quantitative aspects of this case by relying on involved statistical comparisons to describe relatively simple facts. It is unnecessarily awkward, for example, to say that *1058 the April layoff reduced the ratio of Traffic Department employees in the protected class from 50% (six out of twelve) to 40% (four out of ten), and that the mean age in the Traffic Department declined from 39 years to 36.8 years after the layoff. 2 In many cases, a drop in the ratio of older employees from 50% to 40%, or a decline of 2.2 years in the mean age, would indeed be virtually meaningless. In this case, however, these overly formal statistical descriptions obscure the more obvious significance of the facts. Valmont chose to terminate the two oldest employees, ages 51 and 48, in a department in which all but one of the nonmanagerial employees remaining were 33 years old or younger. This fact is certainly not conclusive evidence of age discrimination in itself, but it is surely the kind of fact which could cause a reasonable trier of fact to raise an eye-brów, and proceed to assess the employer’s explanation for this outcome.

Valmont argues vigorously that inferences of discrimination cannot be reliably drawn from two discharges in a nine-iriember department, citing several cases in which statistics based on workplaces of up to 50 employees have been rejected as insignificant. Taken to its logical conclusion, Valmont’s position is that plaintiffs employed in smaller workplaces can never use statistics to establish a circumstantial case of discrimination, since the size of the workplace precludes any pattern observed in the data from proving intentional discrimination to a statistical certainty. This approach would unjustifiably deny employees in smaller workplaces the protection of federal discrimination law. See EEOC v. American National Bank, 652 F.2d 1176, 1194 (4th Cir.1981), cert. denied, 459 U.S. 923, 103 S.Ct. 235, 74 L.Ed.2d 186 (1982). There is no minimum sample size prescribed either in federal law or in statistical theory: the adequacy of numerical comparisons within small sets of data depends on the degree of certainty the factfinder requires, as well as the type of inference the statistics are meant to demonstrate. 3

In this case, MacDissi is not relying on the comparison of the ages of those laid off to the ages of those retained as conclusive proof of intentional discrimination. In this sense, Valmont’s citations to authority rejecting statistics based on larger sample sizes than MacDissi’s Traffic Department do not apply. This is not a case, like Harper v. Trans World Airlines, 525 F.2d 409 (8th Cir.1975), Soria v. Ozinga Bros., 704 F.2d 990 (7th Cir.1983), or Contreras v. City of Los Angeles, 656 F.2d 1267 (9th Cir.1981), cert. denied, 455 U.S. 1021, 102 S.Ct. 1719, 72 L.Ed.2d 140 (1982), in which plaintiffs seek to prove that facially neutral employment criteria have a disparate impact on persons in a protected class. In disparate-impact cases, statistical patterns of disparity are typically the entire basis of the plaintiffs’ claims, whereas in cases like MacDissi’s, statistics are used only as circumstantial evidence which tends to support a specific claim of disparate treatment. Nor is this a case, like Parker v. Federal National Mortgage Association, 741 F.2d 975 (7th Cir.1984), or Holley v. Sanyo Manufacturing, 771 F.2d 1161 (8th Cir.1985), in which a plaintiff seeks to discredit an employer’s proffered non-discriminatory reason for termination solely by demonstrating an overwhelming statistical probability of discrimination. MacDissi provides independent, direct grounds for disbelieving Valmont’s explanation for his layoff, and so his quantitative evidence does not need to reach the degree of certainty required of plaintiffs who present no proof of discrimination besides a statistical pattern.

Valmont objects further that the District Court clearly erred in considering *1059 only layoffs within the Traffic Department, arguing that company-wide layoff statistics do not suggest the same pattern of age discrimination. There is no dispute, however, that the decision to select MacDissi for layoff was made by Phil Williams, the head of the Traffic Department. Evidence concerning layoff decisions made outside the Traffic Department would have relatively little relevance to determining Williams’s discriminatory animus. While company-wide statistics are useful in establishing the presence or absence of a general climate of age bias, we cannot see any clear error in the trial court’s decision to give the most weight to layoffs within the Traffic Department.

Valmont also urges us to find clear error in the District Court’s refusal to believe its proffered reasons for terminating MacDissi. Valmont argues that the District Court ignored several ways in which the introduction of a computerized transportation analysis system reduced the need for MacDissi’s auditing services, and that the transition to mileage-based freight rates simplified MacDissi’s job out of existence, notwithstanding the District Court’s finding that his primary responsibility was the non-mileage-based LTL shipments. On review of the conflicting evidence in the record, we cannot say how much of MacDissi’s auditing function has actually been eliminated, nor is it clear exactly how many of his duties were taken up with LTL shipments. Nor are we presented with a clear understanding of why MacDissi, and not one of the younger employees with less seniority, was laid off. When the record is opaque, we defer to the judgment of the District Court, particularly where its understanding of Valmont’s defense is enhanced (as ours is not) by its observation of Valmont’s witnesses on the stand. We do not have a sufficiently clear reason to say that the District Court made a mistake in refusing to believe Valmont’s reasons for terminating MacDissi.

Valmont objects that, even if its proffered reasons for firing MacDissi were not its true reasons, MacDissi must still prove intentional discrimination, instead of merely discrediting the employer’s defense. Valmont’s approach unjustifiably multiplies the plaintiff’s burden. An employment-discrimination plaintiff “... may succeed in [persuading the court that she has been a victim of intentional discrimination] either directly ... or indirectly by showing that the employer’s proffered explanation is unworthy of credence.” Texas Department of Community Affairs v. Burdine, 450 U.S. 248, 256, 101 S.Ct. 1089, 1095, 67 L.Ed.2d 207 (1981). As a matter of both common sense and federal law, an employer’s submission of a discredited explanation for firing a member of a protected class is itself evidence which may persuade the finder of fact that such unlawful discrimination actually occurred. 4

In this case, the evidence surrounding the 1982 layoff persuaded the District Court that Valmont had intentionally discriminated against MacDissi on account of his age, and we cannot say that the record before us clearly dictates the opposite conclusion. We therefore affirm the judgment for MacDissi on his ADEA claim.

II. The Title VII and § 1981 Claims

On cross-appeal, MacDissi argues that the District Court erroneously held that he had failed to make out a prima facie case of discrimination based on his Lebanese national origin under Title VII. MacDissi argues that each of the McDonnell Douglas criteria for a prima facie case had been satisfied, and that since Val-mont’s explanations for terminating him *1060 were the same as these discredited under the ADEA claim, that he is therefore entitled to judgment. MacDissi’s approach to his Title VII claim suffers from the same unnecessary formalism that characterizes Valmont’s preoccupation with his prima fa-cie case under ADEA. Following Aikens, supra, we are no longer interested on appeal in the individual segments of the proof of MacDissi’s employment discrimination claim, but only with review of whether the record supports the trial court’s findings of discrimination vel non. In this case, the District Court found that there was no discrimination against MacDissi based on his heritage, especially in light of MacDis-si’s admission that the occasional remarks about his origin were made in fun and without malice. We see no clear error in this finding. 5

MacDissi argues further that the District Court mistakenly dismissed his § 1981 claim when it held that the § 1981 protection against racial discrimination does not extend to people of Lebanese descent. We agree that his § 1981 claim should have gone to trial, in light of the Supreme Court’s subsequent decision in Saint Francis College v. Al-Khazraji, 481 U.S. 604, 107 S.Ct. 2022, 95 L.Ed.2d 582 (1987), which held that persons belonging to a distinct ethnic sub-group, like Arabs, are indeed protected under § 1981. We do not agree, however, that there could have been a result before the District Court on MacDissi’s § 1981 claim different from that on his Title VII claim. 6 In some cases, the distinction between a § 1981 claimant’s race and his national origin may prevent an adverse judgment under Title VII from having a preclusive effect as to the § 1981 claim. 7 In MacDissi’s case, however, this appears to be a difference without significance. In ruling on MacDissi’s Title VII claim, the District Court found that Mac-Dissi was not discriminated against because of his Lebanese ethnic background, and this finding appears to apply whether we conceive of his Lebanese descent as “race” or as “national origin.” As a result, we find that MacDissi’s failure to prove his Title VII claim precludes relief on his § 1981 claim.

III. The Remedy

The parties present various objections to the District Court’s remedial order. Valmont contends that the District Court should not have awarded MacDissi front pay, in light of the Court’s decision that subsequent changes in Valmont’s operation had made his reinstatement impracticable. Valmont reasons that, if MacDissi is not currently employable at Valmont, it is unjust to order Valmont to pay him prospective wages for a job which no longer exists. This argument confuses the question of whether MacDissi could presently fit back into Valmont’s operations (after six years of absence) with the question of whether MacDissi would have remained employed if he had never been subject to age discrimination to begin with. These two questions may have different answers. In particular, courts will presume for the purposes of awarding relief that an illegally discharged employee would have continued working for the employer until he or she reaches normal retirement age, unless the employer provides evidence to the contrary. See Gibson v. Mohawk Rubber Co., 695 F.2d 1093, 1101 n. 8 (8th Cir.1982). Valmont presented no such evidence. Although the calculation of a front-pay award necessarily involves some uncertainty, it is a matter of equitable relief which we leave to the sound discretion of the District Court. We see no abuse of discretion in MacDissi’s front-pay award.

*1061 MacDissi, in turn, argues that the District Court clearly erred in failing to find that Valmont’s conduct constituted a “willful violation” which would entitle Mac-Dissi to liquidated damages under 29 U.S. C. § 626(b). According to MacDissi, the record clearly demonstrates that Valmont did not systematically review its employment decisions to ensure compliance with the ADEA. This argument seems to assume that a company’s negligence in identifying possible ADEA violations equates with the “reckless disregard” the law requires in determining willfulness under § 626(b). This is precisely “the broad standard ... [which] would result in an award of double damages in almost every case” rejected in Trans World Airlines v. Thurston, 469 U.S. 111, 128, 105 S.Ct. 613, 625, 83 L.Ed.2d 523 (1985). In this case, we find nothing in the record which suggests the kind of conscious intent to violate the law which would justify a finding of willfulness. See Bethea v. Levi Strauss & Co., 827 F.2d 355, 359 (8th Cir.1987).

MacDissi further argues that the District Court abused its discretion in denying him prejudgment interest, suggesting that prejudgment interest may be presumptively due in ADEA cases. Although prejudgment interest may be appropriate in cases where the award does not otherwise make the victim of discrimination whole, there are a number of reasons — especially the generous amount of MacDissi’s front-pay award — which could lead the District Court to conclude that prejudgment interest was not appropriate here.

Next, MacDissi challenges the propriety of the District Court’s decision to amend its original judgment to strike pension benefits from its award. The District Court acted on Valmont’s post-judgment presentation of evidence which showed that its pension plan had been terminated in 1984, and held that MacDissi would not have been able to accrue additional benefits (beyond the lump-sum settlement of his previously vested benefits paid on termination) under a plan which no longer existed. MacDissi objects that Valmont should not be entitled to relief from the Court’s original judgment when it was in possession of the evidence that the plan had already been terminated at trial. We decline to reinstate the original pension award— which is clearly inappropriate given the termination of the plan — on such a technical ground. Although Valmont clearly should not have waited until after judgment to fill in the trial judge’s understanding of the plan, the District Court acted within its discretion by allowing Val-mont to correct its inadvertent omission on a motion to amend the judgment.

Finally, Valmont protests the District Court’s award of $34,500 to MaeDis-si’s attorneys, on the ground that MacDis-si’s attorneys submitted reconstructed, rather than contemporaneous, records of time expended on MacDissi’s case. Val-mont suggests no particular reason to believe that these reconstructed records overstate the time actually spent. Instead, Val-mont seems to be urging us to adopt a per se rule that the failure to keep contemporaneous time records automatically precludes the recovery of attorneys’ fees. We decline to do so. The question of whether reconstructed records accurately document the time attorneys have spent is best left to the discretion of the court most familiar with the litigation. The maintenance of such records is certainly desirable, and district courts may reduce or eliminate attorneys’ fees awards where the absence of such records leaves the court without a reliable basis on which to award fees. See Hensley v. Eckerhart, 461 U.S. 424, 433, 103 S.Ct. 1933, 1939, 76 L.Ed.2d 40 (1983). In this case, however, the District Court concluded that the attorneys’ records reconstructed from contemporaneous notes did satisfactorily document their time, and that the time expended was reasonable in the context of this litigation. We have no reason to doubt that this assessment was a proper exercise of the District Court’s discretion.

The judgment of the District Court is accordingly

Affirmed.

. The Hon. Lyle E. Strom, Chief Judge, United States District Court for the District of Nebraska.

. For some reason, these figures, adopted by the District Court, include the three management employees, all of whom were over 40 years old, but none of whom was subject to layoff.

. The concept of statistical significance entails an arbitrarily determined threshold of probability that observed disparities are not due to chance. See Craik v. Minnesota State University Board, 731 F.2d 465, 475-76 n. 13 (8th Cir.1984). Even where quantitative evidence does not alone demonstrate discrimination to some judicially created standard of statistical conclusiveness, it is still relevant in conjunction with all other evidence in determining intentional discrimination. See id.

. In this context, we reject Valmont’s related argument that the District Court unjustifiably substituted its business judgment for Valmont’s. It is undeniably true that the ADEA does not empower courts to choose which business strategies should be implemented or which employees hired or fired. The ADEA does, however, require courts to examine critically employer rationales based on business necessity when they are presented as explanation for allegedly discriminatory conduct. Here, the District Court’s comparison of MacDissi’s capability to those of his younger co-workers, as well as its evaluation of the changing requirements of the Traffic Department, was entirely necessary to determine whether Valmont’s explanation for its decision was pretextual.

. Nor is this finding inconsistent with the District Court’s finding of discriminatory animus based on age. Any number of factors — e.g., the contemporaneous firing of a 48-year-old white woman, the demeanor of the witnesses — could have led a reasonable fact-finder to conclude that age, but not ethnicity, was the real reason MacDissi was laid off.

. MacDissi did not request a jury trial for any claim brought as part of this lawsuit.

. For example, it is conceivable that an immigrant from the Chinese community in Cuba could be exposed to racial discrimination actionable under § 1981, even where the same person’s employer had no discriminatory animus against people born in Cuba.