EASTERBROOK, Circuit Judge.

Chicago's Fire Department has five ranks: firefighter, engineer, lieutenant, captain, and battalion chief. Promotions depend on competitive examinations. Chicago developed the 1986 exam for lieutenant (a position open to firefighters and engineers) with care to ensure that it was both non-discriminatory and a valid test of skills. Yet although 29% of those who took the exam were either black or Hispanic, only 12% of those who received the highest 300 scores were in these groups. The Department concluded that this disparate impact could be justified, under the EEOC's Uniform Guidelines on Employee Selection Procedures, 29 C.F.R. § 1607.4, only if the exam were valid for rank-order use — that is, if someone who scores higher on the test is bound to perform better than the person next in line. According to the Department's expert, this examination had a standard error of measurement of 3.5, which is to say that a person who scored 80 and took a similar test again could score as high as 83.5 or as low as 76.5 without implying that his skills and probability of success in the higher position had changed. Convinced that it could not make promotions from the 1986 list in rank-order fashion, the Department established what it called "standardized" lists and what most people would call racially segregated lists: it drew up one list for whites and another for blacks and Hispanics, and then made 29% of all promotions from the minorities-only list. The Department used these lists until 1991, promoting a total of 209 lieutenants from the 1986 exam.

This process meant that the promotion of some white candidates was delayed, and others were not promoted even though minority candidates with lower scores became lieutenants. Some of the disappointed applicants filed suit under 42 U.S.C. § 1983 and Title VII of the Civil Rights Act of 1964. The Department acknowledged that its approach can be sustained only if a compelling interest supports its use of race and ethnicity. See, e.g., Adarand Constructors, Inc. v. Pena, 515 U.S. 200, 115 S.Ct. 2097, 132 L.Ed.2d 158 (1995). It did not argue that either past discrimination or a quest for diversity supports its approach. Cf. Grutter v. Bollinger, 539 U.S. 306, 123 S.Ct. 2325, 156 L.Ed.2d 304 (2003); Petit v. Chicago, 352 F.3d 1111 (7th Cir. 2003). Chicago instead maintains that it had a compelling need to comply with federal regulations that frown on using tests to make promotions in strict sequence. The district court bifurcated liability from damages and conducted a trial before an advisory jury, which concluded that the Department violated both § 1983 and Title VII. In open court the district judge agreed with this conclusion. Two later jury trials (limited to 19 of the plaintiffs) led to substantial awards of compensatory damages plus equitable relief such as front pay. Biondo v. City of Chicago, 2002 WL 982336, 2002 U.S. Dist. LEXIS 3463 (N.D.Ill. Feb. 28, 2002); Cloud v. City of Chicago, 2002 WL 1160930, 2002 U.S. Dist. LEXIS 9817 (N.D.Ill. May 31, 2002).

According to the Department, the jury's (and thus the judge's) finding that the 1986 exam was valid is clearly erroneous, see Pullman-Standard v. Swint, 456 U.S. 273, 102 S.Ct. 1781, 72 L.Ed.2d 66 (1982), even though the trier of fact credited plaintiffs' statistical expert, who supported the exam's validity. Bryant v. Chicago, 200 F.3d 1092 (7th Cir. 2000), sustains rank-order use of a similar exam, making it hard to see how the outcome of this trial could be deemed clearly erroneous. The Department acknowledges that the evidence taken in the light most favorable to the verdict shows the exam's overall validity as a test of skills but maintains that it does not demonstrate the propriety of making promotions in sequence, given the standard error of measurement. Even if the plaintiffs' expert undermined the conclusion that the standard error of measurement is 3.5, the Department insists, the record does not show that the standard error is zero, and without such a finding (which would be impossible, as no exam predicts perfectly) the exam could not have been used to make promotions in rank-order sequence, given the disparate impact that would have ensued.

Let us suppose that the City has the better of the statistical argument on this record (perhaps the one compiled in Bryant did more to show the exam's validity). We shall suppose further that the EEOC's regulations tell employers not to hire or promote in strict sequence when that would cause minority groups to succeed less than 80% as often as whites. But see Paul Meier, Jerome Sacks Sandy L. Zabell, What Happened in Hazelwood: Statistics, Employment Discrimination, and the 80% Rule, 1984 Am. Bar Foundation Research J. 139, 158-70. Still, the premise of the City's argument is that regulations supply a compelling governmental interest in making decisions based on race. How can that be? Then Congress or any federal agency could direct employers to adopt racial quotas, and the direction would be self-justifying: the need to comply with the law (or regulation) would be the compelling interest. Such a circular process would drain the equal protection clause of meaning. Decisions such as Adarand Constructors show that compliance with federal laws cannot automatically be a compelling interest; Adarand Constructors held a federal statute unconstitutional precisely because it required public officials to make use of race, and the statute was not itself supported by a compelling governmental interest. Chicago does not contend that 29 C.F.R. § 1607.4 carries out any compelling governmental interest, and given the holding of Washington v. Davis, 426 U.S. 229, 96 S.Ct. 2040, 48 L.Ed.2d 597 (1976), that disparate impact in hiring or promotion by a public employer does not violate the equal protection clause, it is hard to see how such an argument could be constructed. If avoiding disparate impact were a compelling governmental interest, then racial quotas in public employment would be the norm, and as a practical matter Washington v. Davis would be undone. Congress did not attempt this; to the contrary, it provided in 42 U.S.C. § 2000e-2(j) that an employer's desire to mitigate or avoid disparate impact does not justify preferential treatment for any group.

The Civil Rights Act of 1991 explicitly forbids the dualist response to disparate impact. 42 U.S.C. § 2000e-2( l). (That section also forbids differential validation, under which scores predicting an equal probability of success on the job lead to an equal probability of favorable decision even though this may mean that minorities are promoted with scores lower than those of white applicants. See Albemarle Paper Co. v. Moody, 422 U.S. 405, 95 S.Ct. 2362, 45 L.Ed.2d 280 (1975) (approving use of differential validation under statutes then in force). The City did not attempt differential validation before resorting to standardization.) The 1991 Act does not apply retroactively, see Landgraf v. USI Film Products, 511 U.S. 244, 114 S.Ct. 1483, 128 L.Ed.2d 229 (1994), but new § 2000e-2( l) and old § 2002-2(j) jointly reveal that standardization cannot be an indispensable response to disparate impact. Public employers have other options. One will suffice here. Instead of making rank-order promotions, Chicago could have created bands reflecting the standard error of measurement. For example, the Department could have treated all scores in the range 96-100 as functionally identical and made promotions by lot from that band; when all test-takers with those scores had been promoted, the Department could have pooled scores in the range 91-95 and promoted randomly from that group, and so on. That procedure would have respected the limits of the exam's accuracy while avoiding any resort to race or ethnicity. Given options of this kind, the City's two-list procedure cannot be thought compelled. Indeed, it is hard to credit the Department's assertion that it viewed rank-order promotions as unsupportable when that is how it actually used the exam. After creating racially segregated lists, the Department promoted in rank-order sequence from each list!

No more need be said about the merits. Remedies are more problematic. Each of the two juries was asked six questions: what was the likelihood that, but for the discrimination, each firefighter would have been promoted to captain and then battalion chief (and when each position would have been achieved); what damages represented loss of pay until the trial; and what damages were appropriate for emotional distress. The district judge then ordered the Department to make retroactive the positions of those plaintiffs who had achieved one or more promotions since 1986, and he granted front pay to those who (the jury found) would have been promoted further. The judge also ordered the City to pay prejudgment interest. This table shows most of the results: Probability Date of Probability Date of Promotion Back Damages Years of Promotion of Promotion to Pay for Emotional of Promotion to to Battalion Battalion Distress Front to Captain Captain Chief Chief Pay Peter Biondo John Byrne Richard Cloud Timothy Corcoran Michael Gacki Clifford Gartner Brian Gilhooly George Healy Patrick Joyce Steven Kolecki Thomas Maliska Bernard McDevitt David McElroy Michael Oliver Probability Date of Probability Date of Promotion Back Damages Years of Promotion Promotion of Promotion to Pay for Emotional of to to to Battalion Battalion Distress Front Captain Captain Chief Chief Pay John Piwinski Thomas Tervanis Michael Timothy Victor Walchuk Martin Wirtz

Some of these findings are hard to quarrel with. For example, Michael Timothy, whose promotion to lieutenant was delayed, made it to both captain (from the 1992 exam for that position) and battalion chief (from the 1994 exam); his damages reflect just a short delay in promotion and a bit of aggravation. Gacki and Wirtz became captains after the 1999 examination. But other findings are more problematic: the juries found that most of the plaintiffs were sure to be promoted even though, except for Gacki, Timothy, and Wirtz, they have failed to achieve advancement through the competitive process since becoming lieutenants (a position 15 of them managed to achieve, eventually, on their own: 5 by delayed promotion from the 1986 list, and 10 by promotion from later examinations). The finding that Gacki had a "0%" chance of becoming a captain is puzzling, as he actually achieved that position in 1999. (This finding was the work of the second jury; the first jury concluded that Timothy and Wirtz, the others who already were captains, had a 100% probability of promotion to that position. Neither jury behaved inconsistently.) Then there is the inexplicable verdict concerning Michael Oliver: 90% likely to become a captain and certain to become a battalion chief, even though the latter promotion is impossible without the former.

The City does not dispute the district court's central approach: asking the jury to determine the probability that being held back in 1986 cost the plaintiffs later chances for advancement. This "loss of a chance" method is the best way to handle probabilistic injuries. See Bishop v. Gainer, 272 F.3d 1009, 1015-16 (7th Cir. 2001); Doll v. Brown, 75 F.3d 1200, 1205-07 (1996); Griffin v. Michigan Department of Corrections, 5 F.3d 186, 189 (6th Cir. 1993). If four people competing for one position lost an equal chance to get it, then each should receive 25% of the benefits available. Or if a person would have had a 25% chance of promotion from lieutenant to captain, then preventing that person from becoming a lieutenant should lead to a remedy equal to 100% of the benefits of being a lieutenant plus 25% of the incremental benefits of being a captain. (This assumes that all plaintiffs are risk neutral; in most cases, however, they would pay to reduce risk, so the proper award should be something less than the actuarial value — for promotion is uncertain. Neither side makes anything of this, however, so for simplicity we assume risk neutrality.)

Unfortunately, the juries' estimates do not reflect a plausible appreciation of the lost chances: the juries concluded that every one of the plaintiffs who had not yet achieved a captaincy was certain to have done so — even though four plaintiffs had been unsuccessful in post-1986 attempts to become lieutenants, and 9 of the 15 plaintiffs who reached lieutenant had tried and failed to achieve higher ranks. (Timothy, Gacki, and Wirtz became captains after succeeding on exams; Corcoran, Joyce, and McElroy were promoted independently of the testing process, for reasons that the parties do not relate.) Even in a world of grade inflation, where teachers living far from Lake Wobegon think nothing of rating all students as "above average," it is hard to swallow a conclusion that all candidates held back from promotion to lieutenant in 1986 were sure to become captains. Remember that 209 firefighters were promoted to lieutenant from the 1986 exam. Plaintiffs ranked well below many of those 209, though high enough to have made the cut but for the preference given to minority firefighters. They are not the pick of the class, the people who regularly finish first on every exam. Of the 528 lieutenants who took the 1992 exam for captain (the first that these plaintiffs could have taken had they been promoted from the 1986 exam), only 136 (or 26%) were promoted. That one-in-four chance was the outcome for persons who, on average, had done better on the lieutenants' exam than plaintiffs (for even had they been promoted, they would have been in the bottom half of the 1986 lieutenant class). And of the 137 captains who took the 1994 exam for battalion chief, only 54 (or 41%) were promoted — though these juries concluded that 8 of our 19 plaintiffs had a 100% chance of promotion to battalion chief, and that another four had chances exceeding 41%.

One must take into account the possibility of sitting for multiple exams: 132 lieutenants who were not promoted from the 1992 test for captain applied again in 1998; of these, 29 (or 22%) have been promoted. This suggests that of those lieutenants who take the captain's exam at all, about 33% are promoted either from the first exam or a second try (if they persist through a second exam, and one could extrapolate that around 40% of aspiring lieutenants could achieve promotion by taking the exam) three or more times. Still, this is a long way from the 100% success rate that the juries calculated for our 19 firefighters.

The verdicts imply that, if we were to look at the most comparable firefighters — say, the 25 who ranked immediately above the plaintiffs on the 1986 exam and were promoted to lieutenant as a result — we would find that all became captains no later than November 2000, and that most would be battalion chiefs today. The record does not reveal how these firefighters actually fared, and plaintiffs, who bear the burden of persuasion, suffer from the omission. The best evidence of the chance they actually lost is missing. Nonetheless the verdicts might be sustainable if evidence demonstrated that, despite finishing out of the top hundred who took the 1986 exam, these plaintiffs must have had a bad day and were really more likely to succeed than the average exam-taker. That only 3 of the 19 have become captains by examination (some after two chances, in 1992 and 1998), suggests otherwise.

The record does not contain any evidence comparing these plaintiffs with other firefighters. Each of the 19 elected to present a non-comparative case. Each testified that he loves the work, strives to succeed, and studies hard for tests. Each presented evidence about his education and experience. Most presented family members and friends who testified to the long hours the plaintiff studied, the plaintiff's commitment to the fire department, and so on. This evidence does not permit quantification of the chance lost by being held back in 1986. Firefighters do not strive to meet an absolute standard; they compete against their colleagues. The other firefighters taking tests for higher positions also love the department and prepare thoroughly. Only a demonstration that these plaintiffs are much better at tests than their rivals would support a verdict that the chance lost was 100% or anything like it. Yet the plaintiffs bypassed their opportunity to introduce comparative evidence; and what evidence the record does contain — that these plaintiffs did not excel on the 1986 exam; that they have not done well on tests for promotion since then; that only 33% of those who were actually promoted to lieutenant by 1992, and pursued further promotion, made captain by 2002 — undermines the verdicts. Even giving all of the evidence and inferences a reading favorable to plaintiffs, no reasonable juror could find that all were destined to become captains. The verdicts therefore cannot stand

Because none of the plaintiffs presented comparative evidence, the view most favorable to the plaintiffs as a group is that each would have done as well as the average lieutenant on the 1992 and 1998 exams. This means that, but for the discrimination after the 1986 exam, 33% of the plaintiffs (or 6 of them) would have become captains by 2002 via examination. Three actually reached that level (and Timothy went on to battalion chief), and three more achieved a captaincy in some other way, so the remaining 16 lost only 3 captaincies (or none, depending on how the non-competitive promotions are handled). Perhaps it is inappropriate to hold the promotions of some plaintiffs against the rest in this way; it depends on whether we view these 19 as all of the victims of discrimination (in which event at most 3 promotions were lost) or as a random sample of the victims (in which event the lost-chance figure should be applied person by person). The City takes the latter view, which is the more favorable to the 13 plaintiffs who have stopped short of captain. (These 19 are not the only victims; other firefighters have claims yet to be tried; their disposition has been postponed to await our decision concerning these firefighters.) Accordingly, on remand each of the 13 is entitled to all of the benefits he would have received from a timely promotion to lieutenant, plus 33% of the benefits available from promotion to captain. And as about 41% of captains eventually become battalion chiefs, the award for the 10 plaintiffs who the jury thought likely to achieve that goal could include about 14% of the benefits of that position (a 33% chance of becoming a captain times the 41% chance that a captain eventually will advance again).

Other remedies depend on the unsupported view that all 19 plaintiffs were sure to become captains. Promotion to battalion chief is possible only for captains, so the high probabilities assigned to battalion chief for 11 plaintiffs are untenable. The back pay awards depend on the probabilities (and dates) of these promotions; they too must be vacated. The awards of compensatory damages for emotional distress also are linked to the estimates of promotional likelihoods. As the district judge remarked, this is why the awards to Timothy and Wirtz are relatively low: they suffered an insult and aggravation but were not held back significantly in their careers. The awards appear to be dominated by the jurors' belief that persons who should have been captains, and were stuck for a decade in the rank-and-file, suffered acute mental distress. Recognition that these plaintiffs were not destined for the captaincy doubtless would influence the evaluation of their degree of (justifiable) distress. Finally, the district judge's equitable orders (retroactive seniority and pensions, front pay) assume the validity of the jurors' conclusions. A change in the promotion probabilities and dates requires everything else to be redone. Consequently we vacate the entire judgment and remand for a new trial limited to the calculation of back pay and damages for emotional distress on the assumption that each of the plaintiffs who has yet to reach captain lost a 33% chance of promotion by 2002.

The district judge should see to it that any awards of compensatory damages for mental distress are proportional to the wrongs — and to the caps added by the 1991 Act. They do not apply directly to injury before 1991, but much of the harm post-dates that legislation. No matter how serious the offense, compensatory damages for discrimination covered by the 1991 Act cannot exceed $300,000. See 42 U.S.C. § 1981a(b)(3). The awards in this case reached $232,000 (to Tervanis), yet the emotional distress that these firefighters endured must be considerably less than that suffered by other victims of discrimination — some of whom lose their livelihood by racially motivated discharge, others of whom endure sexually or racially based harassment or assaults that can make a job a source of trauma and misery. Sometimes discrimination produces mental breakdowns and long-term disability. See Wilson v. Chrysler Corp., 172 F.3d 500 (7th Cir. 1999); Williamson v. Handy Button Machine Co., 817 F.2d 1290 (7th Cir. 1987). None of our plaintiffs suffered such substantial injuries. If awards even in these extreme cases are capped at $300,000, the award appropriate to discrimination that simply retards the rate of promotion must be considerably less in order to maintain proportionality. Compare Hennessy v. Penril Datacomm Networks, Inc., 69 F.3d 1344 (7th Cir. 1995), with Fine v. Ryan International Airlines, 305 F.3d 746 (7th Cir. 2002).

This leaves front pay. The district court added 12 years' worth, at the highest position the jury projected for each plaintiff (provided that the plaintiff was still employed at the Department in 2002). The judge explained that this unusually long period was justified by the infrequency of promotional opportunities and the fact that the plaintiffs were too distracted by the litigation (and the wrong done them in 1986) to be effective exam-takers earlier. Front pay must be reduced to match the opportunity lost — so a person who lost a 33% chance to become a captain by 2002 is limited to front pay at 33% of that position's salary until it is achieved by examination. (To be precise, such a person gets the salary and benefits of the position actually held, plus 33% of the difference in salary and benefits between that position and captain.)

As for duration: we agree with the City that 12 years exceeds the scope of a district court's equitable discretion. The goal of front pay is to put the victim in the financial position he should have enjoyed, when circumstances make it inappropriate to direct the employer to promote (or hire) him. See, e.g., Pollard v. E.I. du Pont de Nemours Co., 532 U.S. 843, 846, 121 S.Ct. 1946, 150 L.Ed.2d 62 (2001); Williams v. Pharmacia, 137 F.3d 944, 954 (7th Cir. 1998); Graefenhain v. Pabst Brewing Co., 870 F.2d 1198, 1212 (7th Cir. 1989). This means, we observed in Williams, that front pay cannot extend past the time a reasonable person needs to achieve the same or an equivalent position in the absence of discrimination. Even if the lingering effects of discrimination hindered promotional opportunities until trial in 2002, front pay cannot logically continue after the next unimpeded promotional opportunity — the first post-2002 exam for captain (for those plaintiffs who were lieutenants in 2002) or battalion chief (for those who had become captains). The opportunity to seek promotion then gives each plaintiff everything to which he is entitled; a lieutenant or captain who does not achieve a competitive promotion no longer can blame his status on discrimination that delayed (but did not prevent) promotion to lieutenant. Setting this limit also gives plaintiffs an incentive to compete for promotions, unlike the district court's remedy, which all but guaranteed plaintiffs the highest possible salary through retirement without the need to seek advancement or perform the duties of the higher positions.

Perhaps what we have said will lead the litigants to resolve these remaining issues (and the remaining firefighters' claims) amicably rather than slug it out again in the courtroom. We hope so; this dispute is approaching its third decade. The judgments are vacated, and the case is remanded for proceedings consistent with this opinion.