POSNER, Chief Judge.

Nine white officers of the Chicago Fire Department brought this suit in 1993 against the City and City officials, charging racial discrimination in promotions. After a bench trial, the district judge gave judgement for the defendants and dismissed the suit. All of the plaintiff officers were lieutenants in 1987 and took the exam given that year for promotion to captain. Three of them had done well on the exam only to be passed over in favor of black and Hispanic lieutenants who received lower scores. These three white officers were, however, promoted to captain a year later. The other six plaintiffs did not do as well on the exam and were denied promotion outright, rather than merely being delayed for a year.

We had a pair of similar cases several years ago (though with different plaintiffs), involving a challenge to affirmative action in promotions in the Chicago Fire Department, including promotions to captain. Billish v. City of Chicago, 989 F.2d 890 (7th Cir. 1993) (en banc). Those cases had been decided — prematurely, we ruled — on summary judgment; now we have a decision to review that was based on a full trial. But before getting to the merits, we address two jurisdictional issues.

The first, which actually is frivolous though urged by the appellants as their first ground of appeal, is whether the district judge had jurisdiction, after the appeal was filed, to replace "Chicago Police Department" with "Chicago Fire Department" in one sentence of the opinion. As this was the correction of a clerical error, it could be made by the district judge "at any time," Fed.R.Civ.P. 60(a), even after jurisdiction over the case shifted to this court by virtue of the filing of the notice of appeal. Local 1545 v. Inland Steel Coal Co., 876 F.2d 1288, 1292 n. 4 (7th Cir. 1989); Stein v. Wood, 127 F.3d 1187, 1189 (9th Cir. 1997). Even if this were wrong, and the error could not be corrected while the case was on appeal, the existence of a trivial error, obviously merely typographical, in the district court's opinion would not be a ground for reversing the judgment.

The second jurisdictional issue is substantial, but also not dispositive of the appeal. It concerns the standing of the six plaintiffs who were denied promotion outright. They ranked between 152 and 166 in the scoring of the 1987 exam for captains. Had the City promoted in strict rank order, without any favoritism for members of minority groups, it would have promoted 143 of the examinees and reached number 146 (because three of the top 146 scorers waived promotion); no one ranking below 146 on the exam would have been promoted. It is true that the actual number of promotions made was not 143, but 161, and that the last promotion would have been 166 (because there were two more waivers) had promotions been made in strict rank order. But the additional 18 promotions (161 — 143) were made pursuant to the same affirmative action plan that was responsible for the favored treatment of which the plaintiffs are complaining. Had there been no favoritism, the six low-ranking plaintiffs would not have been promoted, because promotions would have stopped at 146 and the highest-ranking of these plaintiffs was as we said number 152.

A plaintiff who would have been no better off had the defendant refrained from the unlawful acts of which the plaintiff is complaining does not have standing under Article III of the Constitution to challenge those acts in a suit in federal court. Lujan v. Defenders of Wildlife, 504 U.S. 555, 560 (1992); Simon v. Eastern Kentucky Welfare Rights Organization, 426 U.S. 26, 41 (1976); Harp Advertising Illinois, Inc. v. Village of Chicago Ridge, 9 F.3d 1290 (7th Cir. 1993). That appears to be (but, as we are about to see, may not be) the situation of these six plaintiffs. The remaining plaintiffs clearly were hurt by the affirmative action plan, because they ranked in the top 146 but were passed over to make room for minority lieutenants. The harm was temporary, because they were promoted to captain a year later, and therefore limited; but obviously not so limited as to deprive them of standing.

But just as a person may recover damages for emotional distress caused by his having been denied due process of law, even though if he had received due process he would still have lost the case in which the procedural violation occurred, Carey v. Piphus, 435 U.S. 247, 263-64 (1978), so it can be argued that a person denied a benefit on an invidious ground may obtain damages for emotional distress caused by that denial even if he would have been denied the benefit anyway. And Price v. City of Charlotte, 93 F.3d 1241, 1245-48 (4th Cir. 1996), so holds. Neither the Supreme Court nor this court has had occasion to examine the question. The objection to the answer given by Price is that it enormously multiplies the number of potential plaintiffs in equal protection cases; persons who had not been harmed in the usual sense could nonetheless sue for damages resulting from the insult implicit in any denial to a person of the equal protection of the laws. But this is not the case in which to try to wrestle the issue to the ground. Since three of the plaintiffs have standing, the merits issue cannot be avoided, and in these circumstances we may elide the jurisdictional issue. Steel Co. v. Citizens for a Better Environment, No. 96-643, 1998 WL 88044 at *9 (U.S. March 4, 1998). And we may do so for the additional reason that, given the claim of damages for emotional distress, the jurisdictional issue is not whether the plaintiffs have been harmed (Article III) but whether they should be allowed to sue for that harm (zone of interests). The latter type of jurisdictional issue ("prudential standing" as it is sometimes called) may be bypassed in favor of deciding the merits when the outcome is unaffected and the merits issue easier than the jurisdictional issue. Id.

So let us turn to the merits. The parties agree that minority lieutenants received favorable treatment on account of their race (in the case of the blacks) and their ethnicity (in the case of the Hispanics) and that such treatment is a denial of equal protection unless it can be justified under the demanding standard that goes by the name "strict scrutiny." Adarand Constructors, Inc. v. Pena, 515 U.S. 200, 236-37 (1995). They further agree, as they also must under the cases, that one justification that passes muster under this demanding standard is that the favored treatment is necessary to remedy unlawful discrimination in the past by the entity conferring the favor. E.g., People Who Care v. Rockford Board of Education, 111 F.3d 528, 535 (7th Cir. 1997). Whether other justifications are possible is unsettled, United States v. Paradise, 480 U.S. 149, 167 n. 18 (1987); compare City of Richmond v. J. A. Croson Co., 488 U.S. 469, 493 (1989) (plurality opinion); Taxman v. Board of Education, 91 F.3d 1547, 1557 (3d Cir. 1996); and Hopwood v. Texas, 78 F.3d 932, 944 (5th Cir. 1996), with Regents of the University of California v. Bakke, 438 U.S. 265, 311-14 (1978) (opinion of Justice Powell), but this court has held that at least in the area of policing and corrections it is possible to justify nonremedial affirmative action upon a showing — which must however be based on evidence rather than merely on conjecture — that such action is necessary to the accomplishment of important law-enforcement objectives. Wittmer v. Peters, 87 F.3d 916 (7th Cir. 1996).

At argument there was some discussion of whether a similar justification might be available in the case of firefighters, who may lack credibility, and be denied cooperation, in minority neighborhoods if the firehouses in those neighborhoods have only white personnel. Cf. Alexander v. Estepp, 95 F.3d 312, 316 (4th Cir. 1996); Hayes v. North State Law Enforcement Officers Ass'n, 10 F.3d 207, 213 (4th Cir. 1993). But as the City presented very little evidence of this at trial, it is not available to support the judgement on appeal. Racial discrimination cannot survive challenge without compelling evidence; even highly plausible speculation will not do. Id. at 214; Wittmer v. Peters, supra, 87 F.3d at 918-19.

So the only available justification is the remedial one, and it requires proof both that the City engaged in racial and ethnic discrimination in the employment of firefighters in the past and that the remedy is narrowly tailored to the violation, which means, as a practical matter, that it discriminates against whites as little as possible consistent with effective remediation. Shaw v. Hunt, 116 S.Ct. 1894, 1902-03 (1996); Billish v. City of Chicago, supra, 989 F.2d at 893; Contractors Association of Eastern Pennsylvania, Inc. v. City of Philadelphia, 91 F.3d 586, 596 (3d Cir. 1996); Podberesky v. Kirwan, 38 F.3d 147, 153 (4th Cir. 1994). To assess the validity of the remedy in this case requires us to review the evidence on both these fronts (past violation and tailored remedy), giving proper weight to the district judge's findings of fact — a precept the plaintiffs ignore.

Until comparatively recently the City employed very few minority firefighters relative to their proportion of the city's population. As late as 1974, only 4 percent of the firefighting force was black and only one-half of 1 percent was Hispanic (figures essentially unchanged from 1950, the earliest year for which there are statistics in the record), even though blacks and Hispanics accounted for a much higher percentage of the city's population. (Blacks alone were almost a third of the city's population in 1970.) Raw statistical disparities prove little; they certainly do not prove intentional discrimination. A glance at almost any occupation in the United States will reveal disparities, often stark, between the racial, ethnic, religious, and sexual composition of the occupation and that of the American population in general, and not all of these disparities are plausibly attributable to intentional, or for that matter unintentional, discrimination, let alone intentional discrimination by the particular employer that has been sued. Racial, ethnic, religious, and sexual identity are correlated with attitudes, accomplishments, tastes, experiences, aspirations, opportunities, educational background, and other aspects of personality and personal history that bear on eligibility and suitability for, and the choice of, a particular occupation. Some of the differences may of course be due to discrimination, but the only discrimination for which an employer is liable is the discrimination that he engages in.

The statistician who testified for the City tried to correct for some of the nonculpable factors that bear on occupational choice by comparing the racial and ethnic composition of people who applied for jobs with the Chicago Fire Department with the racial and ethnic composition of the city's population in the relevant age group, viewed as a control group. It turned out that at least in the case of blacks the composition of the applicant group was similar to that of the control group, indicating that in the early 1970s blacks were being hired at a lower rate than white applicants, since they were applying at a similar rate yet were underrepresented in the fire department. Even this, of course, was not decisive, for the whites might simply have had better aptitudes. Whites on average have more years of schooling than blacks, and this is important in many jobs — but not, so far as the record shows, in a firefighter's job, where no college is required and the entry-level firefighter's job does not, at least according to this record, require special skills, see also Peightal v. Metropolitan Dade County, 26 F.3d 1545, 1554 (11th Cir. 1994), let alone skills correlated with membership in particular racial or ethnic groups. We note, however, that the statistician's evidence does not suggest discrimination in the hiring of Hispanics, who until the middle 1980s applied for firefighters' jobs at a substantially lower rate than their percentage of the relevant age group.

In 1974, the City settled a discrimination suit by the federal government by agreeing to hire a much higher percentage of blacks and Hispanics. But immediately the City instituted a hiring freeze as a result of which no firefighters of any race were hired for three years. When the freeze ended, the percentage of black and Hispanic firefighters was no higher than it had been in 1950, even though the black and Hispanic percentage of the population of Chicago had doubled.

In 1979, upon motion by the Department of Justice to enforce the consent decree, the district court issued a further order requiring preferential hiring of blacks and Hispanics. The Department also questioned the exam that the City had administered in 1978 for promotion to lieutenant and captain; the exam contained a large subjective component and had not been shown to test for the knowledge and skills actually required in a fire lieutenant or captain. Again the City settled. The City redid the exam; and it was the redone and presumably race-neutral exam that was administered to would-be captains in 1987. Because so few blacks and Hispanics had been hired in the 1960s and 1970s, and because it takes many years for a firefighter to work his way up to lieutenant — and only lieutenants are eligible for promotion to captain — in 1986 only 2.6 percent of the captains were black and only 1 percent were Hispanic, even though by this time a majority of the City's population was black or Hispanic.

There was also direct evidence of racial discrimination in the Chicago Fire Department up to and into the 1980s, much of it concerning the racial policies of senior officials of the department, including one Fire Commissioner. The plaintiffs point out that no minority witness testified that he had applied and been turned down for an entry-level position and that the minority witnesses who had been hired and did testify had been promoted, in some cases to the very top. This was certainly evidence for the district judge to consider. But there was enough contrary evidence to justify the judge's conclusion that until sometime during the 1980s the people running the fire department endeavored with considerable success to make the department uncongenial to blacks and Hispanics. This conclusion in turn justified an inference that the under-representation of blacks and Hispanics in the various ranks, including that of captain, was due, at least in part, to discrimination by the City.

The groundwork thus was laid for justifying on remedial grounds the application of a plan for affirmative action to the results of the 1987 captains' examination. The exam itself is conceded to have been fair, and had promotions been made in strict order of rank on the exam 20 of the 143 lieutenants promoted would have been black (14 percent) and 5 would have been Hispanic (3.5 percent, though the figure in the record is a mysterious 4.7 percent). The affirmative action plan called for increasing the fraction of minority lieutenants promoted to captain to 20 percent black and 5 percent Hispanic, but for doing so with minimum adverse impact on whites. To this end, the City made the 18 additional promotions, for a total of 161, of which 31 were of black lieutenants and 9 were of Hispanic lieutenants. Because there were so few black and Hispanic captains before the new promotions, these promotions, though slanted in favor of the minority lieutenants, did not bring the racial and ethnic composition of the fire captains into conformity with the racial and ethnic composition of either the city population (corrected for age) or the fire department as a whole. In 1992, immediately after the completion of the promotions, the percentage of black captains was 10.8 percent and the percentage of Hispanic captains 3.6 percent. Had it not been for the affirmative action plan, these percentages would have been 5.7 percent and 1.7 percent in a city the population (age adjusted for comparability) of which is, as we said, more than half black or Hispanic.

The increase in the number (as distinct from percentage) of black captains over what it would have been without the affirmative action plan was 11 and of Hispanic captains 4, and it could be argued that the City must demonstrate a compelling interest in such an increase. But that would be the wrong approach in a case in which the justification for affirmative action is remedial. The test is, rather, whether the increase is a plausible lower-bound estimate of a shortfall in minority representation among fire captains that is due to the fire department's intentional discrimination in the past. See Aiken v. City of Memphis, 37 F.3d 1155, 1165 (6th Cir. 1994) (en banc); Peightal v. Metropolitan Dade County, supra, 26 F.3d at 1558; cf. Pasadena City Board of Education v. Spangler, 427 U.S. 424, 434-37 (1976); People Who Care v. Rockford Board of Education, supra, 111 F.3d at 534. And we think it is. The result of the affirmative action plan was to boost the minority proportion of captains to 14.4 percent, which was still far below the minority proportion of the city's population and of the lowest rank in the fire department (29 percent minority). The percentage in the lowest rank may be inflated to some extent by affirmative action. But considering the history of discrimination by the City and the minimal educational requirements for being a firefighter, we think it a reasonable surmise (and no more is practicable, given the nature of the inquiry) that, had it not been for the City's discrimination against blacks and Hispanics in the past, the percentage of black and Hispanic captains would be at least a third of the potential applicant pool; and that is all it is, even with the affirmative action challenged here.

It remains to consider the plaintiffs' contention that the doctrine of judicial estoppel barred the City from trying to prove that it had engaged in intentional discrimination in the past, since in earlier litigation it had, unsurprisingly, denied that charge. The doctrine provides that a party who prevails on one ground in a lawsuit cannot turn around and in another lawsuit repudiate the ground. Davis v. Wakelee, 156 U.S. 680, 689-91 (1895); Chaveriat v. Williams Pipe Line Co., 11 F.3d 1420, 1427-28 (7th Cir. 1993); Astor Chauffeured Limousine Co. v. Runnfeldt Investment Corp., 910 F.2d 1540, 1547 (7th Cir. 1990); Ryan Operations G.P. v. Santiam-Midwest Lumber Co., 81 F.3d 355, 358 (3d Cir. 1996); Patriot Cinemas, Inc. v. General Cinema Corp., 834 F.2d 208, 212-13 (1st Cir. 1987). If repudiation were permitted, the incentive to commit perjury and engage in other litigation fraud would be greater. Chaveriat v. Williams Pipe Line Co., supra, 11 F.3d at 1428. A party envisaging a succession of suits in which a change in position would be advantageous would have an incentive to falsify the evidence in one of the cases, since it would be difficult otherwise to maintain inconsistent positions. The doctrine of judicial estoppel requires, however, that the party sought to be estopped have obtained a favorable judgment or settlement (Kale v. Obuchowski, 985 F.2d 360, 361-62 (7th Cir. 1993)) on the basis of a legal or factual contention that he wants to repudiate in the current litigation. Otherwise it would be inconsistent with the rule that permits inconsistent pleadings. Astor Chauffeured Limousine Co. v. Runnfeldt Investment Corp., supra, 910 F.2d at 1548. The City did convince a district judge, in the first suit brought by the Department of Justice, that it had not discriminated against blacks and Hispanics during part of the period covered by the present litigation. But it abandoned that position on appeal and so never obtained either a favorable judgement or a favorable settlement on the basis of it. The doctrine of judicial estoppel is inapplicable.

The plaintiffs make other arguments, based mainly on evidentiary rulings by the district judge; but these do not have sufficient merit to warrant discussion.

Affirmed.