POSNER, Circuit Judge.
Mary Riordan appeals from a judgment by the district court dismissing her consolidated complaints of sex discrimination. One complaint is based on section 1 of the Civil Rights Act of 1871, now 42 U.S.C. § 1983, and names as the defendants Shirley Randolph and William Kempiners. The other is based on the Equal Pay Act of 1963, 29 U.S.C. § 206(d)(1), and names only Kempiners as a defendant. The judge granted summary judgment for Kempiners on the section 1983 complaint. The section 1983 claim against Randolph was then tried to a jury, but at the close of the plaintiffs case the judge directed a verdict for Randolph. The judge then entered judgment for Kempiners on the Equal Pay Act complaint as well.
The evidence that was allowed in at trial (an important qualification, as we shall see) showed the following. Mary Riordan, a college graduate with a bachelor’s degree in English, went to work in the Sexually Transmitted Diseases unit of the Illinois Department of Public Health in 1976, at the age of 22. Her starting salary was $10,000 a year. Promotion was rapid. By April 1981 she was running the unit and receiving a salary of $22,000 (we round off all dollar figures to the nearest thousand).
Although she was a state employee and the unit was a part of the state department of public health, two of the workers in the unit were federal employees and both of them (Atkinson and another) earned considerably more than she did, as did one of the state employees in the unit. The reason for the admixture of federal employees is that the federal government supports state units for sexually transmitted diseases in one of two ways, at the state’s choice — by direct financial support, or by providing federal employees to work in the unit at federal expense — and Illinois had chosen the second method of support.
Riordan proposed to reorganize the Sexually Transmitted Diseases unit by converting the federal employees to state employees and the mode of federal support from lending employees to giving money. The proposal was approved by Riordan’s immediate supervisor, Dr. Martin (the chief of the Acute Disease Section of the Division of Disease Control), by Martin’s immediate supervisor (the division chief), then by defendant Randolph, who was Assistant Director of the Public Health Department in charge of the Office of Health Services, and finally by Kempiners, the director of the entire department. The reorganization took place in 1982. Riordan’s salary as administrator of the reorganized unit was set at $25,000, which represented a 15 percent raise for her. At her request, the two federal employees whom she supervised were hired as state employees. Despite her raise, and her higher grade, their salaries still were higher than hers ($30,000 for Atkinson, $27,000 for the other); one of the state employees also had a higher salary than she, either $26,000 or $27,000 (the record is unclear which). All three were men. Dr. Martin told Riordan that within two or three years she would be receiving a higher salary than any of her subordinates by virtue of merit increases that she could expect.
But shortly afterward the State of Illinois, as an austerity measure, abolished merit increases, and instead gave all state employees a 5 percent increase. This left Riordan’s salary below the salaries of the three men. She requested a special salary adjustment of $5,000 — a 19 percent raise. Special salary adjustments are available to correct perceived inequities. Riordan’s immediate supervisor, Dr. Martin, approved the request, as did his immediate supervisor. The request then went to defendant Randolph, who turned it down. This action, which blocked the request from going any higher, is the basis of Riordan’s section 1983 claim.
Riordan eventually quit the department in disgust (she has since become a law student), and was replaced as head of the Sexually Transmitted Diseases unit by Atkinson, who by virtue of having thus been promoted was entitled under Illinois' civil service rules to a raise of 10 percent, to $33,000. (This was, however, a smaller percentage raise than Riordan had received when she had been made head of the newly reorganized unit.) Riordan’s claim under the Equal Pay Act is based on the disparity between the salaries that she and Atkinson received in the same job successively, and on the disparity between her salary and that of her three higher-paid male subordinates before she left.
Ordinarily when a woman complains about being paid less than a male co-worker for the same work, she sues the employer, in this case the Illinois Department of Public Health. For reasons unclear to us, Riordan’s lawyer decided to name Kempiners, the director of the department, as the only defendant to her suit under the Equal Pay Act. The word “employer” is defined broadly enough in the Fair Labor Standards Act (of which the Equal Pay Act is an amendment) to permit naming another employee rather than the employer as defendant, provided the defendant had supervisory authority over the complaining employee and was responsible in whole or part for the alleged violation. See 29 U.S.C. § 203(d) (employer within the meaning of the Act includes “any person acting directly or indirectly in the interest of an employer in relation to an employee”);
Patel v. Wargo,
803 F.2d 632, 637-38 (11th Cir. 1986);
Koster v. Chase Manhattan Bank,
554 F.Supp. 285, 290 (S.D.N.Y.1983). Yet at argument Riordan’s counsel was emp
hatic in insisting that the Equal Pay Act complaint was against Kempiners in his official capacity, i.e., against the Department itself. See
Brandon v. Holt,
469 U.S. 464, 471-73, 105 S.Ct. 873, 878-79, 83 L.Ed.2d 878 (1985);
Kentucky v. Graham,
473 U.S. 159, 165-67, 105 S.Ct. 3099, 3105-06, 87 L.Ed.2d 114 (1985). But if so, why not have named the Department instead of Kempiners in that complaint and have made life a little easier for us judges?
Employees complaining about sex discrimination in pay can also bring suit under Title VII of the Civil Rights Act of 1964, 42 U.S.C. §§ 2000e
et seq.,
at least if they are willing to prove intentional discrimination (“disparate treatment”) (the Bennett Amendment limits disparate-impact suits for sex discrimination under Title VII, see 42 U.S.C. § 2000e-2(h), though to what extent we need not consider here, see
EEOC v. Madison Community Unit School District No. 12,
818 F.2d 577, 587, 590 (7th Cir.1987);
American Nurses’ Ass’n v. Illinois,
783 F.2d 716, 723 (7th Cir.1986)), and if the jobs being compared are the same (see
id.
at 718-23). Riordan did not file a Title VII claim, however. Finally, women complaining of pay discrimination can, and Riordan did, sue under 42 U.S.C. § 1983 for violation of the equal protection clause, provided, again, that the discrimination was deliberate, see
Personnel Administrator v. Feeney,
442 U.S. 256, 279, 99 S.Ct. 2282, 2296, 60 L.Ed.2d 870 (1979), and provided that the defendant was acting under color of state law (which Randolph and Kempiners were) but is not itself a state or a state agency, in which event the suit would be barred by the Eleventh Amendment. As Riordan’s section 1983 complaint names Randolph and Kempiners in their personal rather than their official capacities, the suit really is against them and not (as Riordan’s Equal Pay Act suit is) against the Illinois Department of Public Health.
There is another condition on suits under 42 U.S.C. § 1983, and it shows that the judge was right to grant summary judgment for Kempiners: the plaintiff cannot win on a theory of superiors’ liability for the wrongs of the defendant’s subordinates. Just as there is no doctrine of respondeat superior (employers’ “vicarious” liability) under section 1983, so also there is no doctrine of superiors’ liability.
McKinnon v. City of Berwyn,
750 F.2d 1383, 1390 (7th Cir.1984). Riordan therefore had to prove that Kempiners had participated personally and substantially in the alleged discrimination against her, that is, in the refusal to grant her a special salary adjustment that would have brought her pay above that of any of her subordinates. There is no indication that Kempiners even knew about the request for the adjustment, let alone participated in the decision to turn it down. The request stopped with Randolph. And Kempiners can hardly be thought guilty of intentional discrimination because he approved Riordan’s proposal to reorganize the unit she supervised and to award her a large pay raise. Finally, none of the evidence that the judge excluded was offered to show a link between Kempiners’ actions and Randolph’s refusal to give Riordan the special raise.
The judge granted Randolph’s motion for a directed verdict on the claim that she had discriminated against Riordan in violation of section 1983. In defense of the directed verdict Randolph argues that a prima facie case of sex discrimination under the equal protection clause requires some evidence of discriminatory animus. Riordan replies that, as in a Title VII disparate-treatment (= intentional discrimination) case, all that is required to establish prima facie liability, and thus allow the plaintiff to get to the jury, is evidence of a disparity in treatment between equally qualified workers of different sexes, from which discriminatory intent can be inferred. See, e.g.,
United States Postal Service Bd. of Governors v. Aikens,
460 U.S. 711, 714 n. 3, 103 S.Ct. 1478, 1481 n. 3, 75 L.Ed.2d 403 (1983).
Although as an original matter it seems odd that the pleading and proof of liability in a case under the Constitution would be the same as in a case under a statute passed in 1964, this is indeed the teaching of an unbroken phalanx of decisions by this and other courts. See, e.g.,
Dugan v. Ball State University,
815 F.2d 1132, 1135-36 (7th Cir.1987);
Guillory v. St. Landry Parish Police Jury,
802 F.2d 822, 824 (5th Cir.1986);
Molthan v. Temple University,
778 F.2d 955, 961 (3d Cir.1985);
Gairola v. Virginia Dept. of General Services,
753 F.2d 1281, 1285-86 (4th Cir.1985);
Craik v. Minnesota State University Bd.,
731 F.2d 465, 468 n. 5 (8th Cir.1984). These cases hold that the issue of liability and the method of proving liability are the same, though only in a disparate-treatment case; there is no disparate-impact liability under the Constitution, which requires, as we have noted, proof of intentional discrimination. (There are also differences regarding procedure, derivative liability, remedy, and scope.) So when a woman establishes that a public employer refused to hire her and instead hired a male who was no more qualified, she makes out a prima facie case of sex discrimination under the equal protection clause as well as under Title VII, and the burden shifts to the employer to produce evidence that the reason for treating the man and woman differently was not an invidious one. However, the application of this template to a case such as this, where the act of discrimination is not hiring or firing or failing to promote but is instead turning down a request for a raise designed to give a female supervisor a higher wage than her higher-paid male underlings, is not obvious. A prima facie case implies that, more likely than not, the plaintiff’s rights have been invaded; yet there seems no reason to believe that a refusal to grant an extraordinary raise to a female supervisor is, more likely than not, the product of sex discrimination — when by hypothesis the employer placed the woman over the men in the first place.
That, however, is only one way of looking at the facts. Another way is that this is a case where a woman was paid less not only than her male subordinates but also than her male successor; and perhaps this configuration is suspicious enough to throw on the employer a burden of producing some evidence that sex was not the determining factor in the wage disparities. But we need not pursue the interesting question of the elements of the prima facie case in unusual factual settings; as part of her case in chief Riordan called Randolph as an adverse witness, thus giving Randolph a chance to explain why she had turned down the request; so the parties were beyond the prima facie stage. The inquiry then became: (1) Was Randolph’s proffered reason — a noninvidious reason— the true reason? (2) If not, was Riordan’s sex the real reason? Only if the first question was answered “no,” and the second “yes,” was Randolph guilty of sex discrimination. See
United States Postal Service Bd. of Governors v. Aikens, supra; Pollard v. Rea Magnet Wire Co.,
824 F.2d 557 (7th Cir.1987);
Benzies v. Illinois Dept. of Mental Health & Developmental Disabilities,
810 F.2d 146 (7th Cir.1987). Because the judge granted a directed verdict for Randolph, the issue for us becomes whether a rational jury, having heard both the plaintiff’s case and the defendant’s case (put in through the defendant’s own testimony as the plaintiff’s witness), could have concluded that the plaintiff was a victim of intentional discrimination by reason of her sex. In other words, could the jury have concluded that if Mary Riordan had been Mark Riordan, Randolph would have approved the request for an extraordinary raise?
Had the plaintiff been allowed to present all her evidence, we might well uphold the grant of the directed verdict, so negligible is the evidence of discrimination. Riordan was only 28 years old in 1982. She had been promoted rapidly. It was at her own request that the two federal employees in her unit had been converted to state employees, even though they would be receiving salaries higher than hers. She did not by this request consent to be paid less than they; she expected in the near future to catch up and surpass them by merit increases, which, however, for reasons unrelated to anybody’s sex, were abolished right after the reorganization. The special 19 percent raise that Riordan then requested and Randolph denied would have been a huge raise — almost four times as large as the automatic raise that Riordan had just received, along with the other employees,
in lieu of the abolished merit raises — and Randolph testified that she feared that granting the request would spark other requests for special salary adjustments. No doubt there was a possible inequity in Riordan’s being paid less than some of her subordinates; but this is a common situation in civil services, where the dispersion of salaries is so limited that seniority, or possession of an additional credential (e.g., an additional degree), can easily propel a subordinate worker into a higher pay classification than his superior. (And not only in civil services: in the private sector, salesmen who receive commissions as well as salary are often better compensated than their superiors.) Randolph herself was paid less than two of her male subordinates yet never claimed that this was due to sex discrimination. Nor does Riordan argue that either she or her higher-paid subordinates were improperly classified under the Illinois civil service rules. The salary distributions within different grades overlapped, making it possible for an employee in a lower grade to qualify for a higher salary than a less senior employee in a higher grade.
Riordan points out that since the Sexually Transmitted Diseases unit was federally funded and had a surplus, the special salary adjustment that she requested would have cost the state nothing. But most of the employees in the Illinois Department of Public Health are not federally funded, and Randolph would have had to worry about their requests for special salary adjustments if she granted such an adjustment to this very young worker. Finally, although there is no rule that women can’t be guilty of discriminating against other women, cf.
Castaneda v. Partida,
430 U.S. 482, 499-500, 97 S.Ct. 1272, 1282-83, 51 L.Ed.2d 498 (1977), the fact that Riordan’s request was turned down by a woman after being approved by two male supervisors does not strengthen the inference of sex discrimination, cf.
id.
at 515-16, 97 S.Ct. at 1291 (dissenting opinion). Anyway, it is not the law that, whenever a supervisory employee turns down a subordinate’s request for an extraordinary pay raise, the supervisor is buying a lawsuit for sex discrimination.
But all this assumes that Riordan was allowed to put in her whole case, and she was not. The judge granted the defendants’ motion
in limine
(at the threshold) to exclude much of the evidence she wanted to present. He gave no reasons for most of the exclusions. The defendants had argued for the motion for the most part — and for the most part unsoundly — on grounds of irrelevance. The only possibly meritorious ground for most of the exclusions was that the probative value of the excluded evidence would have been substantially outweighed at trial by the potential of the evidence to confuse the jury or extend the length of the trial. See Fed.R.Evid. 403. This is a discretionary determination and we therefore review it under a deferential standard. See, e.g.,
United States v. Beasley,
809 F.2d 1273, 1278-79 (7th Cir.1987);
Kiev v. Commercial Union Ins. Cos.,
808 F.2d 1254, 1258 (7th Cir.1987). But we must be satisfied that the judge exercised his discretion,
United States v. Beasley, supra,
809 F.2d at 1279, and we are not. Moreover, as the judge’s discretion was exercised on a wholesale basis before trial began, rather than in response to the developing course of the trial, careful appellate review is necessary to make sure the judge did not truncate the trial prematurely.
Such review is particularly necessary in a case like this, where the substantive issue is whether there was intentional discrimination in employment. Proof of such discrimination is always difficult. Defendants of even minimal sophistication will neither admit discriminatory animus nor leave a paper trail demonstrating it; and because most employment decisions involve an element of discretion, alternative hypotheses (including that of simple mistake) will always be possible and often plausible. Only the very best workers are completely satisfactory, and they are not likely to be discriminated against — the cost of discrimination is too great. The law tries to protect average and even below-average workers against being treated more
harshly than would be the case if they were of a different race, sex, religion, or national origin, but it has difficulty achieving this goal because it is so easy to concoct a plausible reason for not hiring, or firing, or failing to promote, or denying a pay raise to, a worker who is not superlative. A plaintiff’s ability to prove discrimination indirectly, circumstantially, must not be crippled by evidentiary rulings that keep out probative evidence because of crabbed notions of relevance or excessive mistrust of juries. See, e.g.,
Mathewson v. National Automatic Tool Co.,
807 F.2d 87, 91 (7th Cir.1986);
Allen v. County of Montgomery,
788 F.2d 1485, 1488 (11th Cir. 1986).
The judge excluded the following categories of evidence:
1. Riordan wanted to prove that she was paid less than two male supervisors of other units, comparable to the one she headed up, in the Department of Public Health. Such evidence would have been relevant to her claim of intentional discrimination (though of course not conclusive), and we do not understand why it was excluded.
2. Riordan also wanted to prove that the Department had given exceptional raises to two young men who were in the same position as she — that is, who had been promoted rapidly before they received these raises. The evidence would have tended to rebut Randolph’s testimony that she had denied Riordan the extraordinary raise because she was concerned that granting it would incite “me too” demands throughout the Department, and would thus have suggested (though no more than suggested) that Riordan had been denied such a raise only because she was a woman. It is true that the offer of proof did not indicate that Randolph, the only defendant left in the section 1983 phase of the case, had had anything to do with the other raises. But since one of the workers worked for Randolph, and since apparently Randolph had to approve all raises for her subordinates, Riordan might well have been able to establish Randolph’s responsibility for, and certainly knowledge of, the raises. Therefore we cannot understand, and again the district judge did not explain, why this evidence was excluded.
, 3. Riordan wanted to introduce evidence, both statistical and anecdotal, showing systematic wage disparities between the Department’s male and female employees. The judge excluded all of this evidence, merely commenting in passing that statistics are inadmissible to show intentional discrimination against an individual plaintiff. That is not the view of this or (so far as we know) any other court. See, e.g.,
Powers v. Dole,
782 F.2d 689, 695 (7th Cir.1986);
Craik v. Minnesota State University Bd., supra,
731 F.2d at 471. What sense would it make? All evidence is probabilistic — statistical evidence merely explicitly so. See
DePass v. United States,
721 F.2d 203, 207 (7th Cir.1983) (dissenting opinion). If it turned out that Randolph always recommended higher pay for men than women, this would be
some
evidence that Riordan hadn’t gotten the raise she wanted because she was a woman. The problem with this category of evidence as with the previous one is that no effort was made to pinpoint Randolph’s responsibility for the wages of the male workers whom Riordan wanted to compare with herself. But this is not the ground on which the judge excluded the evidence; and whether he would have been abusing his discretion to exclude it on that or another ground is a question that must abide an exercise of that discretion free of misconceptions about the admissibility of statistical evidence.
4. The judge properly excluded a public statement by Kempiners’ successor acknowledging a study which showed that the Department had in the past “underutilized” women. The evidence was entirely remote from the question of pay equity for Riordan, and potentially prejudicial.
5. The judge excluded all evidence of events subsequent to Riordan’s filing of her claim for discrimination in December 1983 — even though she didn’t quit the Department till August 1984. Proximity in time to the alleged discrimination is a proper consideration in assessing
probative value; but given the importance of circumstantial evidence in proving (and, equally, disproving) employment discrimination, a blanket exclusion of evidence of events that occurred before or after the discrimination is arbitrary. See
Mathewson v. National Automatic Tool Co., supra,
807 F.2d at 91.
Too much probative evidence was arbitrarily excluded; the directed verdict on Riordan’s section 1983 claim cannot stand.
We turn to the Equal Pay Act. The principal though not only pay disparity of which Riordan complains is, again, that between her and her higher-paid male subordinates, especially the two former federal employees. It might seem that the Act, which forbids an employer to pay male and female employees different wages for the same work, could have no possible application to that disparity, since the work of a supervisor and of the workers she supervises is necessarily different; only the supervisor has supervisory responsibilities. But an employer cannot avoid the Act by the simple expedient of loading extra duties onto its female employees — unless it pays them more. See, e.g.,
Hein v. Oregon College of Education,
718 F.2d 910, 917 (9th Cir.1983). So, where a female supervisor is paid less than the workers she supervises and the only difference in work between the supervisor and the supervised is the supervisory responsibilities of the former, the Act applies. The scope of this principle need not be considered further, however. Even if Riordan’s work was identical to that of her male subordinates, the disparity between her pay and theirs probably is not actionable at all, as we shall see. And her work
was
identical to that of Atkinson, when he replaced her as head of the Sexually Transmitted Diseases unit at a higher wage ($33,000 versus $29,000, her ending salary). So there was a prima facie violation of the Act.
However, the Act provides, among other defenses, that there is no violation if the unequal pay was due to “any other factor other than sex.” 29 U.S.C. § 206(d)(l)(iv). That clearly is true in this case, at least so far as the disparity between Riordan and her two highest-paid subordinates (one of whom was also her successor) was concerned. The initial disparity was due to the fact that the two had been federal employees whose wages were not fixed by the Illinois Department of Public Health and who were unwilling to become state employees, as contemplated by Riordan’s proposed reorganization of the Sexually Transmitted Diseases unit, unless they received wages comparable to their wages as federal employees. As it was, Atkinson received a $3,600 pay cut upon switching into the state service. The discrepancy between Riordan’s salary and Atkinson’s when he replaced her reflected the operation of Illinois’ civil service rules: Atkinson was already at the top of his grade, and the rules entitled him to a raise when he was promoted to supervisory rank. Thus the discrepancy between his salary and hers was not due to her sex. See
Strecker v. Grand Forks County Social Service Bd.,
640 F.2d 96 (8th Cir.1980);
EEOC v. Aetna Ins. Co.,
616 F.2d 719, 726 (4th Cir.1980).
This is not to say that a salary difference between a male and a female worker that reflects a previous salary classification is always due to a factor other than sex within the meaning of the statute. For the previous classification might itself have been due to sexual discrimination. See
Corning Glass Works v. Brennan,
417 U.S. 188, 205, 94 S.Ct. 2223, 2233, 41 L.Ed.2d 1 (1974);
Covington v. Southern Illinois University,
816 F.2d 317, 322-23 (7th Cir.1987). If Atkinson had been placed at the top of the pay scale when he entered state service because he was a male, the fact that this classification entitled him to higher pay as Riordan’s successor than she had received would not justify the state in paying these two employees different wages for the identical work. However, Riordan does not argue that Atkinson’s initial classification was due to his sex. It was due to his receiving a high salary from his previous, federal employer, and she makes no argument that his high federal
salary was a consequence of sex discrimination.
We need not decide whether any of the disparities in pay may have been due to the operation of “a seniority system,” another defense under the Act, see 29 U.S.C. § 206(d)(l)(i), and merely mention the issue for completeness. Atkinson had been a federal employee for almost 18 years and had accrued considerable seniority, resulting in a high salary that he carried with him when he became a state employee. The difference in salary between him and Riordan was ultimately a function of Atkinson’s greater seniority, albeit with a different employer. See
EEOC v. Aetna Ins. Co., supra,
616 F.2d at 726 and n. 10. Nor was Riordan prevented by discrimination from accruing seniority. Her lack of seniority relative to Atkinson was due to the differences in their ages. There is only so much seniority that a 28-year-old can accrue.
The rub is that the statutory words “seniority system” may refer to seniority with the employer who is paying the different wages rather than to length of service with a previous employer, though the distinction may be artificial in a case such as this where the employee doesn’t change jobs but only the identity of his employer changes, by virtue of a reorganization. See
EEOC v. Cleveland State University,
28 F.E.P.Cas. 1782, 1788-89, 1794 (N.D. Ohio 1982). But if as we believe the disparity was due to a factor other than sex, it does not matter whether that factor is a “seniority system” within the meaning of the statute.
If the only inequality in pay were that between Riordan and Atkinson (or the other former federal employee), we would affirm the dismissal of her Equal Pay Act claim. And while our analysis of why the two former federal employees were paid more than Riordan may not extend to the third male who was paid more, and who had only slightly more experience than Riordan and less time in the Department, we do not understand Riordan to be making an issue of that employee’s higher salary; if this is incorrect the matter can be pursued on remand — for the Equal Pay Act claim, too, must be remanded because of the district judge’s evidentiary rulings. They prevented Riordan from showing that other male employees in the Department of Public Health did the same work as she and were paid more but without the extenuating circumstances that justified the higher pay of the two former federal employees. We doubt whether the work of those other males (two of whom were accountants, and two supervisors of other units) was work equal to hers within the meaning of the Act, see
EEOC v. Madison Community Unit School District No. 12, supra,
818 F.2d at 580-83; but this is a factual question and the district judge should have first crack at it.
The judgment of the district court is affirmed in part and reversed in part, and the case is remanded for further proceedings consistent with this opinion. Circuit Rule 36 shall apply on remand.
So Ordered.