McMILLIAN, Circuit Judge.
Martha Coble and other female teachers employed by Hot Springs School District No. 6 appeal from a final judgment entered in the District Court1 for the Western District of Arkansas finding in favor of the school district on their individual and class claims of sex discrimination in pay, job assignments and promotions to administrative positions, in violation of Title VII of the Civil Rights Act of 1964, 42 U.S.C. § 2000e et seq. For reversal appellants argue that the district court erred in finding that the school district did not discriminate on the basis of sex in (1) not appointing Martha Coble as director of the teachers center, (2) not appointing Susan Chapman as junior high school counselor, (3) paying lower salaries and extra duty stipends to female coaches, (4) failing to promote qualified female teachers to administrative positions, and (5) assigning higher paying jobs to male teachers. For the reasons discussed below, we affirm in part and reverse in part and remand for further proceedings.
In early August 1978, Martha Coble applied for the position of director of the school district’s newly formed teachers center. The school district appointed a male teacher, Bill Nipper, as director. Coble then filed a charge with the Equal Employment Opportunity Commission in late August 1978. She received a right-to-sue letter and timely filed the present action in December 1979, alleging individual and elasswide discrimination against female teachers in pay, job assignments and promotions to administrative positions. The district court certified the action as a class action under Fed.R.Civ.P. 23(b)(2). The action was tried in November 1980. The class claims were based upon statistical evidence, circumstantial evidence, and specific instances of alleged sex discrimination. The district court found that Coble had established a prima facie case but failed to prove that the school district’s legitimate, nondiscriminatory reasons for not appointing her as director were pretextual; that Susan Chapman, an individual plaintiff, had established a prima facie case but similarly failed to prove that the school district’s legitimate, nondiscriminatory reasons for not appointing her as a junior high school counselor were pretextual; and that the class plaintiffs failed to establish a prima facie case of sex discrimination in pay, job assignments and promotions to administrative positions. Coble v. Hot Springs School District No. 6, No. 79-6055 (W.D.Ark. June 29, 1981).
We will discuss the individual claims first and then the classwide claims. The Supreme Court recently clarified that findings of fact made by the district courts in employment discrimination cases are subject to the clearly erroneous standard of review in the courts of appeals. Pullman-Standard v. Swint, - U.S. -, -, 102 S.Ct. 1781, 1789, 72 L.Ed.2d 66 (1982) (§ 703(h) seniority system exception), rev’g and remanding 624 F.2d 525 (5th Cir. 1980). “A finding is ‘clearly erroneous’ when although there is evidence to support it, the reviewing court on the entire evidence is left with the definite and firm conviction that a mistake has been committed.” United States v. United [724]*724States Gypsum Co., 333 U.S. 364, 395, 68 S.Ct. 525, 542, 92 L.Ed. 746 (1948).
Martha Coble
Coble alleged that she had been discriminated against on the basis of sex when the school district appointed a male teacher as director of the teachers center. In July 1978, the school district received a federal grant to establish a teachers center. The teachers center is a materials resource and media center for teachers within the school district; it is a facility where teachers can improve teaching skills, develop and revise curricula, and experiment with new teaching methods and materials. The teachers center is managed by the director. The director also organizes professional workshops and seminars. According to the grant proposal, the requirements for the position of director included an awareness of the objectives and problems of the Hot Springs school system, a master’s degree, five years teaching experience, some expertise in conducting workshops, experience in pilot testing innovative, practices in the classroom, experience as a regular classroom teacher, and some knowledge of administration, including budgeting.
In late July, the school district learned that the grant had been approved and in early August 1978 the policy advisory board of the teachers center met to discuss the budget and the selection of a director. The policy advisory board nominated ten candidates for the position of director and then appointed a five-member committee to interview the candidates and recommend two to the school district superintendent, E. Bruce Meeks. The interview committee members were Elizabeth Jackson, chairperson of the policy advisory board and the school district’s supervisor for federal programs; Richard Johnson, a high school teacher; Dorothy Threlkeld, a junior high school teacher; Maurice Dunn, assistant superintendent for elementary education; and William Lane, assistant superintendent for secondary education. The interview committee informally interviewed the nominated candidates and recommended Coble and Nipper to the superintendent. Meeks interviewed both candidates and selected Nipper. The selection was later approved by the school board.
Coble then requested an interview with Meeks to discuss the reasons why she was not appointed as director of the teachers center. At Coble’s request, Hal Robbins, assistant executive secretary to the Arkansas Educational Association, accompanied Coble to the August 8, 1978, interview. Dunn was also present. With Coble’s permission Meeks tape-recorded the interview; the tape-recording was not introduced at trial. During the interview Meeks stated that he thought that Nipper was the more “available” and “dedicated” of the two candidates and referred to Nipper’s attendance record. During the interview Dunn referred to the fact that Nipper was not married and had no “immediate family.”
At trial Meeks testified that he had interviewed both candidates recommended by the interview committee and had selected Nipper because Nipper was “the best person for the job.” Meeks testified that Nipper had a reputation for “making things happen” and as an “exciting” teacher, had a good concept of the function of the teachers center, and had a better attendance record. Meeks considered the fact that Nipper “never missed” a day of school and virtually “lived at school” to be an indication of his commitment to the school district.
Coble testified that during her initial interview with Meeks (before the selection was made), Meeks stated that he considered her attendance record to be a “strike against her” and questioned her extensively about her children and whether she would be able to handle both the long and demanding hours required by the teachers center and her family responsibilities. Co-ble further testified that when Meeks called to tell her that he had selected Nipper for the position, Meeks stated that although she possessed qualities that Nipper did not, she had a family.
The district court correctly determined that Coble had established a prima facie case of sex discrimination. See Texas Department of Community Affairs v. Bur-[725]*725dine, 450 U.S. 248, 252-54 & n.6, 101 S.Ct. 1089, 1093-1094 & n.6, 67 L.Ed.2d 207 (1981); McDonnell Douglas Corp. v. Green, 411 U.S. 792, 802, 93 S.Ct. 1817, 1824, 36 L.Ed.2d 668 (1973). Coble established that (1) she belonged to a protected group; (2) she applied for and was qualified for a job for which the school district was seeking applicants; (3) despite her qualifications, she was rejected; and (4) after her rejection, the employer filled the position with a person of her qualifications. The district court reviewed both Coble’s and Nipper’s qualifications and found that Coble had superior educational qualifications because she had more credit hours beyond her master’s degree than Nipper and was certified as an elementary principal and elementary supervisor. The district court, however, considered certain subjective criteria were also relevant and concluded that, after considering leadership qualities, ability to work with students and teachers, understanding of the function of the teachers center, general personality, and experience, the two candidates were equally qualified. Coble v. Hot Springs School District No. 6, No. 79-6055, slip op. at 5.
Although the school district argues on appeal that Nipper was more qualified than Coble, the district court’s findings on the candidates’ qualifications are not clearly erroneous. Both Coble and Nipper are evidently talented and dedicated teachers. As found by the district court, Coble, at the time of trial, had been employed by the school district for thirteen years, had a variety of classroom teaching experience and was currently teaching academically gifted fifth grade students, had thirty credit hours beyond her Master’s Degree in Education, and was certified as an elementary teacher, elementary principal, elementary supervisor, and curriculum specialist for grades kindergarten through high school. She had actively participated in several teachers organizations, participated in varying degrees in several workshops, frequently served as the liaison between the professional workshops and other teachers, and had received several teaching awards and honorary scholarships. She had received excellent evaluations from her principals. Similarly, Nipper was an outstanding teacher. At the time of trial he had been employed by the school district for twelve years and taught government and history at one of the two junior high schools; he was the department chairperson. He was involved in teachers organizations and sponsored student activities at the junior high school. He had been selected as an “outstanding young teacher.” He had organized the junior high school’s “Bicentennial Extravaganza” and a career-guidance seminar for educators. He had eighteen credit hours beyond his Master’s Degree in Education. Nipper had also received excellent evaluations from his principals.
Coble having established a prima facie case and thus raised a presumption of discrimination, the burden then shifted to the school district to “rebut the presumption of discrimination by producing evidence that [Coble] was rejected, or [Nipper] was preferred, for a legitimate, nondiscriminatory reason.” Texas Department of Community Affairs v. Burdine, 450 U.S. at 254, 101 S.Ct. at 1094. As further explained in Burdine,
[t]he defendant need not persuade the court that it was actually motivated by the proffered reasons. [Only the burden of production or of going forward with the evidence shifts to the defendant.] It is sufficient if the defendant’s evidence raises a genuine issue of fact as to whether it discriminated against the plaintiff. To accomplish this, the defendant must clearly set forth, through the introduction of admissible evidence, the reasons for the plaintiff’s rejection. The explanation provided must be legally sufficient to justify a judgment for the defendant.
Id. at 254-55, 101 S.Ct. at 1094 (citations and footnotes omitted). Here, the school district argued that Nipper was clearly the better qualified candidate. Hal Robbins testified that at the August 8 interview Meeks identified the factors he considered in evaluating the two candidates’ qualifications as certification, experience, recommendations, concept of the function of the [726]*726teachers center, ability to work with others, leadership potential, and “availability” as indicated by attendance records. The district court found that Meeks during this August 8 interview emphasized the candidates’ “availability.” Meeks noted that Nipper had accumulated far more sick leave (over sixty days through the 1977-1978 school year) than Coble (ten days through the 1977-1978 school year) and considered that Nipper’s better attendance record indicated dedication. Coble v. Hot Springs School District No. 6, No. 79-6055, slip op. at 3. Our review of the record indicates that Meeks also believed that Nipper had greater experience in organizing workshops than Coble. Both assistant superintendents shared this view of Nipper’s relatively greater experience with workshops.
We agree that the school district satisfied its burden of production by articulating legitimate, nondiscriminatory reasons for Coble’s rejection, although we note that the school district’s explanation of its legitimate, nondiscriminatory reasons could have been clearer and more specific. See Texas Department of Community Affairs v. Burdine, 450 U.S. at 258, 101 S.Ct. at 1096 (“This obligation arises both from the necessity of rebutting the inference of discrimination arising from the prima facie case and from the requirement that the plaintiff be afforded ‘a full and fair opportunity’ to demonstrate pretext.”). Meeks considered Nipper to be the better qualified candidate on the basis of certain subjective factors, workshop experience and “availability” as measured by attendance.
At this point our analysis must focus upon whether Coble established that the school district’s “proffered reason was not the true reason for the employment decision . .. either by directly persuading the court that a discriminatory reason more likely motivated the employer or indirectly by showing that the employer’s proffered explanation is unworthy of credence.” Id. at 256, 101 S.Ct. at 1095, citing McDonnell Douglas Corp. v. Green, 411 U.S. at 804-05, 93 S.Ct. at 1825-1826. Coble argues that the real reason Meeks did not select her as director of the teachers center was because she was a woman with children and he assumed that her family responsibilities would interfere with her performance as director of the teachers center. Coble testified that during the preselection interview Meeks referred to her attendance record, noted the days missed attributable to family illness, discussed her children and their ages, and asked her whether she could handle the long hours required by the position. According to Coble, when Meeks told her that he had selected Nipper, he told her that she had qualities that Nipper did not but that she had a family. Coble further testified that during the postselection interview on August 8 Meeks again referred to her attendance record, described her as “energetic” because she had pursued many activities but “her children had not suffered,” and again discussed her children and their ages. Hal Robbins testified that at the postselection interview Meeks considered Coble “less available” than Nipper and that Dunn referred to the fact that Nipper was single and had “no immediate family.” This testimony was not disputed by the defense.
We have carefully reviewed the record and conclude that the district court’s finding that the school district’s proffered reason was the true reason for the employment decision is clearly erroneous. The evidence showed that superintendent Meeks unfairly emphasized Coble’s family responsibilities and used essentially subjective promotion criteria in selecting the director of the teachers center.
Although not illegal per se, subjective promotion procedures are to be closely scrutinized because of their susceptibility to discriminatory abuse. The mere fact that the subjective process is intended to recognize merit does not necessarily alleviate its susceptibility to discriminatory abuse. When the evaluation is in any degree subjective and when the evaluators themselves are not members of the protected minority, the legitimacy and undiscriminatory basis of the articulated reason for the decision should be subject [727]*727to particularly close scrutiny by the trial judge [and on appeal].
Royal v. Missouri Highway & Transportation Comm’n, 655 F.2d 159, 164 (8th Cir. 1981) (citations omitted). Coble presented substantially unrebutted evidence showing that the superintendent made the employment decision largely on the basis of a sexual stereotype.2 Coble also presented evidence of her superior educational and certification qualifications and evidence showing that the superintendent may have undervalued her workshop experience3 and overestimated her absences from the classroom.4 As noted in Burdine, “[t]he fact that a court may think that the employer misjudged the qualifications of the applicants does not in itself expose [the employer] to Title VII liability, although this may be probative of whether the employer’s reasons are pretexts for discrimination.” 450 U.S. at 259, 101 S.Ct. at 1096-1097 (citations omitted).
That part of the district court’s judgment finding that the school district did not discriminate against Martha Coble on the basis of sex in the selection of the director of the teachers center is reversed and remanded to the district court with directions to grant appropriate relief.
Susan Chapman
Susan Chapman also raised an individual claim of sex discrimination, alleging that in August 1980 the job requirements for the position of counselor at Southwest Junior High School were drafted to specifically fit a male teacher and that the male teacher was in fact preselected for the counselor position. The district court found that the circumstances surrounding the selection of Drew Alexander as a junior high school counselor were “somewhat suspect” but [728]*728concluded that Chapman had not shown that the reason proffered by the school district (junior high school teaching expericnce) was pretextual. Coble v. Hot Springs School District No. 6, No. 79-6055, slip op. at 14.
At the time of trial Chapman had been employed by the school district for ten years; before the 1980-1981 school year she was employed as an elementary school counselor. She had received a master’s degree in counseling in 1976 and had been certified as a counselor for kindergarten through high school in 1979. In early August 1980, the school district informed Chapman that the elementary school counseling positions were being eliminated for the coming school year and offered her an assignment as a remedial mathematics teacher. Chapman approached William Lane, the assistant superintendent for secondary education, for information about available secondary school counseling positions. Lane told her that there was an opening for a junior high school counselor at Southwest Junior High School; Chapman expressed an interest in the position, which paid more money and was an extended contract position (10 month contract; the standard teaching contract is 9Vi months), and was told that she would be called for an interview within a week. After almost two weeks (on August 13) Chapman called Lane’s office to inquire about the counseling position; Lane’s secretary told her that the position might require certification in French and counseling. Lane confirmed the double certification requirement later that day.
Chapman is not certified to teach French. She was not interviewed for the counseling position. The position was filled by Drew Alexander, a male junior high school teacher at Central Junior High School who was certified to teach French and who had just received certification in counseling. At trial Lane admitted that Alexander was the school district’s only teacher with a double certification in French and counseling. Later Chapman filed an administrative grievance about the selection of Alexander; Meeks sent Chapman a letter explaining that Alexander had been selected because he was a junior high school teacher and had experience working with students of this age group.
We have carefully reviewed the record and conclude that the district court’s finding that the school district did not discriminate against Susan Chapman on the basis of sex in the appointment of Drew Alexander as a counselor at Southwest Junior High School is clearly erroneous. There was persuasive evidence of preselection. See Shaw v. Boorstin, 517 F.Supp. 336, 337-41 (D.D.C. 1981). As noted by the district court, the reason for the requirement of certification in French was never explained by the school district. During the 1980-1981 school year, Judy Cole taught ninth grade English and French at Central Junior High School; for the five preceding years she had taught English and French at Southwest Junior High School. Cole testified that on August 14, 1980, the principal of Southwest informed her that she was being transferred to Central and cited financial considerations as the reason. Cole had not wanted to transfer and asked whether Alexander was being transferred to fill the counseling position. Cole testified that the principal told her that Alexander probably would be transferred. Cole further testified that as she passed by her mailbox that day she discovered that her name had been replaced with Alexander’s and that the position requiring certification in French and counseling was not posted until the next day, August 15. It is the practice of the school district to post notices of available positions on bulletin boards in the school buildings. This testimony was not disputed by the school district. The French and counseling certification requirement, the transfer of Cole from Southwest to Central, and the posting of the notice after the position had in fact been filled indicate that the counseling position, in particular the French certification requirement, was carefully tailored to fit Drew Alexander.
This evidence discredited the validity of the French certification require[729]*729ment. Chapman established a prima facie case by showing that she had “applied for an available position for which she was qualified, but was rejected under circumstances which give rise to an inference of unlawful discrimination.” Texas Department of Community Affairs v. Burdine, 450 U.S. at 253, 101 S.Ct. at 1094 (footnote omitted). The school district rebutted the inference of discrimination by articulating a legitimate, nondiscriminatory reason for her rejection. However, the evidence of prese-lection also effectively discredited the school district’s proffered explanation that Alexander was selected because he had junior high school experience. Id. at 255 n.10, 101 S.Ct. at 1095 n.10.
That part of the judgment of the district court finding that the school district did not discriminate against Susan Chapman is reversed and remanded to the district court with directions to grant appropriate relief.
Class Claims
Appellants also raised claims of classwide sex discrimination in pay, job assignments and promotions to administrative positions. Appellants also alleged that the school district paid female coaches lower salaries and lower extra duty stipends than it paid male coaches. Specifically, appellants alleged that the school district paid male teachers more than female teachers and that the salary differential was not attributable to education or experience, that the school district assigned male teachers to more positions with extended term contracts and extra duty stipends than female teachers, and that the school district consistently “underutilized” qualified female teachers by failing to promote them to responsible administrative positions.
The district court found that appellants failed to establish a prima facie case of sex discrimination in pay and promotions and, even assuming appellants established a pri-ma facie case, the school district had rebutted it. Coble v. Hot Springs School District No. 6, No. 79-6055, slip op. at 13. The district court also found that male coaches are paid more on the average than female coaches, that male and female coaches have essentially the same duties, but that the salary differential is at least in part attributable to the longer term contracts and extra duty required by preseason football practice. Id. at 12. The district court thus concluded that appellants had not proven that female coaches were being discriminated against on the basis of sex. However, the district court also cautioned the school district “to strive to bring the salaries of the female coaches more in line with the duties they perform or this issue might be the subject of further scrutiny.” Id. at 15.
We have carefully reviewed the record and conclude that appellants failed to establish a prima facie case of discrimination in pay and promotions to administrative positions. That part of the district court’s judgment is affirmed. Further, we conclude that appellants did prove that the school district discriminates against female coaches because the female coaches have shorter term contracts, lower salaries, and smaller extra duty stipends than male coaches even though male and female coaches perform essentially the same duties. Accordingly, that part of the district court’s judgment is reversed and remanded to the district court with directions to make further findings and grant appropriate injunc-tive relief.
Appellants relied principally upon statistical evidence prepared by expert witness Charles T. Kenny, an industrial psychologist who specializes in employment statistics. Kenny’s statistical analyses were based upon state and school district personnel records obtained from the school district through discovery. Appellants also presented circumstantial evidence and evidence of specific individual incidents of alleged discrimination against Martha Coble and Susan Chapman.5
[730]*730“We have previously found statistics to speak loudly in employment discrimination cases. Statistics, however, must be used carefully.” Eubanks v. Pickens-Bond Construction Co., 635 F.2d 1341, 1347 (8th Cir. 1981) (citations omitted); see Hazelwood School District v. United States, 433 U.S. 299, 307-08, 97 S.Ct. 2736, 2741-2742, 53 L.Ed.2d 768 (1977) (gross statistical disparities alone may in a proper case constitute prima facie proof of discrimination); Teamsters v. United States, 431 U.S. 324, 339-40, 97 S.Ct. 1843, 1856-1857, 52 L.Ed.2d 396 (1977) (statistics are competent evidence in proving employment discrimination); Taylor v. Teletype Corp., 648 F.2d 1129, 1134 (8th Cir.), cert. denied, 454 U.S. 969, 102 S.Ct. 515, 70 L.Ed.2d 386 (1981). “[T]he use of statistics ‘is conditioned by the existence of proper supportive facts and the absence of variables which would undermine the reasonableness of the inference of discrimination which is drawn.’ ” White v. City of San Diego, 605 F.2d 455, 460 (9th Cir. 1979), citing United States v. Ironworkers Local 86, 443 F.2d 544, 551 (9th Cir.), cert. denied, 404 U.S. 984, 92 S.Ct. 447, 30 L.Ed.2d 367 (1971). The plaintiffs have the burden of establishing a prima facie case of sex discrimination; then the burden of production or going forward with the evidence “shifts to the employer to defeat the prima facie showing of a pattern or practice by demonstrating that the [plaintiffs’] proof is either inaccurate or insignificant.” Teamsters v. United States, 431 U.S. at 360, 97 S.Ct. at 1867; cf. Texas Department of Community Affairs v. Burdine, 450 U.S. at 254-55, 102 S.Ct. at 1094-1095 (describing defendant’s burden of production in individual disparate treatment case).
The burden is on the opposing party to clearly rebut statistical evidence; hypotheses or conjecture will not suffice. When a plaintiff submits accurate statistical data, and a defendant alleges that relevant variables are excluded, defendant may not rely on hypothesis to lessen the probative value of plaintiff’s statistical proof. Rather, defendant, in his rebuttal presentation, must either rework plaintiff’s statistics incorporating the omitted factors or present other proof undermining plaintiff’s claims.
Segar v. Civiletti, 508 F.Supp. 690, 712 (D.D.C.1981), citing Davis v. Califano, 198 U.S.App.D.C. 224, 613 F.2d 957, 964 (1979), and Pennsylvania v. Local 542, Operating Engineers, 469 F.Supp. 329, 370-75 (E.D.Pa.1978) (Higginbotham, J., sitting by special designation), aff’d mem., 648 F.2d 922 (3d Cir. 1981) (banc); see also Taylor v. Teletype Corp., 648 F.2d at 1134 (defendant claimed plaintiff’s statistics were inaccurate but failed to show that the figures used by plaintiff produced inaccurate results or that a different result would have been reached using different analytical basis); Trout v. Hidalgo, 517 F.Supp. 873, 881-82 (D.D.C.1981).
As discussed below, at first glance, appellants’ statistical data reveals pay and promotion disparities that are extremely suspect. However, upon further examination, we must conclude that the probative value of appellants’ statistical data is undermined by the small sample size for promotions and the failure to consider the effect of education and experience together on salary. See, e.g, EEOC v. American National Bank, 652 F.2d 1176, 1189-94 (4th Cir. 1981) (analytical problems with small sample size); Sainte Marie v. Eastern R.R. Ass’n, 650 F.2d 395, 400-401 (2d Cir. 1981) (plaintiffs’ statistical analysis produced standard deviations of 4 to 14.5, but statistical evidence underlying standard deviation analysis had little probative value); but cf. Eubanks v. Pickens-Bond Construction Co., 635 F.2d at 1351, 1354-56 (McMillian, J., dissenting in part) (discussion of standard deviation statistical analysis). Neither side used standard deviation analysis. See, eg., Hameed v. Ironworkers Local 396, 637 F.2d 506, 513 n.8 (8th Cir. 1981) (impact case).
It should be noted that the school district’s rebuttal evidence with respect to the pay differences was speculative and hypothetical and thus would have been wholly inadequate to rebut a prima facie case. [731]*731The school district challenged appellants’ statistics on the basis that several relevant factors had been omitted from appellants’ statistical analysis but failed to test its proffered hypotheses. See Trout v. Hidal-go, 517 F.Supp. at 881-82, citing Finkel-stein, Regression Models in Administrative Proceedings, 86 Harv.L.Rev. 1442, 1466 (1973) (“In criticizing models it is possible to speculate endlessly that different data, forms of equation, or explanatory variables would yield significantly different and superior results. Compelling definite calculations will bring such easy speculation down to earth.”); Segar v. Civiletti, 508 F.Supp. at 712. In addition, to a certain extent the deficiencies in appellants’ statistical proof are attributable to the personnel data supplied by the school district. “In an employment discrimination case, in which the defendant employer has the greatest access to potentially relevant employment data, the burden is on it to produce and use such data in developing its statistical case [in rebuttal].” Trout v. Hidalgo, 517 F.Supp. at 886.
Kenny testified that he had prepared several analyses of the school district’s salaries and that in his opinion the salary disparity expressed in the analyses was the product of sex discrimination. At first glance the salary analyses reveal consistent disparities in favor of male teachers. For example, Plaintiffs’ Ex. 4 is an analysis of the average salary by sex, broken down into three professional groups (BA, MA, administrative), for four school years (1977-1978 through 1980-1981). In each year in every group the male teachers and administrators are paid more than the female teachers and administrators. Plaintiffs’ Ex. 5 is an analysis of the average term of contract by sex, showing that the average term of male teachers’ contracts is 10.1 months, the average term of female teachers’ contracts is 9.4 months. Extended term contracts are apparently related to higher pay. Plaintiffs’ Ex. 7 is a “test period approach” analysis; Kenny followed newly hired teachers as a group year by year from 1977-1978 to 1980-1981 to determine whether the salary disparity between male and female teachers was attributable to experience. Ex. 7 shows that in each year in each group of employees the male teachers on the average are paid more and that this salary disparity increased each year during the four year test period.
According to Kenny, Ex. 4 showed that the salary disparity between men and women with similar education or training is not attributable to education and Ex. 7 showed that this salary disparity is not attributable to experience or seniority. We must disagree. Kenny did not analyze the employment data using a method that considered the combined effect of education and experience on the salaries paid male and female teachers. Instead, Kenny analyzed education and experience separately, considering the effect of each factor in isolation. The key question is whether male and female teachers of comparable education and experience are paid the same salary. Appellants’ statistical proof shows only that male teachers are paid more than female teachers with similar degrees on the average but without taking experience into consideration. Similarly, appellants’ statistical proof showed that male teachers are paid more over time than female teachers on the average but without taking education into consideration. Appellants’ statistical proof simply does not show that, after taking into account education and experience, female teachers received lower salaries than male teachers.
We think the deficiency lies in the method of statistical analysis used by appellants. Rather than approaching the variables of education and experience separately, appellants should have used multiple regression analysis.
Multiple regression is a statistical technique designed to estimate the effects of several independent variables on a single dependent variable. Properly used in a case such as this, the methodology provides the ability to determine how much influence factors such as sex, experience, and education each have had on determining the value of a variable such as salary level. The analysis also enables an [732]*732observer to cumulate effects of the various factors so as to determine the degree to which explanation of the dependent variable can be attributed to the independent variables in combination. Regression analysis is well recognized by the literature and the courts in Title VII litigation.
Trout v. Hidalgo, 517 F.Supp. at 877-78 (footnotes omitted); see generally Finkel-stein, The Judicial Reception of Multiple Regression Studies in Race and Sex Discrimination Cases, 80 Colum.L.Rev. 737 (1980); Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. 702 (1980); Note, Beyond the Prima Facie Case in Employment Discrimination Law: Statistical Proof and Rebuttal, 89 Harv.L.Rev. 387 (1975).
As explained by Professor Fisher,
[m]ultiple regression is well suited to answer [the question of sex discrimination in wages] fairly precisely. Moreover, without a multiple regression study it is difficult to see how it could be decided. The raw comparison of average wages for women and for men may make one suspicious, but it cannot tell one anything definite. Indeed, it can be misleading in either direction. For example, it would be entirely possible in a different setting that women are paid on the average just as much as men but that a multiple regression analysis would show that there is indeed discrimination because women are more highly qualified in the measures that account for the variation in male pay.
Fisher, Multiple Regression in Legal Proceedings, 80 Colum.L.Rev. at 721 (footnotes omitted; emphasis in original). For example, in Trout v. Hidalgo, plaintiffs’ expert first established the salary differentials for men and women and then sought to determine, using multiple regression analysis, whether the salary disparity was attributable to education and experience or more likely was attributable to sex discrimination. 517 F.Supp. at 878-79. The district court in Trout v. Hidalgo found that plaintiffs had made a prima facie case of sex discrimination because multiple regression analysis showed that, when education and experience were taken into account, women were paid substantially lower salaries than men. Id. at 879.
Assuming for the purpose of argument that appellants had established a prima fa-cie case using multiple regression analysis, we think that the school district’s rebuttal evidence would have been inadequate to undermine the accuracy or significance of appellants’ statistical evidence. The school district did not counter appellants’ expert analysis with comparable expert analysis but instead offered hypothetical variables unsupported by statistical analysis. In addition to education and experience, superintendent Meeks testified that postgraduate hours, the length of contract, extra responsibilities, and summer school teaching also affected teacher salaries. The effect of these variables on teacher salaries was not supported by any proof. The school district’s failure to include these variables or more refined measurements of the education and experience variables in its own multiple regression analysis “must be interpreted as a concession that these refinements in the specification of the model [used by appellants] would not have affected the rejection of the hypothesis of no discrimination.” Id. at 886; see also Segar v. Civiletti, 508 F.Supp. at 712 (“The fact that Defendants choose not to test their hypotheses leaves their criticisms speculative and incapable of rebutting Plaintiffs’ statistical showing.”).
The weakness in appellants’ statistical proof of sex discrimination in promotion to administrative positions is slightly different. At first glance appellants appear to have an impressive case. Over 70% of the teachers employed by the school district are female. Since the 1972-1973 school year only four of fifteen (26%) promotions (or three of eighteen (17%); the discrepancy in the figures is not explained) to administrative positions went to women. Plaintiffs’ Ex. 8, prepared by expert Kenny, is an analysis of the percentage of males and females in administrative positions in the school district from 1977-1978 through [733]*7331980-1981. The figures were consistent for all four school years: 21-22% of the male teachers held administrative positions; only 1-2% of the female teachers held administrative positions. According to Plaintiffs’ Ex. 9, a comparison of the number of teachers certified as administrators by sex, in 1977-1978 one of four females certified as administrators and fourteen of seventeen males certified as administrators held principal or assistant principal positions; in 1980-1981 one of five females certified as administrators and thirteen of fifteen males certified as administrators held principal or assistant principal positions. These figures did not significantly change during the four year period. On a percentage basis, 82-86% of the males certified as administrators held principal or assistant principal positions, but only 20-25% of the females certified as administrators did. Kenny testified that these statistics showed consistent “underu-tilization” of qualified women in administrative positions and that in his opinion this “underutilization” was the result of sex discrimination.
Appellants supplemented this statistical evidence with circumstantial evidence. Superintendent Meeks testified that the only objective criteria for promotion to administrative positions were certification as an administrator and good health. The school district policy was to fill administrative positions from within the school district (with the possible exception of the superintendent position). Available administrative positions were posted on bulletin boards in school buildings and advertised in local newspapers. The highest level administrative positions were held almost exclusively by males: the superintendent, the three assistant superintendents, the high school principal, the two junior high school principals, the five assistant high school and junior high school principals, and five of the six elementary school principals are male; only one elementary school principal is female. In addition, twelve positions, mostly administrative or supervisory, have never been held by a woman: superintendent, assistant superintendent, business manager, football coach, auto mechanics, driver’s education, assistant principal, coordinator of vocational education and of distributive education, athletic director, maintenance director, and band director. Appellants also presented evidence of specific instances of discrimination.
Upon further examination, however, we conclude that appellants’ statistical evidence lacks probative value. Plaintiffs’ Ex. 8, showing that 21-22% of the male teachers held administrative positions while only 1-2% of the female teachers did so, has little probative value because its underlying statistics include teachers who are not certified as administrators and are thus overinclusive. Certification as an administrator is a prerequisite to consideration for an administrative position. “When special qualifications are required to fill particular jobs, comparisons to the general population (rather than to the smaller groups of individuals who possess the necessary qualifications) may have little probative value.” Hazelwood School District v. United States, 433 U.S. at 308 n.13, 97 S.Ct. at 2742 n.13.
The comparison between the number of male and female certified administrators who in fact held administrative positions in Ex. 9 is relevant, but we conclude that the sample size involved here is too small to support any inference of discrimination in promotion. See Teamsters v. United States, 431 U.S. at 339 n.20, 97 S.Ct. at 1856 n.20 (“Considerations such as small sample size may ... detract from the value of [statistical] evidence.”); Mayor of Philadelphia v. Educational Equality League, 415 U.S. 605, 620-21, 94 S.Ct. 1323, 1333-1334, 39 L.Ed.2d 630 (1974) (sample of thirteen); EEOC v. American National Bank, 652 F.2d at 1193-94 (two managerial level hires in seven years); White v. City of San Diego, 605 F.2d at 461, citing Morita v. Southern California Permanente Medical Group, 541 F.2d 217, 220 (9th Cir. 1976) (sample of eight), cert. denied, 429 U.S. 1050, 97 S.Ct. 761, 50 L.Ed.2d 765 (1977); Harper v. Trans World Airlines, Inc., 525 F.2d 409, 412 (8th Cir. 1975) (sample of five). Determining whether a given number of employment [734]*734decisions is too small to be statistically meaningful is necessarily imprecise. Standard deviation analysis is not particularly helpful in evaluating the statistical significance of small samples. See Eubanks v. Pickens-Bond Construction Co., 635 F.2d at 1355 n.3 (McMillian, J., dissenting in part).
The danger of unfairness to the employer in resting inference of discriminatory employment practices on proof involving small total members of employment decisions is obvious. But there is the countervailing consideration that, given the difficulties of proving discriminatory motive under any circumstances, a too ready rejection of claims solely on this account practically precludes proof of discrimination in circumstances involving local employers with relatively small total work forces. Courts have simply to balance [these conflicting factors] with an eye to protecting against purely speculative findings of discrimination while not cutting off the claims of some employees simply because of the small overall size of the work forces in which they happen to be employed.
EEOC v. American National Bank, 652 F.2d at 1194 (citations omitted).
In the present case, there has been a total of fifteen (or eighteen) promotions to administrative positions since 1972, and four (or three) of those promotions went to women. In addition, there was testimony that since 1975 the school district has reduced the number of administrative positions due to declining enrollments and school closings and that since 1976 there have been two administrative positions available (high school principal, elementary school principal) and one was filled by a woman. Although Ex. 9 shows that qualified males are about four times more likely than qualified females to be promoted to administrative positions, appellant’s statistical evidence also shows that from 1977 through 1980 women constituted about 18-25% of the persons certified to be administrators and from 1972 through 1980 women received either 17% (three of eighteen) or 26% (five of fourteen) of the administrative promotions. We think that a sample of fifteen (or eighteen) employment decisions over a period of more than eight years is simply too small to support any inference of a discriminatory pattern or practice, at least where women received several of the promotions. Compare Teamsters v. United States, 431 U.S. at 342 n.23, 97 S.Ct. at 1858 n.23 (“the inexorable zero”). Even when this statistical evidence is considered against the background of the school district’s substantially male administration and the use of subjective promotion criteria, see, e.g., Royal v. Missouri Highway & Transportation Comm’n, 655 F.2d at 164; Rogers v. International Paper Co., 510 F.2d 1340, 1345-46 (8th Cir.), vacated on other grounds, 423 U.S. 809, 96 S.Ct. 19, 46 L.Ed.2d 29 (1975), we are left with deep suspicions but little else.
We think that appellants’ evidence of disparities between the compensation paid male and female coaches did establish a prima facie case of sex discrimination which was not rebutted by the school district. Appellants showed that male coaches are paid higher salaries and higher extra duty stipends and have longer term contracts than female coaches who the district court found perform essentially the same duties. The school district argued that the male coaches’ greater compensation was the result of extended term contracts and extra duty stipends. However, the assignment of extended term contracts and extra duty stipends to particular coaching assignments is itself subject to employer discrimination on the basis of sex. For example, the male head coach of the boys high school basketball team (who also provides some assistance with football and track) is paid $17,400 (11-month contract, extra duty stipend of $2,700); the female head coach of the girls high school basketball team (who is also the head coach of the girls high school track team) is paid $13,353 (9V4-month contract, extra duty stipend of $1,400).
This part of the district court’s judgment is reversed and remanded to the district court for further proceedings. At trial superintendent Meeks testified that [735]*735the school district was at that time in the process of revising the salary schedule of the coaches. On remand the district court should determine what changes, if any, in the coaches’ salary schedule have been made. Voluntary efforts at correcting past discrimination should be considered by the district court, but will not preclude the granting of injunctive relief. See, e.g., Taylor v. Teletype Corp., 648 F.2d at 1135.
Summary
For the reasons discussed above, we reverse the district court’s findings that the school district did not discriminate on the basis of sex against Martha Coble, Susan Chapman, and the female coaches and remand for the grant of appropriate relief. We affirm the district court’s findings that appellants failed to establish a prima facie case of discrimination on the basis of sex in pay and promotions to administrative positions.
Accordingly, the judgment of the district court is affirmed in part, reversed in part and remanded for further proceedings.