
    Billy Ray EUBANKS, Appellee, v. PICKENS-BOND CONSTRUCTION CO., Appellant. Billy Ray EUBANKS, Appellant, v. PICKENS-BOND CONSTRUCTION CO., Appellee.
    Nos. 79-2005, 79-2029.
    United States Court of Appeals, Eighth Circuit.
    Submitted June 10, 1980.
    Decided Dec. 5, 1980.
    
      James W. Moore, Little Rock, Ark., for appellant/cross-appellee, Pickens-Bond Const. Co.
    James E. Youngdahl, Little Rock, Ark., for appellee/cross-appellant, Billy Ray Eu-banks.
    . Before BRIGHT, HENLEY and McMIL-LIAN, Circuit Judges.
   BRIGHT, Circuit Judge.

Billy Ray Eubanks, a black cement finisher, brought this class action suit against Pickens-Bond Construction Co. (Pickens-Bond or the Company), a commercial construction contractor with offices in Little Rock, Arkansas, alleging discrimination against the black cement finishers who worked for the Company on the construction of the First National Bank Building in Little Rock from August 1, 1973 to August 31, 1975. Following a three-day trial, the district court found that the Company had discriminated against black cement finishers in the assignment of menial and unpleasant tasks, allocation of straight-time and overtime work, layoffs, and promotions to foremen. The district court awarded both monetary and injunctive relief. The injunction, in part, ordered the Company to employ essentially equal numbers of black and white cement finishers as foremen for a two-year period and thereafter until the Company promulgated and received court approval of nondiscriminatory objective standards for the promotion of cement finishers to foreman positions.

On appeal Pickens-Bond contends that the district court erred in the factual findings necessary to the court’s determination of liability and, in the alternative, abused its discretion in ordering the limited quota for foreman positions. Eubanks cross-appeals complaining of the inadequacy of the attorneys’ fee award and of the trial court’s failure to award him monetary damages on an individual basis rather than solely as a member of the class. We affirm the district court’s award of attorneys’ fees and its determination of liability for discriminatory work assignments, allocation of overtime, and layoffs, but we reverse its finding of discrimination in the promotion of blacks to cement finisher foremen.

I. Background.

Pickens-Bond is a major construction contractor based in Little Rock, Arkansas. The Company employs a small permanent work force primarily consisting of managerial, supervisory, technical, and clerical workers. Jt hires the balance of its work force as needed for particular construction projects. On January 1, 1973, the Company’s permanent work force of 101 employees included three blacks. On January 1, 1976, four of its 110 permanent employees were black; only one of the four held a supervisory position.

In the course of constructing the thirty-one floor First National Bank Building and its parking deck, Pickens-Bond employed fifty-seven cement finishers to pour floors and lay steps. The Company also employed four or five cement finisher foremen to supervise the cement finishers’ work. A foreman’s responsibility also included the assignment of overtime work to particular cement finishers. Jack Cooper, the Company’s cement finisher superintendent, exercised sole discretion in the appointment of foremen to the project. Both Cooper and the foremen hired and laid off cement finishers as the requirements of the job dictated. Neither Cooper nor the foremen used objective criteria in the appointment of foremen, the assignment of overtime, or the laying off of cement finishers.

Primarily on the basis of disputed oral testimony and inferences from statistical evidence susceptible to varying interpretations, the district court concluded that the Company had discriminated against the plaintiff class in making work assignments, in assigning overtime work, in determining layoffs, and in promoting cement finishers to foreman positions. The district court ordered Pickens-Bond to distribute $16,854 to members of the plaintiff class, enjoined Pickens-Bond from assigning tasks, jobs, layoffs, or promotions on the basis of race, and required the Company to employ essentially equal numbers of black and white cement finishers as foremen for a two-year period.

II. Title VII Liability.

A. Discriminatory Work Assignments, Allocation of Overtime, and Layoffs.

Pickens-Bond challenges as clearly erroneous the factual findings supporting the district court’s determination of liability under Title VII for discrimination in work assignments, assignment of overtime, and layoffs. The Company chiefly contends that the testimony and statistical evidence adduced at trial belie the district court’s findings of unlawful discrimination. We have carefully reviewed the record in this case and conclude that it supports contradictory inferences. We cannot say, therefore, that these findings of fact are clearly erroneous.

With respect to work assignments on the job, the record discloses that after pouring a concrete floor a cement finisher’s duties include “rodding” (pulling a straight edge), hand troweling, “jitterbugging” (tamping down gravel with an instrument called a jitterbug), “bullfloating” (smoothing a cement surface with a metal plate attached to a long rod), and running a trowel machine. The district court found that “rodding,” “jitterbugging,” and “bull-floating” were among the most onerous duties of a cement finisher, and that one foreman on the First National Bank job had discriminatorily assigned these tasks to black cement finishers until the project was nearly completed. No witness characterized “bullfloating” as an undesirable task, but the testimony of several witnesses supports the finding that black cement finishers were assigned the most menial and unpleasant tasks on the job. From the record before us, therefore, we cannot say that the district court erred in failing to credit contrary testimony that no cement finishers were assigned specific duties on the job or that black and white cement finishers performed all tasks equally.

With respect to the assignment of overtime and determination of layoffs, Pickens-Bond primarily relies on the unrebutted testimony of its statistical expert and the statistical analysis of relevant data proffered by him. The Company argues that the district court, in computing the wage differential between white and black cement finishers, erroneously included the wages earned by cement finisher foremen who received a fifty-cent per hour premium for their additional responsibilities. A comparison of the wages earned and number of pay periods worked by white and black cement finishers, omitting foremen, it points out, reveals no significant statistical disparity from which discrimination may be inferred. Nevertheless, given the district court’s findings that Pickens-Bond discriminated against black cement finishers in assigning overtime at undesirable hours, and allowing for the differing inferences to which the statistical evidence was susceptible, the district court did not clearly err in finding the total wage differential attributable to the Company’s discriminatory acts.

B. Promotion to Foreman.

The district court also concluded that Pickens-Bond had discriminated against blacks in the promotion of cement finishers to cement finisher foremen. In reaching this conclusion, the court found the following evidence sufficient to establish a prima facie case of discrimination:

(a) The defendant’s permanent work force of 101 employees included only three blacks on January 1, 1973. By January 1, 1976, four of its 110 employees were black. Only one of the four blacks was a supervisory employee; that black was a cement finisher foreman.
(b) Only one black, Robert Brown, Sr., has held a permanent job as a foreman with Pickens-Bond since 1964.
(c) Pickens-Bond has no objective standards for hiring, layoffs, or promotion to the position of cement finisher foreman. These foremen were customarily selected by the cement finisher supervisor, Jack Cooper, a white.
(d) On the First National Bank job, defendant employed 29 black cement finishers, or 50.88% of the total employed, and 28 white cement finishers, or 49.12% of the total employed. The cement finishing work on the First National Bank job lasted from 1973 to 1975.
(e) In 1973, defendant employed a total of five cement finisher foremen in Little Rock, Arkansas, one of whom was black; therefore, the percentage of black cement finisher foremen was 20% of the total. In 1974, defendant employed a total of seven cement finisher foremen in Little Rock, one of whom was black; therefore, the percentage of black cement finisher foremen was 14.29% of the total. In 1975, defendant employed a total of eight cement finisher foremen in Little Rock, two of whom were black; therefore, the percentage of black cement finisher foremen was 25% of the total.
(f) Black cement finishers are uniformly as well qualified to be foremen as the white cement finishers. [Finding of Fact No. 7.]

Pickens-Bond contends that these findings fail to make out a prima facie case of discriminatory promotions. We agree.

The district court first cited as evidence of discrimination the number of blacks on the Company’s permanent work force. The issue here, however, is not the number of blacks serving on Pickens-Bond’s permanent work force, but the number of blacks serving as cement finisher foremen on the First National Bank job. Thus, finding (a) does not directly contribute to the establishment of a prima facie case of discrimination for this plaintiff class. Moreover, Pickens-Bond maintains a permanent work force that primarily consists of managerial personnel, not construction workers or foremen. For the 1973-1975 period in question, the Company’s permanent work force included only two cement finisher foremen, Jack Cooper (a white) and Robert Brown (a black). Of those cement finisher foremen on the permanent work force, therefore, an equal number of whites and blacks worked for the Company. See Plaintiff’s Exhibit No. 20, pp. 99-100. See also Plaintiff’s Exhibit No. 6 (Company’s permanent personnel on April 17, 1974).

The district court noted in finding (b) that only one black has held a permanent job as a foreman with Pickens-Bond since 1964. Because this finding focuses on foremen, it more directly bears on the issue before us. Again, however, this finding speaks to the number of black foremen on the permanent work force rather than to the number on the First National Bank job. This finding also addresses a more general question than the one before us by focusing on the number of black foremen rather than on the number of black cement finisher foremen. As noted above, only a few foremen serve on the Company’s permanent work force, and an equal number of blacks and whites served in the two cement finisher foremen positions on the permanent work force for the 1973-1975 period in question and, in fact, since 1968. See Plaintiff’s Exhibit No. 20, pp. 99-100.

Finding (c), unlike findings (a) and (b), directly focuses on the position of cement finisher foreman. Nevertheless, the Company’s use of subjective criteria in promoting cement finishers to foreman positions does not in itself supply an inference of discrimination. See Rogers v. International Paper Co., 510 F.2d 1340, 1345 (8th Cir.) (subjective employment selection standards not unlawful per se), vacated and remanded on other grounds, 423 U.S. 809, 96 S.Ct. 19, 46 L.Ed.2d 29 (1975). Cf. id. at 1345-46; United States v. N.L. Industries, Inc., 479 F.2d 354, 371-72 (8th Cir. 1973) (use of subjective criteria in awarding promotions relevant to the establishment of prima facie case of discrimination). In the absence of other evidence of discrimination, the use of subjective criteria does not alone make out a pattern or practice of discrimination. Likewise, a finding of equal qualifications, as in finding (f), does not alone make out a prima facie case of discrimination in promotions.

We come, then, to findings (d) and (e) as a foundation for the district court’s determination of discriminatory promotions. The court found that blacks comprised 20% (one of five) of the cement finisher foremen employed by the Company in Little Rock in 1973,14.29% (one of seven) in 1974, and 25% (two of eight) in 1975. By comparison, blacks accounted for 50.88% (twenty-nine of fifty-seven) of the cement finishers who worked on the First National Bank job at some time during the three-year project.

We have previously found statistics to speak loudly in employment discrimination cases. E.g., United States v. N.L. Industries, Inc., 479 F.2d 354 (8th Cir. 1973); Parham v. Southwestern Bell Telephone Co., 433 F.2d 421 (8th Cir. 1970). See Ha-zelwood School District v. United States, 433 U.S. 299, 307-08, 97 S.Ct. 2736, 2741-2742, 53 L.Ed.2d 768 (gross statistical disparities alone may in a proper case constitute prima facie proof of discrimination); International Brotherhood of Teamsters v. United States, 431 U.S. 324, 339, 97 S.Ct. 1843, 1856, 52 L.Ed.2d 396 (1977) (statistics competent in proving employment discrimination). Statistics, however, must be used carefully. “[T]hey come in infinite variety and, like any other kind of evidence, they may be rebutted. In short, their usefulness depends on all of the surrounding facts and circumstances.” International Brotherhood of Teamsters v. United States, supra, 431 U.S. at 340, 97 S.Ct. at 1856. Having carefully evaluated the numbers and percentages cited by the district court in the context of this case, we believe that the data fails to establish a prima facie case of discrimination by Pickens-Bond in the promotion of cement finishers to the position of foreman on the First National Bank project.

In 1973, as the district court noted, only one of five (20%) cement finisher foremen were black. Two of the five, however, were the only two cement finisher foremen on Pickens-Bond’s permanent work force when it began the First National Bank project. See Plaintiff’s Exhibit No. 20, pp. 99-100. These men — Jack Cooper (a white) and Robert Brown (a black) — quite naturally would be named foremen on this job, and we find nothing in the record suggesting race as a factor in their selection. Larry Cooper (a white) obtained his promotion to foreman in 1973 solely because of his familial relationship to Jack Cooper, although nepotism was not pervasive in the Company, not because of his race. Finding of Fact No. 13. Thus, the essence of plaintiff’s claim of discrimination in 1973 reduces itself to the promotion of two men.

Of the twenty-two cement finishers (twelve blacks and ten whites) eligible for these positions in 1973, two whites were promoted, T. Tyler and O.D. Holland. We cannot say, however, that the selection of two white cement finishers rather than one white and one black or two blacks establishes a prima facie case of discrimination. As a practical matter, a difference of one may represent only a random result, not proof of discrimination. Selection processes characteristically produce short-term results that do not hold true over the long run. Small deviations from the “ideal” number arise naturally and harmlessly in many selection processes operating in a completely evenhanded manner. See generally D. Baldus & J. Cole, Statistical Proof of Discrimination 287-321 (1980); Shoben, Differential Pass-Fail Rates in Employment Testing: Statistical Proof Under Title VII, 91 Harv.L.Rev. 793 (1978). See also note 10 infra.

Similarly for those promotions which occurred in 1974, the district court’s findings do not establish a prima facie case of employment discrimination. The court noted that blacks comprised only 14.29% (one of seven) of the cement finisher foremen in 1974. This percentage statistic, however, is misleading. The 14.29% figure corresponds to the percentage of black cement finisher foremen employed by Pickens-Bond for all of its Little Rock projects in 1974, not for the First National Bank job alone. The record indicates that the Company maintained no more than four or five cement finisher foremen on the First National Bank job at any point in time, that two of the foremen in 1973 (both white) left that project in 1974, and that the two foremen selected to take their place were both white. See note 6 infra. Thus, the percentage of black foremen on the First National Bank job in 1974 does not differ from that in 1973.

More importantly, the district court’s focus on the number of blacks among all foremen is misplaced. Because the complaint alleges that Pickens-Bond discriminated in the promotion of blacks to cement finisher foremen, the more telling figure is the number of blacks among the new foremen selected in 1974. It is for the group of new foremen, i. e., those first promoted to that position in 1974, to which the complaint speaks and in which the observed and expected number of blacks promoted should not differ to a significant degree if race were not a factor in the Company’s selection process. Viewing the employment data from this perspective, we find that Pickens-Bond promoted only two men to the foreman’s position in 1974 — A.L. Starrett and G. Pater, both of whom were white. See Defendant’s Exhibit No. 8, Attachment 8-4. At the time of their promotions, nine whites and six blacks were eligible for the foreman’s position. Although two whites were promoted, we again cannot say that a difference of one provides prima facie evidence of employment discrimination.

In 1975 Pickens-Bond promoted one black and two white cement finishers from an eligible pool of six blacks and six whites still employed on the First National Bank job at that time. As for the years 1973 and 1974, the employment data for 1975 indicates no significant disparity between the number of black and white cement finishers promoted to foremen. In addition, the three promotions apparently applied to projects other than the First National Bank job. Thus, these promotions do not shed light on the promotions to which the complaint speaks — those for the First National Bank job.

Arguably, the promotion of only one black during a three-year period in which seven promotions occurred could provide some evidence of discrimination even if the promotions, when viewed on a year-by-year basis, would not give rise to an inference of discrimination. Neither the record nor the district court’s findings of fact, however, contain any appropriate statistical or legal analysis of the three-year employment data that would permit such an inference to be drawn. The district court merely stated the actual number and percentage of black foremen for the three years in question and the number and percentage of black cement finishers on the First National Bank job during that period. To infer a prima facie case of discrimination from these numbers and percentages, the court needed to go beyond presentation of the raw data to application of an appropriate and recognized test for determining the statistical significance of any observed disparity. See Mayor v. Educational Equality League, 415 U.S. 605, 619-20, 94 S.Ct. 1323, 1332-1333, 39 L.Ed.2d 630 (1974) (rejecting as unreliable “simplistic percentage comparisons” of the racial composition of a thirteen-member nominating panel and the population of Philadelphia). Cf. Hazelwood School District v. United States, supra, 433 U.S. at 308-09 n.14, 97 S.Ct. at 2741-2742 n.14 (standard deviation analysis). See also D. Baldus & J. Cole, supra at 322-28 (selected methods for calculating tests of statistical significance). For a number of reasons, moreover, we entertain serious doubt that this case lends itself to reliable statistical analysis.

First, the promotion of only four cement finishers to foremen for the First National Bank project (or even the entire ten cement finishers employed by the Company as foremen on all of its Little Rock projects from 1973 to 1975) may constitute too small a sample size to establish a prima facie finding of discrimination through statistical analysis. See, e. g., Mayor v. Educational Equality League, supra, 415 U.S. at 621, 94 S.Ct. at 1333 (smallness of sample size is important flaw in straight percentage comparisons); Harper v. Trans World Airlines, Inc., 525 F.2d 409, 412 (8th Cir. 1975) (“[Statistical evidence derived from an extremely small universe, as in the present case, has little predictive value and must be disregarded.”). See also Shoben, supra at 801 & n.37 (when the sample size is small, a test based upon the assumption of normality may not be used). Although a gross statistical disparity may be evidence of discrimination for purposes of Title VII, “[considerations such as small sample size may, of course, detract from the value of such evidence.” International Brotherhood of Teamsters v. United States, supra, 431 U.S. at 339-40 n.20, 97 S.Ct. at 1856-1857 n.20. Cf. Castaneda v. Partida, 430 U.S. 482, 497 n.17, 97 S.Ct. 1272, 1281 n.17, 51 L.Ed.2d 498 (1977) (large sample of 870). Here, the promotion of one less or one more black among the seven promotions that occurred during the period in question would result in over a fourteen percent change in the racial, composition of the foremen group. Cf. Mayor v. Educational Equality League, supra, 415 U.S. at 610-11, 94 S.Ct. at 1328-1329 (upholding district court’s rejection of data as unreliable, in part, because the change of one position in a thirteen-member panel would alter racial composition by eight percent).

Second, tests of significance require some assurance of homogeneity among the candidates in the pool from which the selections occurred. See Shoben, supra at 801. See also D. Baldus & J. Cole, supra at 181-286 (discussion of the implications of gaps between the assumptions and the characteristics of the data as a recurring problem). Here, according to some evidence in the record, the pool of cement finishers on the First National Bank job may have included persons who had no interest in the available foreman positions or who lacked sufficient experience or other legitimate credentials to become foremen.

In contrast to these threats to the reliability of the data, the record is devoid of any expert analysis on behalf of plaintiff to indicate that discrimination may be inferred from the statistics cited by the district court in its findings. Although we do not rely on the contrary testimony of the Company’s expert witness, the data in this case seems too inconclusive under any analysis to justify an inference of discrimination against blacks in the selection of foremen.

Moreover, the plaintiff class did not bolster its case with any evidence of specific instances of discrimination against black cement finishers seeking promotion. Cf. Teamsters v. United States, supra, 431 U.S. at 338, 97 S.Ct. at 1856 (statistical proof of discriminatory pattern-or-practice buttressed by forty specific instances of discrimination); United States v. Hazelwood School District, 534 F.2d 805 (8th Cir. 1976) (showing of specific discrimination against sixteen black applicants for teaching positions), vacated and remanded on other grounds, 433 U.S. 299, 97 S.Ct. 2736, 53 L.Ed.2d 768 (1977).

Accordingly, we hold that the district court’s finding that Eubanks and the plaintiff class established a prima facie case of discriminatory promotions must be set aside as without adequate support in the record. Thus, the relief based on that finding, which requires the assignment of equal numbers of black and white cement finishers to foremen positions, must be vacated.

D. Cross-Appeal.

The district court awarded $15,-823.64 in attorneys’ fees and costs rather than the $29,703.14 requested by plaintiff class. The court also awarded Eubanks, as the named plaintiff, $904.81 as his apportioned share of the class monetary judgment, but elected not to grant Eubanks additional damages on an individual basis for the specific finding of discrimination against him. Having reviewed the record, we find no error or abuse of discretion in these rulings and affirm on the issues raised in the cross-appeal.

III. Conclusion.

For the reasons discussed above, we affirm the district court’s award of attorneys’ fees and findings of discrimination in work assignments, assignments of overtime, and layoffs, but reverse its finding of discrimination in the promotion of blacks to cement finisher foremen.' Accordingly, we direct the district court on remand to vacate paragraph three of its judgment and order, dated October 23, 1979, and filed October 30, 1979.

The district court on remand shall also determine and award appellee/cross-appel-lant reasonable attorneys’ fees and seventy-five percent of costs on this appeal.

Affirmed in part and vacated in part. Remanded for proceedings consistent with this opinion.

McMILLIAN, Circuit Judge,

dissenting in part.

I concur in that part of the majority opinion affirming the district court’s finding of employer liability for discriminating on the basis of race in work assignments, allocation of overtime and layoffs and the award of attorneys’ fees. Like the majority (see at 1345 n.3), I have reservations about the district court’s formula for distribution of the backpay award. However, as noted by the majority, no party has challenged the distribution formula on appeal.

For the reasons discussed below, I do not agree that plaintiffs below failed to establish a prima facie case of racial discrimination in the promotion of cement finisher foremen on the First National Bank project. In particular, I disagree with the majority’s treatment of plaintiffs’ statistical evidence. Therefore, I dissent from part 11(B) of the majority opinion.

First, I disagree with the majority’s treatment of plaintiffs’ nonstatistical evidence. Although I agree that any single factor, such as the use of subjective promotional criteria or a finding of equal qualifications, does not in itself make a prima facie case, I cannot understand why the majority considers and then rejects each factor in isolation in the present case. Plaintiffs did not seek to establish on the basis of any single factor that the employer was discriminating on the basis of race in promoting or selecting cement finisher foremen. In my opinion, the employer’s management structure and racial composition, the method of promotion, the absence of objective promotional criteria, and a finding of equal qualifications are clearly relevant. E.g., 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), reinstated with modification on other grounds, 526 F.2d 722 (8th Cir. 1975); United States v. N.L. Industries, Inc., 479 F.2d 354, 367-69 (8th Cir. 1973); Rowe v. General Motors Corp., 457 F.2d 348, 357-59 (5th Cir. 1972).

The district court found that in 1976 only four of the employer’s permanent employees were black (included among the four were the chauffeur and the yardman), only one of the permanent supervisory employees was black (permanent cement finisher foreman Robert Brown, Sr.), the superintendent (who was white) selected the foremen without using objective criteria, the foremen were generally selected from the cement finishers working on the job, all the cement finishers were equally qualified to be foremen, and 51% (28 of 57) of the cement finishers on the First National Bank project were black. While not conclusive, these factors, especially when considered together, do support the district court’s conclusion that a prima facie case had been made.

Further, I do not agree that plaintiffs necessarily needed to “bolster” their evidence with specific instances of racial discrimination in order to make a prima facie case. Evidence of specific instances of racial discrimination undoubtedly “[brings] the cold numbers convincingly to life,” Teamsters v. United States, 431 U.S. 324, 339, 97 S.Ct. 1843, 1856, 52 L.Ed.2d 396 (1977), but such evidence certainly is not essential to a prima facie case. In fact, the “state of the art” of employment discrimination law has, in my opinion, evolved considerably beyond the point where plaintiffs are able in most cases to produce specific instances of overt racial discrimination. Employers, often acting upon the advice of expert legal counsel, will very rarely openly subject plaintiffs to displays of intentional racial discrimination. I perceive, however, little diminution in employment discrimination. The violation and the proof of the violation have become more subtle and more sophisticated. As has been observed elsewhere, “[i]n many cases the only available avenue of proof is the use of racial statistics to uncover clandestine and covert discrimination by the employer . . . . ” Teamsters v. United States, supra, 431 U.S. at 340 n.20, 97 S.Ct. at 1857 n.20 (citations omitted).

I note preliminarily that I would not consider plaintiffs’ statistical evidence apart from plaintiffs’ other evidence. I do not understand plaintiffs’ case to be based on statistical evidence alone.

Because this case involves selection of employees, I would begin the analysis by looking at the Uniform Guidelines on Employee Selection Procedures, 29 C.F.R. Part 1607 (1979). The Uniform Guidelines have been adopted by the EEOC, the Office of Personnel Management (formerly the Civil Service Commission), the Department of Labor, the Department of Justice, and the Treasury Department’s Office of Revenue Sharing and incorporate a single set of principles designed to assist employers, labor organizations, employment agencies, and licensing and certification boards in complying with the requirements of federal employment discrimination law. These guidelines were a response by administrative agencies to the unfortunate development of varying and sometimes inconsistent approaches to the problem of discrimination in employee selection. The Uniform Guidelines attempt to present employer and employee alike with a framework for analysis of at least some kinds of discrimination problems. I believe consideration of the Uniform Guidelines is critical to the development of some degree of predictability in the enforcement of federal employment discrimination law and to giving employers a tool for judging their own employment practices and complying with the law without active intervention by the government. The Uniform Guidelines have been adopted not only by one administrative agency but by all those federal agencies having significant responsibility in the area of antidis-crimination law. They are therefore entitled to great deference as a matter of ordinary principles of administrative law. See Griggs v. Duke Power Co., 401 U.S. 424, 433-34, 91 S.Ct. 849, 854-855, 28 L.Ed.2d 158 (1971).

The Uniform Guidelines contain a helpful “rule of thumb,” the “four-fifths rule”:

A selection rate for any race, sex, or ethnic group which is less than four-fifths (Vs) (or eighty percent) of the rate for the group with the highest rate will generally be regarded by the Federal enforcement agencies as evidence of adverse impact, while a greater than four-fifths rate will generally not be regarded by Federal enforcement agencies as evidence of adverse impact .... Greater differences in selection rate may not constitute adverse impact where the differences are based on small numbers and are not statistically significant .... Where the user’s evidence concerning the impact of a selection procedure indicates adverse impact but is based upon numbers which are too small to be reliable, evidence concerning the impact of the procedure over a longer period of time and/or evidence concerning the impact which the selection procedure had when used in the same manner in similar circumstances elsewhere may be considered in determining adverse impact.

29 C.F.R. § 1607.4(D).

For purposes of my analysis, I will accept, arguendo, the majority’s characterization of the evidence. I summarize the figures in a table

Year Cement finishers eligible for promotion Cement finishers promoted Total foremen employed

1973 12 black 0 black 1 black

10 white 2 white 4 white

1974 6 black 0 black 1 black

9 white 2 white 6 white

1975 6 black 1 black 2 black

6 white 2 white 6 white

I note initially that in each year the differential in selection rates shows discrimination under the four-fifths rule, but in each year a change of only one selection would lead to a different result. Therefore, taking each year separately, I do not consider the numbers large enough to apply the four-fifths rule meaningfully. The Uniform Guidelines point to two additional criteria for such cases. First, the guidelines suggest consideration of the impact of the selection procedure over longer periods of time. Over the entire three-year period the selection rates for qualified blacks (4.5%) was well below four-fifths of that for qualified whites (24.0%). Second, the guidelines suggest consideration of “the impact which the selection procedure had when used in the same manner in similar circumstances elsewhere.” 29 C.F.R. § 1607.4(D). The type of “subjective and ad hoc promotion decisions” at issue here has been a familiar target in employment discrimination cases. See Davis v. Califano, 613 F.2d 957, 965-66 & nn.49-51 (D.C. Cir. 1979), citing Rogers v. International Paper Co., supra, 510 F.2d 1340. This type of promotion procedure is “highly suspect and must be closely scrutinized because of [its] capacity for masking unlawful bias.” Davis v. Califano, supra, 613 F.2d at 965 (footnote omitted).

I would therefore conclude that the selection rates in this case met the four-fifths rule and were presumptively discriminatory. The Uniform Guidelines, however, suggest one further step in the analysis: such a difference in selection rates may not be discriminatory if based on small numbers and not statistically significant. The Uniform Guidelines do not preclude in all cases a finding of discrimination in the absence of a showing of statistical significance, but permit the fact finder to reject the other evidence of discrimination if there is absence of statistical significance. I observe that this provision would lend itself to treatment as a defense of lack of statistical significance which could be offered by an employer to rebut a four-fifths rule showing, depending on how persuasive the statistics were in showing a high probability that the evidence of discrimination could be explained by non-discriminatory factors. In this case the statistics suggest, if anything, that the failure to select blacks was unlikely in the absence of discrimination.

The small amount of data allows only the most rudimentary statistical analysis. One can say that the proportion of blacks chosen to be foremen is somewhat less than the proportion of blacks among those eligible. One can say, with the majority, that this discrepancy would not have occurred if in each year one of the white foremen chosen was black. Finally, if one assumes that race was not a factor at all in the selection process (the “null hypothesis” approach), one can calculate the probability that the selections would have occurred as they did: a chance of about 20 in 100 (one in five) that two whites would be chosen in 1973, a chance of about 35 in 100 (one in three) that two whites would be chosen in 1974, and a 50-50 chance of choosing at least two whites in 1975. The chance that no more than one black would be chosen and that the one black choice would occur in the final year is about 3.4 in 100; the chance that no more than one black would be chosen among the seven choices during the whole period is about 6 in 100.

In my view there is no doubt that this data supports quite strongly the inference that discrimination took place. The majority makes much of the fact that we are only looking at two promotions a year in 1975 and 1976: that is, if one of the two foremen selected had been black, the percentages would thus indicate no discrimination. But the fact is that all four of the foremen selected were white. None of the four selected was black. The employer could have selected black foremen but did not.

What is the “significance” of this data? The concept of statistical “significance” is one that seems particularly susceptible to misunderstanding. The critical question is, of course, significant for what purpose? Social scientists regard a hypothesis as true if it shows less than a 5% or perhaps 1% chance of error; they then regard a statistical test of the data confirming a hypothesis to a 95% or 99% certainty as scientifically true. Businesses sometimes disregard as much as a 10% chance of error in deciding whether a hypothesis is a valid basis for action. See T. Dyckman & J. Thomas, Fundamental Statistics for Business & Economics 366-471 (1977). In a eour„ of law in my view there is no arbitrary cutoff point for statistical significance; while statistics regarded by social scientists as sufficient to prove a fact are obviously entitled to great weight, less definite statistical results also have some significance, especially in light of other circumstances.

Here the hypothesis is that the employer did (or did not) discriminate in promotions. Assuming that promotions were made by totally race neutral means from the pool of qualified employees, there is only about a 6% chance of one or fewer blacks being chosen. Moreover, there is less than a 4% chance that no more than one black foreman would be chosen and that that one black foreman would be chosen in the third and final year of the project. These probabilities tend to confirm the district court’s conclusion that discrimination took place.

This approach only measures the probability of discrimination under a random selection procedure, as if the selection had been made by picking names out of a hat. To me it is evident that this assumption underestimates the probability of discrimination. Employers do not pull names out of hats. When the job started this employer had a permanent workforce of more than 100 that was white except for a black yardman, a black chauffeur and one black job foreman. The foremen it had when it started the job were white with the exception of the one black permanent foreman. The first two years of the job, the employer selected all white foremen. Therefore, not only does the promotion data meet the test of showing less than a 5% chance of nondiscriminatory selection, but the data is part of a broader pattern indicative of discrimination. Therefore, although the number of selections is few, I have a great deal of difficulty with the majority’s emphasis on the small number of promotions as a weakness in plaintiffs’ case.

“Title VII tolerates no discrimination, subtle or otherwise.” McDonnell Douglas Corp. v. Green, 411 U.S. 792, 801, 93 S.Ct. 1817, 1823, 36 L.Ed.2d 668 (1973). True, there is no open discrimination in promotion here. Considering each single promotion decision separately and considering the whole pattern in isolation from the persuasive national experience of racial discrimination in employment that gave rise to the need for Title VII, each promotion decision appears innocuous enough. But the district court looked at the whole picture in finding these promotions discriminatory. I would affirm the district court’s judgment. 
      
      . Eubanks, a former employee of Pickens-Bond, alleged violations of section 1 of the Civil Rights Act of 1866, 42 U.S.C. § 1981 (1976), and Title VII of the Civil Rights Act of 1964, 42 U.S.C. §§ 2000e to 2000e-16 (1976) (amended 1978).
      Pursuant to Fed.R.Civ.P. 23(b)(2), the District Court for the Eastern District of Arkansas certified the plaintiff class as consisting of all black cement finishers who worked for Pick-ens-Bond on the First National Bank job from August 1, 1973 to August 31, 1975. In its judgment and order, the district court identified the members of the class in conformity with Fed.R.Civ.P. 23(c)(3).
     
      
      . The district court also found that Pickens-Bond had discriminated against Eubanks in laying him off. See Finding of Fact No. 11.
     
      
      . The district court ordered the distribution to proceed according to a formula under which those black cement finishers who worked longest on the project would be compensated most. Although this distribution has the seeming irony of awarding the greatest sums to those least injured by the Company’s discrimination, no party challenges the formula on appeal. To some extent the formula compensates for discrimination in work assignments on the job.
     
      
      . The district court also found that Pickens-Bond then “changed the practice and required that work be equitably assigned.” [Finding of Fact No. 9.]
     
      
      . Although 57 cement finishers ultimately worked on the three-year project, only 29 had been employed by the end of 1973. Of these 29, Jack Cooper, Robert Brown, and Larry Cooper must be excluded from consideration for the two foremen promotions in question because they already held that rank. Four additional men should be excluded from consideration because they were transferred to another job on the same day they were hired for the First National Bank project.
      For purposes of this discussion, we have assumed that the pool of candidates eligible for promotion to foreman consists of those cement finishers employed on the First National Bank job, even though it is possible that there may have been persons who were eligible and considered for the foreman position who were not on-site workers. This assumption, however, is consistent with the district court’s comparison of the number of black foremen to the number of black cement finishers on this project in finding a prima facie case of discrimination. We also recognize that these figures may be underinclusive of the number of cement finishers eligible for promotion to the extent that the Company’s discrimination in work assignments, allocation of overtime, and layoffs may have affected the number of finishers, black and white, still on the job when it came time for awarding promotions. This effect, however, is speculative on the record of this case and the district court itself did not consider or rely upon other discriminatory acts in formulating its percentage figures or making its percentage comparisons of black foremen and black cement finishers. See also note 13 infra.
      
     
      
      . The exact date that Pickens-Bond promoted Starrett and Pater is not entirely clear from the record. We may reasonably infer, however, that because the Company would need a fairly constant number of foremen throughout the job, these promotions occurred in March of 1974 when two of the 1973 foremen, T. Tyler and L. Cooper, transferred to another project. See Defendant’s Exhibit No. 17. This inference also finds support in another exhibit, which indicates that Pater worked as a nonsuperviso-ry employee for only two weeks from the date he was first hired for the First National Bank job, March 13, 1974. See Defendant’s Exhibit No. 14.
     
      
      . These figures assume that only those cement finishers who worked on the job before Pick-ens-Bond hired Starrett and Pater were eligible for the two positions. See note 5 supra. If we include all cement finishers who worked on the First National Bank job at any time in 1974, the work force eligible for promotion would consist of 13 whites and 11 blacks. See Defendant’s Exhibit No. 17.
      As before, see note 5 supra, these numbers exclude those workers who had been laid off, transferred to another project, or already named foreman before the vacancies arose.
     
      
      . Pickens-Bond may have promoted a fourth foreman in 1975, R. Brown, Sr., (a black cement finisher) but the record is not entirely clear whether he and the “R. Brown” listed as foreman in 1973 and 1974 are the same person. See Defendant’s Exhibit No. 17. The denomination of “Sr.” after the name and the listing of both a R. Brown, Sr., and R. Brown, Jr., on the list of cement finishers employed on the First National Bank job in 1975 may or may not refer to the same individual.
      Given the uncertainty in the record and our analysis that a difference of one is not statistically significant, however, we have not considered him to be an additional promotion for purposes of this discussion. Our approach, moreover, casts the district court findings in their most favorable light, for the promotion of R. Brown, Sr., would mean that the Company promoted two blacks and two whites in 1975 instead of one black and two whites.
     
      
      . Defendant’s Exhibit No. 8, Attachment 8-4, indicates that the new foremen in 1975 consisted of L.M. Weddell (a white), J. Shaw (a black), and D. Fultz (a white). Weddell’s name, however, does not appear on the payroll list of those cement finishers employed on the First National Bank job. Shaw’s name appears on the list, but this list indicates that he had been transferred to another project in Little Rock before 1975. Fultz’s name also appears on the payroll, but the list indicates that he worked on the First National Bank job only one week before transferring to another project. Compare Defendant’s Exhibit No. 8, Attachment 8-4, with Defendant’s Exhibit 17.
     
      
      . For a discussion of the small sample prob- ' lem as a source of unpredictability, instability, and variability in selection processes, see D. Baldus & J. Cole, supra at 297-303.
      Even in a selection process which is completely neutral as to group status, * * * intangible influences may in the short run produce results suggesting bias, when the long-run result would be a perfectly evenhanded treatment of minorities. For example, assume four black and four white women with equal formal qualifications applied for three positions as a personal secretary with a corporate official, and 1 black and 2 whites were hired. The small numbers of applicants would lend but little support to the proposition that the results reflected bias on the defendant’s part rather than reflecting the impact of subjective and impressionistic factors which would favor neither blacks [n]or whites over the long run. [Id. at 299.]
     
      
      . [W]hen the assumptions of randomness or representativeness are challenged, proper interpretation requires a preliminary inquiry into the degree to which the data in fact fail to meet the assumption at issue. [D. Baldus & J. Cole, supra at 289-90 (emphasis in original).]
     
      
      . Although we have determined that plaintiff did not make out a prima facie case of discriminatory promotions, we leave intact another portion of the order that enjoins the Company from assigning work, laying off, and promoting cement finishers on the basis of race. This employment remedy may be justified by the discrimination found in the Company’s work practices in general.
     
      
      . The majority points out that these figures may not truly represent those employees who sought promotion, because it is not established in the record that all qualified employees wanted to be foremen. Although I agree this deficiency may reduce slightly the probative value of the data, I do not think it warrants disregarding it. The evidence was that all of the qualified employees were eligible for promotion. There is no reason to suppose that black employees would be any less likely to accept a promotion — and a pay increase — than white employees.
      Finally, the data rests on assumptions that minimize any discrimination shown. For example, three whites were promoted in 1973, not two as the table shows. The majority disregards the third because he was promoted on the basis of his familial relationship to a management employee. See at 1347. The majority considers this reason for promotion nondiscriminatory, despite the fact that management was nearly all white, so that the usé of nepotism as a criteria for promotion would have an obvious discriminatory impact. Thus, these data are conservative and may well omit factors which would strengthen the support for the district court’s finding of discrimination.
     
      
      . The “conditional probability” that two successive events will occur is the product of the probabilities of each occurrence. See J. Freund, Modem Elementary Statistics 134-40 (1973); T. Wonnacott & R. Wonnacott, Introductory Statistics 40-41 (1969). The probability of selecting the first white foremen in 1973 from a pool of 10 whites and 12 blacks was 10/22; the probability of selecting a second white from a pool of 9 whites and 12 blacks was 9/21; and the probability of selecting 2 whites was 90/462 or about 20%. Other probabilities were similarly computed.
      I have not used the formula for “standard deviation” set forth by the Supreme Court in Hazelwood School Dist. v. United States, 433 U.S. 299, 311 n.17, 97 S.Ct. 2736, 2743 n.17, 53 L.Ed.2d 768 (1977), and Castaneda v. Partida, 430 U.S. 482, 496-97 n.17, 97 S.Ct. 1272, 1281-1282 n.17, 51 L.Ed.2d 498 (1977). That formula was developed to estimate probabilities in cases involving large numbers. But if the numbers are small enough, an exact computation, rather than the “standard deviation” analysis, should be used. See D. Baldus & J. Cole, Statistical Proof of Discrimination 323 (1980). The “standard deviation” analysis has built into it certain approximations which are insignificant in analyzing large numbers and simplify enormously the calculation of the probability of a given result. As the sample gets smaller, the appoximations built into the analysis increasingly affect the significance of the result, while the calculation of an exact result becomes simpler. See also note 3 infra.
      
     
      
      . Until approximately 1978, the statistical analysis in employment discrimination cases was “largely intuitive.” Mitchell v. Rose, 570 F.2d 129, 133 n.4 (6th Cir. 1978) (jury discrimination case). However, two footnotes in two 1977 Supreme Court cases seem to have launched the lower courts upon what I would describe as an unnecessarily narrow course. Hazelwood School Dist. v. United States, supra, 433 U.S. at 311 n.17, 97 S.Ct. at 2743 n.17; Castaneda v. Partida, supra, 430 U.S. at 496-97 n.17, 97 S.Ct. at 1281-1282 n.17; see, e. g., EEOC v. United Virginia Bank/Seaboard Nat’l, 615 F.2d 147, 149-54 (4th Cir. 1980); Gay v. Waiters’ and Dairy Lunchmen’s Union, Local No. 30, 489 F.Supp. 282, 307-11 (N.D.Cal.1980). Recently, a panel of this Circuit used a statistical significance/standard deviation type analysis in Hameed v. Iron Workers Local 396, 637 F.2d 506 (8th Cir. 1980) (large sample, applicant flow data). In my opinion, undue emphasis on “statistically significant” employment disparities under the “standard deviation” analysis in order to make a prima facie case constitutes yet another barrier of proof in the case of small samples, as in the present case, or in the case of large samples when there is “some” minority representation. See, e. g., Mayor v. Educational Equality League, 415 U.S. 605, 619-21, 94 S.Ct. 1323, 1332-1334, 39 L.Ed.2d 630 (1974) (sample of thirteen); Adams v. Reed, 567 F.2d 1283, 1287 (5th Cir. 1978) (sample of five); Turner v. Texas Instruments, Inc., 555 F.2d 1251, 1257 (5th Cir. 1977) (sample of eight); Harper v. Trans World Airlines, Inc., 525 F.2d 409, 412 (8th Cir. 1975) (sample of five). But cf. Bilingual Bicultural Coalition v. FCC, 595 F.2d 621, 648 n.84 (D.C.Cir.1978) (en banc) (Robinson, J., dissenting in part) (“As a general matter, the value of a statistical showing depends not merely upon the size of the sample but also upon the extent of the expected observation and the degree of disparity — element unknowable until the commission looks at the particular group’s statistics. In short, as the application of the logic of statistics that it purports to be, the court’s conclusion that statistics for small groups are always unreliable is simply wrong.”). See also note 2 supra.
      
      I read neither Castaneda nor Hazelwood to mandate a statistical significance/standard deviation type analysis in all cases regardless of the appropriateness of their use. In fact, the last sentence in footnote 17 in Hazelwood states: “These [statistical significance] observations are not intended to suggest that precise calculations of statistical significance are necessary in employing statistical proof, but merely to highlight the importance of the choice of the relevant labor market area.” 433 U.S. at 311-12 n.17, 97 S.Ct. at 2743-2744 n.17 (definition of relevant labor market at issue).
     
      
      . Although this case seems to me to involve disparate impact of the employer’s selection procedures on black employees, the evidence may also be susceptible to analysis under the McDonnell Douglas test for race-based disparate treatment. In 1973 and 1974, black employees were in the pool of qualified employees and were in effect among those treated as job applicants, in each case the applicant was not chosen and instead a white applicant was chosen. Although it is true that the employer considered at the same time the entire pool of employees qualified for promotion and the position did not remain open after rejection of the black applicants, the fact is that the black applicants were passed over and similarly situated white employees were chosen for a job that had been held by whites in overproportion to their presence among qualified persons. See Furnco Constr. Co. v. Waters, 438 U.S. 567, 579, 98 S.Ct. 2943, 2945, 57 L.Ed.2d 957 (1978) (relevance of statistics in McDonnell Douglas disparate treatment case). The selection of a black through the same procedure in 1975 would, of course, be evidence that discriminatory intent was not the sole determinant for the selection of foremen. But the question in a disparate treatment case is whether discriminatory intent was a factor in the selection process, not whether race was'the only factor. McDonald v. Santa Fe Trail Transp. Co., 427 U.S. 273, 282 n.10, 96 S.Ct. 2574, 2580 n.10, 49 L.Ed.2d 493 (1976).
     
      
      . Given the evidently nondiscriminatory posture of the employer with respect to hiring (51% of the cement finishers on the First National Bank project were black; 37% of the journeymen finishers/plasterers in the Little Rock SMSA were black), I would have, however, favored a less intrusive remedy for the promotion-to-foreman discrimination than the hiring order devised by the district court.
     