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 *1352the 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 *1353understand 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 1
*1354Year 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 *1355that no more than one black would be chosen among the seven choices during the whole period is about 6 in 100.2
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.3
Here the hypothesis is that the employer did (or did not) discriminate in promotions. *1356Assuming 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).4 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.5
. 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 obser*1356vation 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.