851 F.2d 1447

#### Lawton FRAZIER, et al., Appellants, v. CONSOLIDATED RAIL CORPORATION, et al.

No. 87-7198.

United States Court of Appeals, District of Columbia Circuit.

Argued April 28, 1988.

Decided July 15, 1988.

Squire Padgett, with whom Grover Han-kins, General Counsel for the National *221Ass'n for the Advancement of Colored People, Washington, D.C., and Edward Hailes, Jr., Asst. General Counsel, Philadelphia, Pa., were on the briefs, for appellants.

Michael J. Ossip, with whom Mark S. Dichter, Kathryn Wikman, Philadelphia, Pa., and Thomas E. Reinert, Jr., Washington, D.C., were on the brief, for appellee Consolidated Rail Corp.

Pamela D. Walker, Little Rock, Ark., for appellee United Transp. Union.

Before MIKVA and SENTELLE, Circuit Judges, and EDMUND L. PALMIERI,* Senior District Judge for the Southern District of New York.

*

 Sitting by designation pursuant to 28 U.S.C. § 294(d).

Opinion for the court filed by Senior District Judge PALMIERI.

PALMIERI, Senior District Judge:

Lawton Frazier, Odyssey E. Gray, William Harling, Beachey S. Thompson, Denise McClone, Samuel Cox, and three branch offices of the National Association for the Advancement of Colored People (“NAACP”) appeal from a final judgment entered by the United States District Court for the District of Columbia (Flannery, J.) in favor of the Consolidated Rail Corporation (“Conrail”) and the United Transportation Union (“UTU”). The appellants, plaintiffs below, attack two orders of the district court. In the first, entered July 31, 1986 and adhered to upon reconsideration in an order entered December 23, 1986, the district court denied class certification, pursuant to Rule 23 of the Federal Rules of Civil Procedure. In the other, entered August 28, 1987, following a five week bench trial, the district court entered judgment— accompanied by a 113 page opinion — in favor of the defendants on all counts.

Because we find that the district court’s decision was supported by the record and is in no way clearly erroneous, we affirm. The district court’s findings of fact and conclusions of law are, for the most part, self explanatory and need not be set out in detail. Several issues, however, require elaboration.

I. Background

Conrail, a railroad company, formerly employed the individual appellants. UTU, the labor union of some of Conrail’s employees, formerly represented the six individual appellants, under the Railway Labor Act, as amended, 45 U.S.C. §§ 151-188 (1980 & Supp. I 1981).

Conrail employs both firemen and engineers to operate its locomotives. Conrail’s engineers are promoted exclusively from the ranks of firemen who have successfully completed Conrail’s Engineer Training Program. During the relevant time period, the Training Program involved three phases. Phase I consisted of both classroom and on-equipment instruction. Phase II consisted of on-the-job training. And in Phase III, the trainee learned the physical characteristics of a particular territory. Each phase culminated in an examination, the successful completion of which was a prerequisite to advancement to the next. Absent exceptional circumstances, each trainee had only two chances to pass each test. Failure resulted in termination of the employee’s participation in the training program.

All of the individual plaintiffs/appellants failed to complete either Phase I or Phase II of the Training Program. After unsuccessful attempts to achieve reinstatement to the Training Program, they filed suit alleging race and sex based discrimination in violation of, *inter alia,*Title VII of the Civil Rights Act of 1964, as amended, 42 U.S.C. §§ 2000e to 2000e-17 (1982),1 and 42 *222U.S.C. § 1981 (1982).2 Claims alleging other violations of law were dismissed with prejudice at the outset of the trial, pursuant to a stipulation among the parties. The plaintiffs attempted to prove their case under theories of purposeful discrimination and adverse impact.

The shifting burdens of proof employed in the trial of discrimination claims are quite familiar. In order to prove a case of race discrimination under Title VII, and under Section 1981,3 a plaintiff has the initial burden of establishing a prima facie case of discrimination, after which the burden shifts to the defendant to demonstrate legitimate reasons for the discriminatory treatment. Finally, the plaintiff then has a chance to rebut that showing with proof that the legitimate reasons invoked by the discriminator were merely pretextual. *See, e.g., Connecticut v. Teal,*457 U.S. 440, 446-47, 102 S.Ct. 2525, 2530, 73 L.Ed.2d 130 (1982) (adverse impact); *Texas Department of Community Affairs v. Burdine,*450 U.S. 248, 253-56, 101 S.Ct. 1089, 1093-95, 67 L.Ed.2d 207 (1981) (disparate treatment); *McDonnell Douglas Corp. v. Green,*411 U.S. 792, 802-05, 93 S.Ct. 1817, 1824-26, 36 L.Ed.2d 668 (1973).

After a five week trial on the issue of liability, the district court made extensive findings of fact and conclusions of law. Fed.R.Civ.P. 52. It held that the plaintiffs’ Title VII claims against UTU were barred because they had not met the requirement of 42 U.S.C. § 2000e-5(f)(1) that certain administrative remedies be exhausted before suit can be brought in the district court. In addition, it held that their Section 1981 claims, as well as their Title VII claims, against UTU failed on the merits because UTU “achieved substantial success in getting changes in the program” and because where UTU did not pursue racial grievance claims, “that assessment [was] within the realm of judgment that the union has a right to make.” Although UTU is named in the notice of appeal, the appellants do not assert the district court committed any errors with regard to dismissal of the claims against UTU. Indeed the argument made in UTU’s brief to this Court were not even answered by the appellants.

With regard to the claims against Conrail, the district court held that none of the plaintiffs had made out a prima facie case of discrimination on the basis of race or, in the case of the one female plaintiff, sex. It further held that, assuming they had made out prima facie cases, Conrail had proved the business necessity for its Training Program. And it held that the plaintiffs had failed to prove by a preponderance of the evidence that the legitimate, nondiscriminatory reasons asserted by Conrail for the terminations were pretext for intentional discrimination. The district court therefore entered judgment in favor of Conrail and UTU with respect to all of the plaintiffs’ claims.

The appellants raise three issues that require elaboration. First, they argue that the district court erred in rejecting the statistics they proffered to prove their prima facie case on the theory of disparate treatment as well as those proffered to *223prove their prima facie case on the theory of adverse impact. A number of corollary assertions flow from this proposition, all of which are designed to lead to the conclusion that “clear error” has been made by the district court in its conclusion that the plaintiffs failed to make out a prima facie case under either theory.

Second, the appellants point out that in response to their prima facie case, the defendants sought only to show that the examinations in Phases I and II were legitimately job related, and thus valid. They assert that the district court should have required Conrail to make that showing with a “job validation study” encompassing the entire Training Program. The absence of a validation study of the Training Program as a whole is asserted as a ground for reversal.

Finally, the appellants assert that the district court abused its discretion by refusing to certify the individuals as class representatives pursuant to Rule 23 of the Federal Rules of Civil Procedure.

II. The Plaintiffs’ Statistics

The use of statistical analysis has become routine in discrimination cases. *See generally*D. Baldus & J. Cole, *Statistical Proof of Discrimination*(1980 & Supp. 1987); *see also*Shoben, *Differential Pass Fail Rates in Employment Testing: Statistical Proof Under Title VII,*91 Harv.L. Rev. 793 (1978). But as well accepted as statistical analysis has become in these cases, it is not without important limitations. A statistical calculation relies upon a number of underlying assumptions, the validity of which can often be assessed only by those with experience in the field.

The plaintiffs in this case attempted to make out their prima facie cases with statistics, buttressed by other kinds of evidence. They chose well in advance of trial not to rely upon the testimony of an expert witness to do so, but to submit the statistics with the expectation that they would have independent persuasive value. In a pre-trial scheduling order, the district court set a cutoff date of September 16, 1986 for the designation of plaintiffs’ experts. Fed. R.Civ.P. 26(b)(4). In an undated statement which was docketed in the district court on September 17,1986, the plaintiffs explained that it was their position that in their “employment discrimination action brought pursuant to Title VII of the Civil Rights Act of 1964 ... and based upon an adverse impact theory of liability, the plaintiffs in order to establish a prima facie case of discrimination need not utilize expert testimony____” The plaintiffs also chose not to rely upon expert testimony to make out a prima facie case on their disparate treatment theory, in as much as they did not designate any expert before the September 16, 1986 deadline set by the district court. In its order of November 24, 1986, the district court for that reason precluded the plaintiffs from using expert testimony to prove any part of their prima facie case.

Having chosen to forgo the use of an expert to explain the statistical calculations to the district court, the plaintiffs submitted, in addition to the raw data, several results of numerical calculation performed by counsel. These calculations were all done according to widely known methods previously reported in other discrimination cases, regulations and commentaries. They included a simple comparison of the pass rate for blacks with that for whites (and a similar comparison of pass rates for men and women)4 and a “Z statistic.”5

A. *Adverse Impact.*

To make out their prima facie case showing the adverse impact of the Engineer Training Program, the plaintiffs set out a raw statistical comparison of the per*224centage of blacks who failed the tests in the Training Program with the percentage of whites who similarly failed.6 According to the district court’s assessment of the evidence, “of the 497 white students ... 432 or 87% passed. Of the 70 black students ... 42 or 60% passed. Thus the black pass rate was 69% of the white pass rate.” *Cox v. Conrail,*Civil Action No. 85-0514, slip op. at 79 (D.D.C. August 26, 1987) (¶ 208).7

In addition to the raw numbers, which by themselves are not very instructive, counsel performed several calculations for the district court, only one of which it accepted and used.

As is clear from the district court’s opinion, the district court took note of the plaintiffs’ comparisons of the pass rate for blacks with the rate for whites. The district court also took note of the EEOC’s *Uniform Guidelines on Employee Selection Procedures,*29 C.F.R. § 1607.4(D), which sets out the “four fifths rule” that a minority pass rate which is less than four fifths (or 80%) of the majority pass rate “will generally be regarded by the Federal enforcement agencies as evidence of adverse impact----” The four fifths rule has also been used with caution by some courts, *e.g., Black, supra,*831 F.2d at 133-35 (6th Cir.1987); *Fudge, supra,*766 F.2d at 657-59 (1st Cir.1985); *Guardians Association of the New York City Police Department, Inc. v. Civil Service Commission of the City of New York,*630 F.2d 79, 87 (2d Cir.1980), *cert. denied,*452 U.S. 940, 101 S.Ct. 3083, 69 L.Ed.2d 954 (1981), but has been criticized as unreliable and overly simplistic. Shoben, *supra,*91 Harv.L.Rev. at 805-06. In the final analysis, the district court rejected application of the four fifths rule because it found the samples of women (11) and blacks (70) too small to support a finding that the figures are “statistically significant.” *Cox v. Conrail, supra,*slip op. at 81 (¶ 214). That conclusion is consistent with the Uniform Guideline statement that “differences [of more than 80%] in selection rate may not constitute adverse impact where the differences are based on small numbers and are not statistically significant____” 29 C.F.R. § 1607.4(D). The Uniform Guidelines do not establish a hard and fast rule for federal agencies, and the courts have followed suit, especially where small numbers are involved. *Black, supra,*831 F.2d at 133-35 (6th Cir.1987); *Fudge, supra,*766 F.2d at 657-59 (1st Cir.1985).

The district court, in rejecting the comparisons, had very little choice. The plaintiffs as a matter of trial strategy decided to avoid an extensive exploration of the use and meaning of statistical analysis by way of expert testimony. The district court was well within the bounds of reason in making an intuitive decision, rather than attempting independently to educate itself in the field of statistical analysis. *See Fudge, supra,*766 F.2d at 657 (1st Cir.1985).

Second, the plaintiffs performed and proffered a “Z statistic” calculation. The Z statistic, or test for differences between independent proportions, is one of several used in proving discrimination cases as a way of assessing the probability that a discrepancy in statistical sample — varying pass rates, for example — is due to chance, rather than discrimination or some other explanation. Baldus & Cole, *supra,*§ 9A.1. As the size of a statistical sample decreases, the probability that a discrepancy is due to chance is heightened. *See Segar v. Smith,*738 F.2d 1249, 1282-83 (D.C.Cir.1984), *cert. denied,*471 U.S. 1115, 105 S.Ct. 2357, 86 L.Ed.2d 258 (1985). There is always a possibility that a discrepancy is due entirely to chance, but with enough repetition of a test, the likelihood of that possibility diminishes. *Ibid.*The Z *225statistic is a calculation which measures in a uniform way the degree of the discrepancy in relation to the size of the sample. It thus allows comparison, from study to study, of the probability that a numerical discrepancy is due entirely to chance. *See*Shoben, *supra,*91 Harv.L.Rev. at 799-800.

This information was not communicated to the district court by an expert, because the plaintiffs believed that judicial notice could be taken of it. While that may be true, the district court was not required to take notice of it. Fed.R.Evid. 201(d) (“a court shall take judicial notice if ... supplied with the necessary information”). The issue apparently was not articulated to the district court in a manner that could permit it to make the legal determination as to whether the level of chance implicated by the plaintiffs’ statistics was significant pursuant to the teachings of *Palmer v. Shultz, supra,*815 F.2d at 90-97. There is always some possibility that a statistical discrepancy is due to chance. But that does not keep the courts from relying on statistics, when shown to be significant, for the establishment of a *prima facie*case. The question — the legal question — is what degree of certainty the courts require for a *prima facie*case to be established. The 5% level — that is to say in 5 (or fewer) out of 100 samples a given numerical discrepancy is, on average, due to chance — is commonly accepted among statisticians as an acceptable degree of uncertainty, and was adopted in *Palmer v. Shultz, supra,*815 F.2d at 96. This “5% or less” figure corresponds to a Z statistic figure of 1.96 or greater. Shoben, *supra,*91 Harv.L.Rev. at 803 & n. 41.

In apparent response to the Z statistic proffer, the district court- wrote,

> There is no indication whether, or to what degree, the pass rate percentages are meaningful. Plaintiffs chose not to use a statistical expert and were thus barred at trial from introducing complicated statistical evidence. Such evidence is meaningless to the lay person____ The court is not in a position to evaluate the appropriateness of the formulas selected, whether they are complete and accurate, how they should be applied to the evidence in the instant case, or what the significance of their results are. Without such information, plaintiffs have failed to prove that the pass rate ratios are meaningful.

*Cox v. Conrail, supra,*slip op. at 81-82 (¶ 216). The appellants press the argument that the district court’s refusal to accept the proffered Z statistic was reversible error. We disagree. We do not believe that the adoption of the “5% or less” legal standard relieves plaintiffs of the burden of making their statistical analyses intelligible to the finder of fact. Statistical calculations performed on data in discrimination cases are not probative of anything without support from an underlying statistical theory. *See*Baldus & Cole, *supra,*§ 10. Generally, the plaintiffs present statistics about themselves — for instance, their own performance in a selection device alleged to have an adverse impact on members of a protected minority group. Shoben, *supra,*91 Harv.L.Rev. at 797-98. For purposes of analysis, those who actually claim discrimination serve merely as a statistical sample of the relevant population: if the disparity between majority and minority cannot be explained by legitimate reasons, discrimination against the entire minority population at large may exist. *Ibid.*But the numerical disparity can sometimes be explained by reasons other than discrimination. For instance, the sample itself may not be representative of the population at large, or sufficiently random, or, as the district court in this case observed, the sample may be so small that a slight change in the data would present an entirely different result. *Id.*at 801. In addition, disparities may be explainable by other factors, such as uncomprehensive 'statistical treatment, varying levels of qualifications among applicants, errors in definition of groups, inappropriate sampling methods, errors in measurement, or even clerical and computational errors. Baldus and Cole, *supra,*§ 10.112. Perhaps the leading commentators on the use of statistical analysis in discrimination litigation have warned that the statistical tests they recommend

> *226provide a good point of departure, but the accuracy of the inferences they suggest (even when assumptions of random sampling are perfectly satisfied) can be no greater than the validity of the basic research design and methodology that produced the measure of disproportionate impact____ Regardless of the legal theory of the case, the issues of measurement and research design that we have discussed throughout this book should be considered in assessing the “pedigree” and weight of the critical numbers. *Id.*at 337.

While plaintiffs in a discrimination case need not set up and demolish every conceivable straw man in making out a prima facie case, the statistics must be made meaningful to the finder of fact in order to permit the plaintiffs to carry their burden of showing that their statistics are significant.

We are not prepared to say that the Z statistic calculation is so simple and straight forward that an expert is never required to explain it to a finder of fact. Nor do we wish to be understood as holding that an expert is always required. We leave both possibilities open because it would be impossible to anticipate the impact of this theory upon every conceivable factual situation. We believe that in the factual context of this case, the district court made a valid finding that the plaintiffs’ proffered statistics were not sufficiently presented to make out a prima facie case of adverse impact.

B. *Disparate Treatment.*

In disparate treatment cases, where the plaintiff must prove purposeful discrimination, statistics may be useful in making a prima facie case. As overt racial discrimination has waned, the courts have made it clear that individuals must still be protected from more subtle forms of discrimination. *See Segar v. Smith, supra,*738 F.2d at 1278-79. “In many cases the only available avenue of proof is the use of racial statistics to uncover clandestine and covert discrimination....” *Ibid, (quoting International Brotherhood of Teamsters v. United States,*431 U.S. 324, 340 n. 20, 97 S.Ct. 1843, 1856-57 n. 20, 52 L.Ed.2d 396 (1977)). In particular, the courts have looked to disparities between the composition of a workforce and that of the population from which it is drawn. *E.g. Hazelwood School District v. United States,*433 U.S. 299, 310, 97 S.Ct. 2736, 2743, 53 L.Ed. 2d 768 (1977).

In *Segar v. Smith, supra,*738 F.2d at 1277-79, we rejected the notion that a “gross disparity” between the racial composition of a workforce and the population from which it is drawn is required. We recognized that statistical evidence may be more or less finely tuned to the relevant labor pool, and that the requisite level of disparity and the amount of supporting anecdotal evidence would vary accordingly. But the statistical evidence must, in some way, accurately reflect the gravamen of the plaintiffs’ case.

The appellants introduced statistical evidence regarding the total number of black engineers and firemen employed by Conrail in its Eastern Region during the years 1977 to 1984.8 The following table summarizes their evidence (Joint 289-91). Appendix 278-80;

Year Black Total *%*

1977

Engineers 12 1290 0.9%

Firemen 25 357 7.0%

1978

Engineers 17 1319 1.2%

Firemen 50 366 13 %

1979

Engineers 16 1270 1.2%

Firemen 63 334 18 %

1980

Engineers 20 1139 1.7%

Firemen 51 295 17 %

1981

Engineers 18 1080 1.6%

*227Year Black Total *%*

Firemen 26 441 5.8%

1982

Engineers 14 951 1.4%

Firemen 22 335 6.5%

1983

Engineers 19 737 2.5%

Firemen 7 65 10 %

1984

Engineers 15 698 2.1%

Firemen 9 110 8.1%

These raw numbers, the appellants contend, reflect “underutilization of blacks as engineers.” The district court disagreed, holding that the “statistics are meaningless without evidence regarding the makeup of the available work-force or applicants.” *Cox v. Conrail, supra,*slip op. at 47 (¶ 121).

The district court found that firemen entered the Engineer Training Program in order of seniority in their respective districts as the need for engineers in a particular district arose. *Cox v. Conrail, supra,*slip op. at 28 (H 76). Promotion therefore depended upon the particular circumstances of each set of firemen in each district. The statistical evidence did not reflect this situation. It made no distinction between firemen and firemen who were eligible for promotion. Moreover, it did not reflect the number of firemen actually promoted, and therefore did not reveal anything about the number of blacks who were promoted from year to year. Instead, the statistics were most likely affected by other factors. For instance, the statistics above reflect the fact that there was substantial attrition among Conrail’s firemen and engineers. There is, therefore, no basis in the record for comparison between the number of blacks who were eligible for promotion at any given time with the number of blacks actually promoted at that time. Indeed, the numbers of black and white firemen and engineers reveal nothing about either of those two figures which could have permitted a meaningful comparison in support of the plaintiffs’ prima facie case. These statistics simply did not tell a sufficiently complete story for the district court to give them any weight in its consideration of the plaintiffs’ prima facie case.

In sum, the district court’s assessment of the statistical evidence presented by the plaintiffs below was not clear error. Additionally, the court sufficiently addressed their historical and anecdotal evidence. Its conclusion that the plaintiffs did not make out a prima facie case is affirmed.

III. Validation

Upon demonstration by a plaintiff of a prima facie case of employment discrimination in violation of Title VIPs prohibition of selection devices which adversely impact a protected minority, the defendant has an opportunity to demonstrate that the selection device has “a manifest relationship to the employment in question.” *Connecticut v. Teal, supra,*457 U.S. at 446-47, 102 S.Ct. at 2530 (1982) *(quoting Griggs v. Duke Power Co.,*401 U.S. 424, 432, 91 S.Ct. 849, 854, 28 L.Ed.2d 158 (1971)). Put another way, the employer may demonstrate that the selection device is “valid.” Despite its conclusion that no prima facie case had been made out, the district court proceeded to make an exhaustive investigation of the Engineer Training Program, and concluded that “Conrail has proved the business necessity of its program.” *Cox v. Conrail, supra,*slip op. at 104 (¶ 40). There was no dispute in this case that the instruction and examinations which constituted Phases I and II, the only phases which parties to this litigation failed, were wholly job related. Instead of attacking that finding, the appellants attack the fact that no validation study was performed for the Training Program as a whole. Their position in this Court is that “the entire program as opposed to its component parts (individual tests), must be shown to be job-related and required by business necessity.”

In essence, this case presents the inverse of the problem addressed by the Supreme Court in *Connecticut v. Teal,*457 U.S. 440, 102 S.Ct. 2525, 73 L.Ed.2d 130 (1982). In that case, the Court held that where a particular portion of a selection device has an adverse impact on a minority group, the fact that the entire device has no *overall*adverse impact does not prevent the em*228ployees from establishing a prima facie case, nor does it provide the employer with a defense to such a case. *Id.*at 442, 102 S.Ct. at 2528. “Title VII does not permit the victim of a facially discriminatory policy to be told that he has not been wronged because other persons of his or her race or sex were hired,” the Court explained. *Id.*at 455, 102 S.Ct. at 2535. The Court therefore rejected the notion that a neutral “bottom line” is a defense to a discriminatory component of a selection device. Here, in contrast, the appellants do not and cannot attack the district court’s amply supported findings with respect to any given component of the selection device; instead, they assert that the district court should have required Conrail to validate the Engineer Training Program as a whole.

The analysis in *Teal,*rejecting the argument that a separable component that has an adverse impact may be insulated from scrutiny where the selection device as a whole is numerically non-discriminatory, does not speak to the issue of whether assessment of the validity of the selection device as a whole is required. The argument presented and rejected in *Teal*was only that a “good bottom line” could protect an admittedly discriminatory component. *Teal*did not speak to the method of assessing a discriminatory selection device.

Like the Fifth Circuit in *Rivera v. City of Wichita Falls,*665 F.2d 531, 539 (5th Cir.1982), we decline the invitation to require validation of an entire selection device in every litigation in which a prima facie case has been made out as to a single, separable component. Such a requirement would place an unreasonable burden upon district courts deciding adverse impact cases without adding any real substantive protection for the allegedly aggrieved parties in those cases. The purpose of Title VII is to ensure that individuals are not denied employment opportunities on the basis of a prohibited discrimination. *See Connecticut v. Teal, supra,*457 U.S. at 453, 102 S.Ct. at 2533. If a separable component of a selection device cannot be shown to have an adverse impact upon members of a protected minority, the degree of job relatedness of that component will generally be irrelevant.

In this case, the district court addressed the job relatedness of the Engineer Training Program in spite of its finding that no prima facie case had been made out. Where possible, the district courts should retain the flexibility to address only those components of selection devices that are implicated by a prima facie case on the part of the plaintiffs.

There are, of course, circumstances which will prevent district courts from neatly, sorting out and isolating each of a selection device’s component elements in order to assess its validity. For example, where two or more components interact to cause an adverse impact, validation of the separate components may not suffice to validate the selection device. *See Griffin v. Carlin,*755 F.2d 1516, 1524-25 ((11th Cir.1985) *(citing Gilbert v. City of Little Rock, Ark.,*722 F.2d 1390, 1396-98 (8th Cir.1983), *cert. denied,*466 U.S. 972, 104 S.Ct. 2347, 80 L.Ed.2d 820 (1984)). Similarly, some components of a selection device may be so subjective that seriatim validation will be impossible. *Ibid.*In this case, the component phases of the Engineer Training Program were all easily separable and no allegation that they interact in a discriminatory way was made or supported. The district court also found that the risk of promotions being affected by subjective criteria in this case was minimal. In addition, the district court did in fact review the Training Program as a whole, without the benefit of a formal validity analysis of the entire program (only Phases I and II were assessed), and its finding that the entire Program was intimately related to the necessities of the very exacting and dangerous work task performed by Conrail’s engineers was supported in the record despite the lack of such a study. That aspect of the district court’s decision holding the Engineer Training Program valid is therefore affirmed.

IV. Class Certification

The appellants attack the district court’s refusal to certify them as repre*229sentatives of a class pursuant to Rule 23 of the Federal Rules of Civil Procedure. Such a refusal may be reversed only if it resulted from the application of incorrect legal criteria or if it constituted an abuse of discretion. *Wagner v. Taylor,*836 F.2d 578, 586 (D.C.Cir.1987).9 The district court based its decision on the putative representatives’ failure to satisfy Rule 23(a)’s numerosity and adequacy of representation criteria. Fed.R.Civ.P. 23(a)(1), 23(a)(4).

Rule 23(a)’s requirements necessitate an examination by the district court of the facts of each case. Here the district court did examine the facts and reasonably concluded that the prospective class members were not sufficiently numerous to warrant certification. It properly based its decision not only on the small number of prospective class and sub-class members,10 but also on the more important consideration that the plaintiffs had failed to carry their burden of demonstrating that joinder was impracticable. It properly held that there was no difficulty in identifying the prospective members and that there was no “color-able evidence supporting plaintiffs’ claim that the other class members may be reluctant to personally appear____” In addition, it properly considered the fact that all the prospective class members resided within a reasonable distance of the district court. Finally, the district court correctly refused to consider hypothetical future employees for purposes of numerosity analysis. The plaintiffs attacked only the Training Program as it stood before being reformed, and not as it stands today. *Cf. Mathews v. Diaz,*426 U.S. 67, 71 n. 3, 96 S.Ct. 1883, 1887 n. 3, 48 L.Ed.2d 478 (1976).

The district court’s finding that the NAACP had associational standing in this case was irrelevant to its analysis of the propriety of class certification pursuant to Rule 23. The two kinds of joint action implicate different advantages and disadvantages. *See International Union, United Automobile, Aerospace, and Agricultural Implement Workers of America v. Brock,*477 U.S. 274, 288-90, 106 S.Ct. 2523, 2532-33, 91 L.Ed.2d 228 (1986). The doctrine of associational standing does not subsume Rule 23. Indeed, the Supreme Court has rejected the argument that “associational standing and Rule 23 are ‘designed to serve precisely the same purpose.’ ” *Id.*at 288, 106 S.Ct. at 2532. We therefore affirm the district court’s denial of class certification pursuant to Rule 23(a)(1) of the Federal Rules of Civil Procedure, and need not reach the district court’s holding on the issue of adequacy of representation.

V. Conclusions

The appellants’ other contentions require little discussion. They rely on the requirements set out in *Albemarle Paper Co. v. Moody,*422 U.S. 405, 425, 95 S.Ct. 2362, 2375, 45 L.Ed.2d 280 (1975), and elaborated upon in *Segar v. Smith, supra,*738 F.2d at 1278-79, that the employer tailor its response to the plaintiffs’ proof and that the court consider whether the selection device is a mere pretext for discrimination. Those requirements are not implicated here, where the district court correctly held that no prima facie case had been made out. In addition, the district court properly held that the plaintiffs did not demonstrate that Conrail could have administered another training program with a less disparate impact while maintaining its legitimate objective of ensuring the safe and efficient oper*230ation of the railroad. *Cox v. Conrail, supra,*slip op. at 112 (11 52). That holding was supported by the record.

Similarly, because the plaintiffs failed to prove a pattern or practice of discrimination, the Supreme Court’s discussion in *International Brotherhood of Teamsters v. United States, supra,*431 U.S. at 361-62, 97 S.Ct. at 1867-68, is not relevant to the plaintiffs’ individual claims.

In sum, the district court’s findings of fact were supported by the record, and its conclusions of law were correct.

*Affirmed.*

1

. Section 703(a) of that Act, as amended, provides:

§ 2000e-2. Unlawful employment practices (a) Employer practices

It shall be an unlawful employment practice for an employer—

(1) to fail or refuse to hire or to discharge any individual, or otherwise to discriminate against any individual with respect to his compensation, terms, conditions, or privileges of employment, because of such individual's race, color, religion, sex, or national origin; or

(2) to limit, segregate, or classify his employees or applicants for employment in any way which would deprive or tend to deprive any *222individual of employment opportunities or otherwise adversely affect his status as an employee, because of such individual's race, color, religion, sex, or national origin.

2

. That provision provides:

§ 1981. Equal rights under the law

All persons within the jurisdiction of the United States shall have the same right in every State and Territory to make and enforce contracts, to sue, be parties, give evidence, and to the full and equal benefit of all laws and proceedings for the security of persons and property as is enjoyed by white citizens, and shall be subject to like punishment, pains, penalties, taxes, licenses, and exactions of every kind, and to no other.

3

. Section 1981 allows recovery only for purposeful discriminatory treatment. *General Building Contractors Association, Inc. v. Pennsylvania,*458 U.S. 375, 391, 102 S.Ct. 3141, 3150, 73 L.Ed.2d 835 (1982). Otherwise, the showing required to recover for employment discrimination in violation of Section 1981 is coextensive with that required to show a violation of Title VII once a Title VII claim has properly reached the district court. *Carter v. Duncan-Huggins, Ltd.,*727 F.2d 1225, 1232 (D.C.Cir.1984); *Lewis v. University of Pittsburgh,*725 F.2d 910, 195 n. 5 (3d Cir.1983) (citing cases), *cert. denied,*469 U.S. 892, 105 S.Ct. 266, 83 L.Ed.2d 202 (1984); *Cf. Oates v. District of Columbia,*824 F.2d 87, 90 (D.C.Cir.1987) (section 1983 claim).

4

. *E.g. Black v. City of Akron, Ohio,*831 F.2d 131, 133-35 (6th Cir.1987); *Fudge v. City of Providence Fire Department,*766 F.2d 650, 657-59 (1st Cir.1985); EEOC’s *Uniform Guidelines on Employee Selection Procedures,*29 C.F.R. § 1607.4(D) (1987); Shoben, *supra,*91 Harv.L. Rev. at 805-06.

5

. The "Z statistic," known among statisticians as the test for differences between independent proportions, is discussed in Shoben, *supra,*91 Harv. at 799-803, and in Baldus *Si*Cole, *supra,*at §§ 9A.11-9A.12. Its use was noted in *Palmer v. Shultz,*815 F.2d 84, 93 n. 8 (D.C.Cir.1987).

6

. They made the same comparison for females and males but do not attack the district courts’ dismissal of their case on these grounds.

7

. The appellants presented slightly different numbers to the district court, which it rejected. The appellants' position appears to be that — at least for black trainees — a trainee who dropped out of the program without failing the examinations should be treated as halving failed. The differences appear to be minor (78 black students instead of 70) and would not change our analysis.

8

. Conrail's Eastern Region was the only one served by the engineer training program operated at Wilmington, Delaware, which is the training center all the individual appellants attended.

In their brief, appellants presented statistics purporting to reflect employment figures for all of Conrail's firemen and engineers, and not just for those employed in the Eastern Region. Conrail asserted, and the appellants conceded in their reply brief, that those statistics were not in the record. Only the figures for the Eastern Region may be considered on appeal.

9

. *Accord McCarthy v. Kleindienst,*741 F.2d 1406, 1410 (D.C.Cir.1984); *National Association for Mental Health, Inc.*v. *Califano, 717*F.2d 1451, 1459 (D.C.Cir.1983), *cert. denied sub nom. Wagshal v. Crozer-Chester Medical Center,*469 U.S. 817, 105 S.Ct. 85, 83 L.Ed.2d 32 (1984); *Bermudez v. United States Dep't of Agriculture,*490 F.2d 718, 725 (D.C.Cir.), *cert. denied,*414 U.S. 1104, 94 S.Ct. 737, 38 L.Ed.2d 559 (1973).

10

. The appellants assert that there were 39 prospective class members. This district court found that there were 28. In either case, certification of such a small class is best left to the sound discretion of the district court. *See General Telephone Co. of the Northwest v. Equal Employment Opportunity Commission,*446 U.S. 318, 330 n. 18, 100 S.Ct. 1698, 1706 n. 18, 64 L.Ed.2d 319 (1980) (approving of denial of certification of classes with similarly sized classes). We are aware of no appellate decision reversing a district court's denial of class certification on grounds of inadequate numerosity where fewer than 40 prospective members exist.