Full opinion text
Opinion for the Court filed by EDWARDS, STARR and D.H. GINSBURG, Circuit Judges. PER CURIAM: This case involves a class action brought by eight black construction workers alleging racial discrimination under 42 U.S.C. § 1981 (1982) and Title VII of the Civil Rights Act of 1964, 42 U.S.C. § 2000e et seq. (1982). The plaintiffs charged that various requirements for admission to Local 201 of the Iron Workers Reinforced Rodmen (the “Union”) and to its international union, the International Association of Bridge, Structural and Ornamental Iron Workers (the “International”), discrimina-torily denied black rodmen the benefits of union membership. The plaintiffs sued both the Union and the International, along with the Union’s Apprenticeship Committee, the National Iron Workers and Employers Training Program, and the Construction Contractors Council/AGC Labor Division, Inc. (“CCC”), an organization that negotiated collective bargaining agreements with the Union on behalf of most employers of rodmen in and around Washington, D.C. Suit was filed in 1975, the case went to trial in 1981, and the District Court rendered its decision in 1985. The trial court’s decision raises very difficult problems for this court. Having received over eighty-five pages of proposed findings of fact and conclusions of law from the plaintiffs in December 1981, the District Court adopted them almost verbatim after a lapse of three and a half years, retaining most of the plaintiffs’ typographical errors but deleting their references to the record, thus hampering review. Not surprisingly, the District Court found all of the defendants liable for all of the alleged violations of Title VII and section 1981. The court’s subsequent remedial order again copied the plaintiffs’ proposal; an amended order was issued in April 1986. The defendants then appealed from the trial court’s decision and order. In considering this appeal, we embrace the view expressed by a number of circuits in strongly disapproving the procedure followed by the trial court in reaching judgment in this case. See, e.g., EEOC v. Federal Reserve Bank, 698 F.2d 633, 639-42 (4th Cir.1983), rev’d on other grounds sub nom. Cooper v. Federal Reserve Bank, 467 U.S. 867, 104 S.Ct. 2794, 81 L.Ed.2d 718 (1984), and cases cited therein. While “the fact that the trial judge has adopted proposed findings does not, by itself, warrant reversal,” “it does raise the possibility that there was insufficient independent evaluation of the evidence and may cause the losing party to believe that his position has not been given the consideration it deserves.” Photo Elecs. Corp. v. England, 581 F.2d 772, 777 (9th Cir.1978). Indeed, this is surely the case here, where the appellants assert that “the District Court’s uncritical acceptance of plaintiffs’ proposed findings ... did not provide the even-handed consideration of the evidence necessary to a fair adjudication of the issues.” Brief of Appellant CCC at 28. After ten years of litigation in District Court, the appellants had a right to expect something more. Because of the state of the record on appeal, this case was needlessly given “complex” treatment under this circuit’s Case Management Plan, and then it required more exacting scrutiny than this court ordinarily accords the District Court’s findings. A meticulous review of the District Court’s conclusions uncovered a number of clearly erroneous factual findings and several errors of law. We therefore reverse in part and affirm in part. I. Background Rodmen are construction workers who handle and position steel rods for reinforcing concrete and other building materials. Most of the rodmen who worked in the Washington, D.C., area in the early 1970s were employed through referrals by the Union, pursuant to a series of collective bargaining agreements between the Union and CCC. The Union’s referral system ensured that Union members were offered employment opportunities before jobs were made available to non-Union permit workers who applied for positions through the Union’s hiring hall. When Washington’s commercial construction industry entered a slump in 1975, Union membership became especially valuable, because limited employment prospects prevented permit workers from obtaining regular employment. Admission to the Union always has been by examination. The prerequisites for taking the entrance exam, however, have varied over time. Four different sets of prerequisites are relevant to the adjudication of this case. 1965-February 1971. During this period, there were two routes to the exam: (1) by permission of the Union’s Executive Committee, which could grant leave, in its discretion, to rodmen with an unspecified amount of experience to take the exam; and (2) by completion of the two-year Apprenticeship Program, admission to which was contingent upon the applicant (a) being between 18 and 31 years of age, (b) possessing a high school diploma, and (c) being physically fit. February 1971-June 1971. There were also two paths to the exam at this time. (1) Those who completed the Apprenticeship Program could take the usual exam. (2) All rodmen with two years’ experience were eligible to take a special exam — not the one given by the Union before or after this period — which had a higher failure rate. Because all experienced rodmen were allowed to sit an entrance exam without participating in a Union-supervised educational program, these five months are referred to as the “Open Period.” June 1971-September 1972. Throughout this period, only rodmen who had completed the Apprenticeship Program were permitted to take the exam. September 1972-October 1975. Once again, there were two tracks to the exam. (1) Completion of the Apprenticeship Program, whose admission requirements were unchanged, provided a ticket to the exam. (2) In addition, the Training Program was established, primarily to allow experienced minority workers to gain access to the exam and thus to Union membership. Although race was not a criterion for admission, special efforts were made to recruit minority workers. Participants in the Training Program had to be (a) physically fit, and (b) 31 years of age or older. Unlike the Apprenticeship Program at that time, the Training Program lacked a high school diploma requirement. Moreover, trainees —but not apprentices — were sometimes allowed to take the exam after a minimum of six months’ instruction, even though both educational programs were two years long. Throughout this entire period, no applicant was guaranteed acceptance into an educational program merely because he met the entrance requirements. Enrollment was always limited, and many who qualified for admission were either rejected or not permitted to apply. Once admitted to either program, however, no enrollee was asked or compelled to discontinue participation because of unsatisfactory performance; the only ground for expulsion was failure to meet one of the enrollment requirements described above. Completion of one of the programs resulted almost automatically in Union membership, since virtually all examinees, with the exception of those tested during the Open Period, passed the test. Prior to the passage of Title VII, the membership of Local 201 was overwhelmingly white. The Union’s racial composition changed after 1965, when Title VII took effect, but only gradually. At the start of 1967, only four of approximately 200 members were black. Trial Finding (“Tr.F.”) 63. Four years later, the Union numbered 16 black and 260 white members. Tr.F. 63. Although black membership increased markedly with the advent of the Open Period, the percentage of blacks entering the Union was plainly smaller than the percentage of blacks among non-Union permit workers. For example, while approximately 40% of those admitted to the Union in the three years prior to the initiation of this suit were black, Brief of Local 201 at 18, blacks comprised 50-60% of nonUnion workers seeking employment through the Union’s hiring hall. Brief of Appellees at 7. On October 21, 1975, six black rodmen who were not members of the Union filed suit against all of the defendants, alleging racial discrimination in violation of Title VII and section 1981. The gravamen of the plaintiffs’ complaint is that the preconditions to Union membership in the early 1970s — in particular, the requirement that apprentices hold a high school diploma and the requirement that rodmen enroll in either the Apprenticeship or the Training Program before taking the Union’s entrance exam — excluded a higher proportion of blacks than whites from Union membership and that these preconditions could not be justified on work-related grounds. Pursuant to a pretrial stipulation by the parties, the District Court certified the following two classes on July 26, 1976: 1) All black persons who have applied for or sought, from representatives of Local 201 or the International, membership in Local 201 and, in connection therewith, the International or who have applied for or sought, from representatives of Local 201, the Apprenticeship Program or the Training Program, admission to the Apprenticeship Program and/or the Training Program and who have been or might be excluded from Local 201 and, in connection therewith, the International or the Apprenticeship Program or the Training Program or any of the above by the alleged discriminatory practices of the Defendants and who could have filed timely charges with the EEOC when their class representatives filed such charges or who could have filed timely lawsuits when the class representatives filed the instant lawsuit. 2) All black persons who have been referred for employment by Local 201 or who have applied to Local 201 for referral for employment by any means, including filling out a referral slip, or presenting themselves at Local 201 and requesting representatives of Local 201 to refer them for work, and who have been or might be discouraged from applying for membership in Local 201 and in connection therewith, the International, and/or Apprenticeship Program and/or the Training Program by the allegedly racially discriminatory practices of the Defendants and who could have filed timely charges with the EEOC when their class representatives filed such charges or who could have filed timely lawsuits when their class representatives filed the instant lawsuit. Order (July 26, 1976), reprinted in Record Excerpts (“R.E.”). On March 6, 1978, the District Court allowed the six named plaintiffs to amend their complaint by adding two black rod-men — Simmons and McMillian — because it appeared that the original six might not include suitable representatives of both classes. Memorandum and Order (Mar. 6, 1978), reprinted in R.E. After the District Court denied separate motions for summary judgment by the Apprenticeship Committee, the International, and CCC, the case was tried during the summer and fall of 1981. More than three and a half years later, on June 7, 1985, the District Court issued its Trial Findings, merely copying, with insignificant changes, the proposed findings of fact and conclusions of law submitted by the plaintiffs on December 18, 1981. The court found all of the defendants liable for the allegedly racially discriminatory admissions policies of the Union dating back to at least 1965, as well as for retaliatory actions alleged to have been taken against several named plaintiffs. The District Court issued a remedial order on December 10, 1985, which was superseded by an Amended Order handed down on April 10, 1986. The Amended Order increased access to the entrance exam or to Union membership for experienced black rodmen, and provided for future proceedings to determine awards of backpay and damages. In addition, the court ordered the Union to return money collected from class members pursuant to a uniform levy on all Union workers to pay the Union’s legal expenses in connection with this action. The defendants appeal from the District Court’s Trial Findings and Amended Order, contending that many of the District Court’s findings of fact were clearly erroneous, that a large number of its legal conclusions were also mistaken, and that the court abused its discretion in choosing remedies. II. Standard of Review Federal Rule of Civil Procedure 52(a) provides in part that “[findings of fact... shall not be set aside unless clearly erroneous, and due regard shall be given to the opportunity of the trial court to judge the credibility of the witnesses.” The appellants have asked this court, however, to perform a more rigorous appraisal of the District Court’s findings than this rule prescribes. They contend that the trial judge’s almost verbatim transcriptions of the plaintiffs’ extensive proposals do not constitute genuine “findings” of fact, and thus that de novo review, or something approaching it, is appropriate in the unique circumstances of this case. We disagree. Although we strongly disapprove of the District Court’s wholesale adoption of the plaintiffs’ proposed findings — especially in a case of this magnitude—we adhere to the view that de novo review “would be wholly inconsistent with the function of an appellate court.” Southern Pac. Communications Co. v. AT & T, 740 F.2d 980, 984 (D.C.Cir.1984), cert. denied, 470 U.S. 1005, 105 S.Ct. 1359, 84 L.Ed.2d 380 (1985). An appellate court is generally ill-equipped to conduct de novo review, and the assignment of that task to courts of appeals would waste judicial resources. De novo review, moreover, would be contrary to the plain meaning of Rule 52(a), which requires courts to set aside factual findings only if they are “clearly erroneous.” It would also contravene the settled doctrine of this court and the Supreme Court. As the Court recently stated: We, too, have criticized courts for their verbatim adoption of findings of fact prepared by prevailing parties, particularly when those findings have taken the form of conclusory statements unsupported by citation to the record.... Nonetheless, our previous discussions of the subject suggest that even when the trial judge adopts proposed findings verbatim, the findings are those of the court and may be reversed only if clearly erroneous. Anderson v. City of Bessemer City, 470 U.S. 564, 572, 105 S.Ct. 1504, 1510, 84 L.Ed.2d 518 (1985). We adhere to the standard of review enunciated in Rule 52(a) and shall set aside factual findings only if we deem them clearly erroneous; however, we note that the function of appellate review that we must undertake in a case of this sort is substantially different (and more difficult) than what is normally required. In Southern Pacific, we explained that when a trial judge “abdicate^] to a party his duty to provide a reasoned explanation for his decision” and merely copies submitted proposals, it is incumbent on this court to check the adopted findings against the record “with particular, even painstaking, care.” 740 F.2d at 984; see also Sealy, Inc. v. Easy Living, Inc., 743 F.2d 1378, 1385 n. 3 (9th Cir.1984) (“special scrutiny”); EEOC v. Federal Reserve Bank, 698 F.2d 633, 639-42 (4th Cir.1983) (“careful scrutiny” is necessary and findings must be “more narrowly” examined when trial court merely reprints proposed findings), rev’d on other grounds sub nom. Cooper v. Federal Reserve Bank, 467 U.S. 867, 104 S.Ct. 2794, 81 L.Ed.2d 718 (1984); Amstar Corp. v. Domino’s Pizza, Inc., 615 F.2d 252, 258 (5th Cir.), cert. denied, 449 U.S. 899, 101 S.Ct. 268, 66 L.Ed.2d 129 (1980). The “special care” we devote to reviewing “findings [that] were not initially penned by the district judge,” Valentino v. United States Postal Serv., 674 F.2d 56, 60 n. 2 (D.C.Cir.1982), differs from that which we ordinarily display, not in the test that we apply to a particular finding of fact — individual findings will only be reversed if clearly erroneous — but in the volume of evidence we sift in judging the correctness of such findings and in the number of discrete findings we review without benefit of express, thoroughly supported allegations of error by the opposing party. Although we undertake with great reluctance what in this case has proven a Herculean task, the District Court’s inexplicable failure to reason independently and to address the defendants’ leading arguments leave us no choice. III. Class Action Issues The appellants raise four objections to the District Court’s certification of the two plaintiff classes in this case. First, they argue that the District Court failed to ascertain whether the requirements of Federal Rule of Civil Procedure 23(a) were met with respect to each of them, and that this court should therefore remand for such a determination. Second, the appellants contend that the District Court failed to set temporal boundaries to the classes, and that a remand is necessary to allow it to do so. Third, the appellants submit that none of the named plaintiffs adequately represents the class of those black permit workers who were allegedly injured by the Apprenticeship Program’s high school diploma requirement in violation of Title VII and section 1981. Fourth, they claim that the plaintiff class should be limited to those who actually applied for admission to the Union or one of its educational programs, and that the second class of “discouraged” nonapplicants should be decertified. We find the first two arguments merit-less. We agree, however, that none of the named plaintiffs is a suitable representative of those who were allegedly wronged by the Apprenticeship Program’s high school diploma requirement; accordingly, that portion of the plaintiffs’ case should not have been permitted to go forward to judgment, and the District Court’s decision on this issue must be reversed. The appellants’ fourth objection is misguided insofar as it applies to the plaintiffs’ remaining challenge to the Union’s requirement that experienced rodmen enroll in an educational program before taking the entrance exam. A. Possible Failure to Conduct a Rule 23(a) Analysis The Supreme Court has stated that “a Title VII class action, like any other class action, may only be certified if the trial court is satisfied, after a rigorous analysis, that the prerequisites of Rule 23(a) have been satisfied.” General Tel. Co. v. Falcon, 457 U.S. 147, 161, 102 S.Ct. 2364, 2372, 72 L.Ed.2d 740 (1982). The two classes in this case were conditionally certified upon stipulation by all the parties. Order (July 26,1978), reprinted in R.E. It is unclear, however, whether the District Court performed the required analysis regarding the prerequisites of Rule 23(a). Nevertheless, we find the District Court’s failure to articulate the reasons for its decisions an insufficient reason to remand. The defendants have not pointed to any possible errors in the District Court’s definition of the classes apart from those considered separately below. See Brief of Local 201 at 34-39. Nor have we identified any ourselves. In the absence of any reason to suspect that the District Court’s decisions in this regard were mistaken, we cannot label the court’s silence an abuse of discretion warranting a remand. See Postow v. OBA Fed. Sav. & Loan Ass’n, 627 F.2d 1370, 1380 n. 24 (D.C.Cir.1980). B. Temporal Limits to the Classes The appellants contend that, while the class definitions contained in the 1976 certification order and reproduced in the Amended Order “purport[ ] to accord with relevant time frames,” the District Court’s “numerous errors in construing what such limitations actually mean ... require a remand for the purpose of specifically defining both backward and forward class limitations by date, even if [the] class definition[s] (at least as to time) [are] to be adhered to.” Brief of Local 201 at 49. We reject this contention as frivolous. The appellants, by their own admission, do not find fault with the temporal aspects of the District Court’s class definitions. Hence, there is no reason at all to remand on this point. To the extent that the appellants quarrel with the District Court's application of this unquestionably acceptable definition in specifying remedies, their objections are properly considered under that heading. C.The Ability of Class Representatives to Challenge the Apprenticeship Program’s High School Diploma Requirement As the appellants correctly note, the plaintiffs’ challenge to the Apprenticeship Program’s admission requirements focused exclusively on the rule that apprentices hold high school diplomas, and the District Court found this requirement alone to violate Title VII and section 1981. See Brief of Local 201 at 41; Tr.F. 85-100. No objections were raised to the requirements that apprentices be physically fit or that they be under thirty-one years of age. The Supreme Court has “repeatedly held [that] a class representative must be part of the class and ‘possess the same interest and suffer the same injury’ as the class members.” East Tex. Motor Freight Sys. v. Rodriguez, 431 U.S. 395, 403, 97 S.Ct. 1891, 1896, 52 L.Ed.2d 453 (1977) (quoting Schlesinger v. Reservists Comm. to Stop the War, 418 U.S. 208, 216, 94 S.Ct. 2925, 2930, 41 L.Ed.2d 706 (1974)). The plaintiffs may therefore only bring a class action challenging the high school diploma requirement of the Apprenticeship Program if one of the named plaintiffs was allegedly injured by that requirement. Hence, this part of the action may not proceed unless one of the eight named plaintiffs was, during the Title VII or section 1981 limitations period, refused admission to the Apprenticeship Program or discouraged from applying for admission because he lacked a high school diploma, notwithstanding his having met the age and fitness requirements at that time. Two of the named plaintiffs — McMillian and Simmons — graduated from high school, and thus may not serve as class representatives. Tr.F. 18-19. The other six did not have high school diplomas at times relevant to this suit. All of them, however, either were too old to challenge the high school diploma requirement within the statutory limitations periods, or were not injured by it because they had no interest in enrolling in the Apprenticeship Program during those periods. A plaintiff may bring a Title VII action only if he has first filed charges with the Equal Employment Opportunity Commission (“EEOC”) within 180 days of suffering injury in violation of the statute. 42 U.S.C. § 2000e-5(e) (1982). Thus, one of the six named plaintiffs without a high school degree could challenge the Apprenticeship Program’s high school diploma requirement only if, within 180 days of filing charges with the EEOC, he was under 31 years of age and physically fit, and if he was at that time denied admission or was discouraged from applying because he lacked a high school diploma. Plaintiff Lewis was the first of the named plaintiffs to file charges with the EEOC. He did so either in December 1974, or on January 13, 1975, depending upon which of the District Court’s inconsistent findings is correct. Compare Tr.F. 5 with Tr.F. 15. The 180-day limitations period under Title VII therefore began to run in June or July 1974. However, Lewis had turned 31 prior to that date, as had all of the other plaintiffs who had not graduated from high school. None of the six could therefore demonstrate that he was injured by the Apprenticeship Program’s high school diploma requirement after June 1974, because all were or would have been excluded by virtue of their age, and the age limit was never challenged or found to be discriminatory. Hence, the class action challenge to the high school diploma requirement may not be brought under Title VII. Although section 1981 does not contain its own statute of limitations, 42 U.S.C. § 1988 requires that the most nearly analogous District of Columbia statute of limitations be applied to suits under section 1981. In Banks v. Chesapeake & Potomac Tel. Co., 802 F.2d 1416 (D.C.Cir.1986), this court held that the most closely analogous period for purposes of section 1981 is that set by D.C. Code Ann. § 12-301(8) (1981), which specifies a three-year limitations period for personal injury claims not specified elsewhere in the D.C. Code. In reaching that decision, however, the court faced only the question whether the three-year statute of limitations for specified personal injury claims or the one-year period provided by the D.C. Human Rights Act, D.C. Code Ann. § 1-2544 (1981), applied to suits under section 1981. The court did not determine whether the one-year limitations period for most intentional torts, D.C. Code Ann. § 12-301(4), or the three-year period applicable to other personal injuries is more appropriately applied to section 1981 suits. Although a majority of the three-judge panel argued at length that the three-year limitations period applied, see Banks, 802 F.2d at 1426-29, the majority recognized that it was “considerpng] the issue in dicta only, and only in order to answer” objections raised in a concurring opinion. Id. at 1427. The question whether a one-year or a three-year limitations period governs actions under section 1981 therefore remains open. We need not resolve it now. Regardless of which limitations period is correct, none of the six original plaintiffs fulfilled the requirements for challenging the high school diploma requirement under section 1981. If the one-year period should be applied, then the plaintiffs’ suit was plainly foreclosed, because all six turned thirty-one prior to October 21, 1974, the date one year before suit was filed. See note 4 supra. If the three-year period is proper, then five of the six would once again have been excluded from the Apprenticeship Program by reason of age; only Lewis was under thirty-one years of age on October 21, 1972. However, Lewis testified at trial that he did not apply for admission to the Apprenticeship Program between October 21, 1972, and his thirty-first birthday, and he never claimed that he would have applied had there been no high school diploma requirement. He apparently only thought of applying to the Apprenticeship Program after he had turned thirty-one: “I didn’t apply for the apprenticeship program because they said you couldn’t get in over 30.” Transcript (“Tr.”) at 471. Since Lewis did not allege injury by the Apprenticeship Program’s high school diploma requirement, he cannot serve as the representative of a plaintiff class contesting the lawfulness of that requirement under section 1981. The class action challenging the high school diploma requirement under both Title VII and section 1981 therefore fails for want of a suitable representative. The plaintiffs did, however, satisfy the jurisdictional prerequisites for assailing the Union’s requirement that experienced workers enroll in either the Apprenticeship or the Training Program before taking the entrance exam. Although the appellants have not denied that the named plaintiffs adequately represent the class of those allegedly injured by this requirement, and thus have waived any objection they might have raised, see Banks, 802 F.2d at 1427, we note that Kirkland was experienced, physically fit, and desirous of taking the entrance exam within six months of the filing of charges with the EEOC on his behalf and within six months of the filing of this law suit. See Tr.F. 6, 14. Because this requirement was in effect since the end of the Open Period in June 1971, all class members who were injured by the requirement after that date may join in this suit and any recovery under it. D. Should Discouraged Nonapplicants Be Part of the Plaintiff Class? The appellants contend that the class definition should not have included “persons who have never actually applied for membership in Local 201, or for admission to the training or apprenticeship programs.” Brief of Local 201 at 46. This objection apparently derives from a misunderstanding of at least part of the plaintiffs’ theory of liability. The plaintiffs’ contention with respect to the educational prerequisite for taking the entrance exam, if not with regard to the Apprenticeship Program’s high school diploma requirement, is that compelling experienced rod-men to participate in some educational program for six months or more before allowing them to take the exam was itself unlawful. There is no reason why those who were unwilling to subject themselves to an allegedly unlawful regimen, but who would have taken the exam had this obstacle not been placed iii their path, may not challenge that requirement. We therefore see no merit in the appellants’ objection insofar as it applies to the plaintiffs’ attack on the instructional prerequisite to sitting the entrance exam. IV. Plaintiffs’ Statistical Proof of Discrimination A. Legal Framework 1. Law Governing Establishment of a Prima Facie Case by Statistical Proof The legal framework governing plaintiffs’ claims is straightforward. Under now-familiar Title VII law, plaintiffs bear the burden of establishing a prima facie case of racial discrimination. See Texas Dep’t of Community Affairs v. Burdine, 450 U.S. 248, 101 S.Ct. 1089, 67 L.Ed.2d 207 (1981); United States Postal Serv. Bd. of Governors v. Aikens, 460 U.S. 711, 103 S.Ct. 1478, 75 L.Ed.2d 403 (1983). In seeking to prove that facially neutral employment criteria fall more harshly on blacks than whites, plaintiffs may rely on statistical evidence tending to show that those criteria “select applicants for hire or promotion in a racial pattern significantly different from that of the pool of applicants.” Albemarle Paper Co. v. Moody, 422 U.S. 405, 425, 95 S.Ct. 2362, 2375, 45 L.Ed.2d 280 (1975); see also Griggs v. Duke Power Co., 401 U.S. 424, 430-31, 91 S.Ct. 849, 853, 28 L.Ed.2d 158 (1971). If the racial makeup of a selected group (in this case, those who became Union members) differs significantly from that of the pool from which it was selected, discrimination may be inferred. See International Bhd. of Teamsters v. United States, 431 U.S. 324, 335 n. 15, 339 n. 20, 97 S.Ct. 1843, 1854 n. 15, 1856 n. 20, 52 L.Ed.2d 396 (1977); Hazelwood School Dist. v. United States, 433 U.S. 299, 307-08, 97 S.Ct. 2736, 2741, 53 L.Ed.2d 768 (1977). Once this is demonstrated, a prima facie case has been established, and the burden shifts to defendants to articulate a legitimate non-discriminatory reason for the challenged employment practice. Burdine, 450 U.S. at 253, 101 S.Ct. at 1093; McDonnell Douglas Corp. v. Green, 411 U.S. 792, 802-03, 93 S.Ct. 1817, 1824, 36 L.Ed.2d 668 (1973); McKenzie v. Sawyer, 684 F.2d 62, 71 (D.C.Cir.1982). Defendants may also attempt to undermine plaintiffs’ prima facie case by attacking the validity of plaintiffs’ statistical evidence, or by introducing statistical evidence of their own showing that the challenged practice did not have racially disproportionate results. See Teamsters, 431 U.S. at 360, 97 S.Ct. at 1867; Segar v. Smith, 738 F.2d 1249, 1268 (D.C.Cir.1984), cert. denied, 471 U.S. 1115, 105 S.Ct. 2357, 86 L.Ed.2d 258 (1985). The degree of disparity between the expected and actual racial composition of the membership necessary to support an inference of discrimination is commonly viewed in terms of standard deviations. The Supreme Court has stated that a difference of approximately two to three standard deviations between expected and actual results is sufficient to demonstrate that the disparity is the product of something other than chance. See Castaneda v. Partida, 430 U.S. 482, 496 n. 17, 97 S.Ct. 1272, 1281 n. 17, 51 L.Ed.2d 498 (1977). There is, however, a large gap between two and three standard deviations, and thus the Supreme Court’s teaching in this respect has not been entirely dispositive. Under this circuit’s precedent, in most instances a difference of just under two standard deviations has been deemed sufficient to establish a prima facie case. See Palmer v. Shultz, 815 F.2d 84, 91-92 (D.C.Cir.1987). As we shall presently see, the statistical disparities in this case are well above two standard deviations; we are thus not faced with nettlesome issues such as whether 1.96 or 2 standard deviations ought to be the standard, or whether a one-tailed or two-tailed statistical test applies. A difference of two standard deviations corresponds roughly to a five percent probability that the disparity is the result of chance; this five percent standard is commonly referred to as the “.05 level of significance.” See Segar v. Smith, 738 F.2d at 1282-83. Thus, if the likelihood that a fluctuation from expected results occurred by chance is five percent or less, a statistically significant difference is proved, and a prima facie case of discrimination is established. It is, of course, elementary that intent to discriminate is a necessary element of Title YII disparate treatment cases. But here too, statistics play an important part. The basic office of statistical proof is to seek to eliminate non-discriminatory explanations for racial disparities; thus a statistically valid showing of a substantial disparity between expected and actual results may give rise to an inference of discriminatory intent. Teamsters, 431 U.S. at 336 n. 15, 97 S.Ct. at 1854 n. 15; Hazelwood, 433 U.S. at 307-08, 97 S.Ct. at 2741. What is more, this court has squarely held that, even absent specific anecdotal evidence of discrimination, statistical proof alone may establish a prima facie case of intentional discrimination. Segar v. Smith, 738 F.2d at 1277-79, 1286-87; see also McKenzie v. Sawyer, 684 F.2d 62, 71 (D.C.Cir.1982) (“Statistics alone will suffice to show illegally discriminatory treatment, if they are condemning enough.”) In the course of enunciating this rule, the court in Segar rejected the proposition that to establish intentional discrimination a showing of “gross disparities,” rather than mere statistically significant disparties, is required. Segar v. Smith, 738 F.2d at 1277-78. 2. Law Governing Rebuttal of a Statistically Based Prima Facie Case Once plaintiffs establish a prima facie case, the burden shifts to the defendants to articulate a legitimate non-discriminatory explanation for the statistical disparity. See Segar v. Smith, 738 F.2d at 1267-68; Davis v. Califano, 613 F.2d 957, 961-62 (D.C.Cir.1979). Alternatively, defendants can also meet their burden by demonstrating that plaintiffs’ statistics are so flawed as to be meaningless, or by introducing alternative statistical evidence. See Dothard v. Rawlinson, 433 U.S. 321, 331, 97 S.Ct. 2720, 2727, 53 L.Ed.2d 786 (1977) (“If the employer discerns fallacies or deficiencies in the data offered by the plaintiff, he is free to adduce countervailing evidence of his own.”); Segar v. Smith, 738 F.2d at 1267-68. We hasten to observe that, as the Supreme Court has stated, “statistics are not irrefutable.” Teamsters, 431 U.S. at 340, 97 S.Ct. at 1856. After plaintiffs put on a statistical case, it remains open on rebuttal for defendants to rely on all types of evidence, statistical and otherwise. In the typical case, defendants’ rebuttal case will likely focus on (1) inadequacies in the plaintiffs’ statistical case, (2) an alternative, more favorable statistical case, or (3) a non-discriminatory explanation for the statistical disparities. Since the ultimate burden of persuasion rests always with plaintiffs, see United States Postal Serv. Bd. of Governors v. Aikens, 460 U.S. 711, 716, 103 S.Ct. 1478, 1482, 75 L.Ed.2d 403 (1983); Segar v. Smith, 738 F.2d at 1284, district courts must weigh in the balance all the evidence — both plaintiffs’ statistical proof and defendants’ rebuttal — in reaching the ultimate conclusion as to liability. Again, we draw from the Supreme Court’s teaching: Whether ... a [statistical] analysis does carry the plaintiffs’ ultimate burden will depend in a given case on the factual context of each case in light of all the evidence presented by both the plaintiff and the defendant. However, as long as the court may fairly conclude, in light of all the evidence, that it is more likely than not that impermissible discrimination exists, the plaintiff is entitled to prevail. Bazemore v. Friday, 478 U.S. 385, 106 S.Ct. 3000, 3009, 92 L.Ed.2d 315 (1986) (per curiam) (Brennan, J., concurring for a unanimous Court on this point) (emphasis added). B. Plaintiffs’ Statistical Challenge to the Union's Educational Prerequisite to the Journeyman Exam In light of the governing legal framework, we turn again to the case at hand. At bottom, plaintiffs claim that (1) unnecessary barriers to taking the journeyman exam prevented (or delayed) the entry of otherwise qualified workers into Union ranks, and (2) those barriers disproportionately disadvantaged black rodmen. The proportion of eligible blacks who became Union members, plaintiffs contend, is much lower than the proportion of eligible whites who did. Inasmuch as the high school diploma requirement is no longer part of the case, see supra Part III.C., the employment practices we are called on to consider are two: the Apprenticeship Program and the Training Program. It bears emphasis that plaintiffs’ challenge is simple and direct: An educational prerequisite — either Apprenticeship or Training — discrimina-torily delayed union membership for qualified black permit workers. According to plaintiffs the educational prerequisite is utterly unnecessary if one can pass the journeyman exam. Under this theory, since those workers who are sufficiently experienced to take (and pass) the exam without undergoing an educational program are disproportionately black, delaying membership by means of such a program is unlawfully discriminatory. 1. The Experienced Pool Analysis To prove that the Union’s educational prerequisite to the journeyman’s exam is discriminatory, plaintiffs were required to demonstrate that during the period in question the percentage of eligible blacks who became Union members — i.e., took and passed the journeyman exam— was significantly lower than the percentage of eligible whites. To accomplish this, plaintiffs relied on an expert in labor economics, Professor Sheldon Haber, to construct a statistical model, which he dubbed the “experienced pool” analysis. Plaintiffs counted and identified as to race both potential and actual journeyman examinees, and compared the racial composition of the two groups. Complicating the analysis was the fact that the Union does not keep statistics regarding applications to take the exam; thus, a proxy for identifying potential applicants had to be devised. One important fact was deemed to be of especial relevance in crafting a proxy. During the 1971 Open Period, the Union itself relied on two years’ experience as the criterion for determining who was qualified to take the journeyman exam. In identifying a proxy for potential applicants, Professor Haber therefore took two years as the benchmark for sufficient experience to take the exam. With this benchmark in hand, Haber sought to identify non-member permit workers with at least two years’ experience in order to determine the pool of “likely examinees.” This pool was then compared to the pool of workers who actually took the exam, who were referred to as “actual examinees.” Another complication arose, however, in completing the analysis: Union pension records reflect workers’ experience in terms of hours, not years. In consequence, Professor Haber had to calculate (or estimate) the number of hours that closely approximates two years’ experience in the rodman trade. To determine this figure, Haber referred back to the examinees during the Open Period in 1971, and calculated the average number of hours worked by those examinees in the two years (1969 and 1970) prior to the Open Period. Professor Haber then calculated three averages: one for whites, one for blacks, and a combined white and black workers average. With these three averages, he chose the lowest average figure of 2150 hours (for whites) as the benchmark. This conservative methodology assured that the pool of likely examinees would not be skewed in favor of blacks. Plaintiffs then determined the percentage of workers with 2150 hours of experience who took the entrance exam (almost all of whom passed, and therefore became Union members, see infra n. 15). For the three-year period prior to the filing of this lawsuit (October 22, 1972 through October 21, 1975), the proportion of experienced whites who became Union members was 33.6 percent (49 out of 146); however, the proportion of experienced blacks who became members was a much lower 15.3 percent (19 out of 124). See Tr.F. 82; see also Brief of Appellees at 63. The possibility that a difference of this magnitude would occur by chance is approximately one in 1000, well over the requisite two standard deviations. See Tr.F. 82. Plaintiffs also introduced evidence relating to Union admissions going back to 1967, all of which showed a statistically significant disparity between the proportion of eligible blacks and whites who became Union members. See id. Dr. Haber’s experienced pool analysis thus showed that the difference in the proportion of eligible white and black achievement of Union membership far surpassed the five percent significance level. The District Court concluded, accordingly, that plaintiffs had established a prima facie case of racial discrimination. Tr.F. 81. The burden of rebutting the prima facie case then shifted to defendants. C. Defendants’ Objections to Plaintiffs’ Statistical Case As in the District Court, defendants launch several broad attacks on plaintiffs’ statistical case. At least one contention, namely that Professor Haber was not a credible witness, can readily be disposed of. Credibility determinations are obviously for the trial court to make; our function on appeal is inconsistent with revisiting the credibility of an expert witness unless his testimony was “patently unsound.” 9 C. Wright & A. Miller, Federal Practice and Procedure § 2586, at 740 (1971). We are unpersuaded that Professor Haber’s testimony can fairly be dismissed on the basis of that daunting standard. However, defendants’ other arguments merit fuller treatment. 1. The Nature of Defendants’Burden on Rebuttal As we indicated above, to demonstrate that statistical evidence is invalid, the challenger must present “credible evidence that the statistical proof is defective” and “a plausible explanation of how the asserted flaw is likely to bias the results against his or her position.” Palmer v. Shultz, 815 F.2d 84, 101 n. 13 (D.C.Cir.1987) (quoting D. Baldus & J. Cole, Statistical Proof of Discrimination vii (1986 Supp.)). In Bazemore v. Friday, 478 U.S. 385, 106 S.Ct. 3000, 3009, 92 L.Ed.2d 315 (1986) (per curiam), Justice Brennan, concurring for a unanimous Court on this point, stated: [I]t is clear that a [statistical] analysis that includes less than “all measurable variables” may serve to prove a plaintiffs case. A plaintiff in a Title VII suit need not prove discrimination with scientific certainty; rather, his or her burden is to prove discrimination by a preponderance of the evidence. To rebut, the challenger must adduce proof specifically showing that an alleged defect actually biases the results. Mere conjecture or general assertions of inadequacies in the opponent’s statistical case, without demonstrating their effect on the results, will not suffice. Palmer v. Shultz, 815 F.2d at 101 (discussing Bazemore’s “dictates”). With this additional background, we turn to defendants’ specific attacks on plaintiffs’ prima facie statistical case. 2. Defendants’ Objections to Plaintiffs’ Statistical Results Defendants argue that the experienced pool analysis is so flawed and divorced from reality as to be meaningless. Their challenges are several. According to defendants, the chosen benchmark of 2150 hours did not represent the cumulative experience of the Open Period examinees, but rather their average hours for the two pri- or years. Thus, the actual experience of the Open Period examinees most likely exceeded 2150 hours by a large margin. Defendants argue that this inaccuracy renders the experienced pool analysis useless, because 2150 hours has no basis either as a measure of the actual experience relied on by the Union during the Open Period, or as an objective approximation of the amount of experience which qualifies one to take the exam. This asserted shortcoming in the experienced pool analysis, however, biases the results in defendants’ favor. Professor Haber testified that he purposely chose the lower (white) average in order to maximize the number of whites in the pool. According to defendants themselves, “the lower the ‘benchmark,’ the lower the proportion of blacks in the ‘experienced pool.’ ” Brief of Local 201 at 106. Given the unmanipulable nature of the figure of actual new members, the ultimate percentage of eligible workers selected for Union membership depends entirely on the number of individuals in the eligible pool: the more blacks in the pool of “likely examinees,” the lower the percentage of black “actual examinees” as a proportion of black “likely examinees” will be, and vice versa. Because white non-members on average had worked fewer hours than black non-members, a lower benchmark increases the number of white “likely examinees” relative to the number of black “likely examinees.” The respective percentages of “likely examinees” who became “actual examinees,” it follows, are correspondingly lower for whites and higher for blacks. Thus, rather than demonstrating that the 2150 hour benchmark biases the results in favor of plaintiffs, defendants’ analysis suggests exactly the opposite. Defendants also argue that the failure to segregate the experienced pool by age (that is, to analyze separately the group over age 30 and the group under age 31) fatally skewed the results in favor of plaintiffs. Defendants maintain that there was no reason not to segregate by age when available data made it possible to do so; this failure is significant, they argue, because the respective paths to Union membership in the Apprenticeship and Training contexts were so different. Defendants maintain that in the under-31 group, blacks as a proportion of the total experienced pool constituted 24.8 percent (the black “availability rate”), while black participation in the Apprenticeship Program was 30.3 percent, a figure obviously greater than the black “availability rate.” See Brief of Local 201 at 98. Similarly, in the over-30 pool, blacks constituted 75.8 percent of the experienced workers, with their participation rate in the Training Program at 96.2 percent, again a figure well above the black “availability rate.” Id. Defendants maintain that the failure to segregate the experienced pool analysis by age therefore skewed the results in plaintiffs’ favor. This argument suffers from a fatal flaw. Defendants’ analysis focuses on black participation in the Apprenticeship and Training programs, rather than black completion of the programs (i.e., achievement of Union membership). If one focuses on black participation as a whole — not segregating by age, the asserted flaw in plaintiffs’ analysis — the results are also favorable to defendants. Thus, the failure to segregate by age had no independent demonstrable effect on the validity of the experienced pool analysis. This suggests that defendants’ challenge to plaintiffs’ statistical case is in fact an attack on the premises underlying the statistical model, to which we now turn. 3. Defendants’ Attack on the Premises of the Experienced Pool Analysis Defendants attack several fundamental aspects of the experienced pool analysis. Among their objections are: (1) that the concept of “access” to the exam relied on by plaintiffs is flawed; (2) that not all experienced rodmen, i.e., “likely examinees,” were interested in becoming journeymen; (3) that no specific procedure within the Apprenticeship or Training Program was shown to be discriminatory; (4) that the Union cannot be forced to increase the size of its membership in order to achieve racial balance; and (5) most centrally, that the experienced pool did not account for “minimum objective qualifications” of the trade because an educational prerequisite is, and always has been, uniformly required. (a) The “False” Concept of “Access” First, defendants argue that it is wrong to regard only those workers who completed one of the educational programs and took the exam as having "access” to the exam. Rather, they maintain, the proper definition of “access” to membership includes those who were admitted to either the Apprenticeship or Training programs. As defendants see it, the only “requirement” to gain access to Union membership is to secure admission to one of the two educational programs; no one “fails” or is otherwise forced out of either program, as evidenced by the fact that virtually all who finished their respective program passed the exam. To survive the educational component of either program, defendants maintain, all one needed to do, in effect, was show up for class. Therefore, the real source of disparity in minority membership, the argument goes, is the disproportionate number of minority dropouts from the Apprenticeship Program and, especially, the Training Program. According to defendants, those who chose not to complete one of the educational programs had no less access to Union membership than did their colleagues who diligently completed their particular programs. If we accepted defendants' invitation to focus on the rate of participation in the educational programs, rather than the rate of completion, then plaintiffs’ prima facie case would indeed disappear. Defendants are correct that the racial disparity in admissions to Union membership can be explained entirely by the disproportionate rate of minority dropouts from the educational programs. In defendants’ view, it follows that, inasmuch as there were no allegations of discrimination in the administration of the programs, the experienced pool analysis is of no relevance in determining the ability of minorities to gain Union membership. With all respect, the Union’s focus on participation in the educational programs misses plaintiffs’ fundamental point. Plaintiffs challenge the very existence of a training program as a discriminatory barrier to Union membership. So viewed, the higher dropout rate for blacks obviously buttresses plaintiffs’ case: Absent rebuttal, the fact of disproportionately high minority dropouts demonstrates the effectiveness of the barrier to membership represented by Local 201 ’s educational requirements, particularly the Training Program. Defendants are thus fundamentally in error in grandly asserting that plaintiffs cannot “simply cite the requirement of training as the discriminatory practice at issue.” Brief of Local 201 at 97 n. 106. The requirement of “training” is precisely the issue. See infra Part IV.C.3.e. (b) The Assumption That All Experienced Workers Desired Union Membership Defendants contend, next, that the experienced pool analysis is inadequate because it assumes that all experienced workers desired to become Union members. According to defendants, the record demonstrates that experience does not necessarily make one a “likely examinee.” During the early 1970s, permit workers and journeymen alike were fully employed. In those circumstances, there was little incentive for a rodman to undergo the considerable inconvenience (and expense) of becoming a Union member. Indeed, several of the named plaintiffs chose to forgo opportunities for training which would have led to Union membership. See Brief of Local 201 at 93. Defendants urge that the experienced pool analysis is therefore flawed by virtue of this unfounded assumption. To hold to the contrary, they maintain, would be to require the Union somehow to force workers to join the Union in order to achieve racial balance, a result that is indisputably not required (and indeed forbidden) by Title VII. See Brief of Local 201 at 94. We can safely assume that, for one reason or another, some experienced workers did not desire Union membership. But it does not follow that the experienced pool analysis is thereby rendered inadequate, or that the Union is somehow being required to force workers to join in order to achieve racial balance. Rather, we believe defendants demand more than the law requires from plaintiffs’ statistical case. As a theoretical matter, the experienced pool analysis is not undermined by the (partially) inaccurate assumption that experienced workers were “likely examinees.” The reason is this: There is no reason to assume that blacks were as a general matter less interested in Union membership than whites. Certainly, defendants have presented no persuasive arguments or evidence to the contrary. Indeed, during the Open Period, when blacks and whites had equal opportunity to take the exam, they did so in relatively equal proportions. Absent a showing that the demonstrated inadequacy in the experienced pool analysis biased the results in plaintiffs’ favor — that is, that black nonmembers did not desire to be members in roughly equal numbers as whites — defendants fail to undermine the prima facie case. (c) The Failure to Identify Specific Discriminatory Procedures, or to Demonstrate that Defendants’ Conduct Caused the Racial Disparities Defendants next urge that plaintiffs failed to identify the specific test, procedure or practice alleged to have had a discriminatory impact, or (under a disparate treatment or section 1981 analysis) to demonstrate that discrimination was the Union’s “standard operating procedure.” See Brief of Local 201 at 94-96. Defendants argue that a prima facie case discrimination was therefore not made out. These arguments can be dealt with in short order. Quite simply, defendants once again decline to recognize the thrust of plaintiffs’ challenge. The "specific procedure” under challenge is the requirement of classroom training before experienced workers may qualify to enter the ranks of journeymen rodmen. That theory of the case fully satisfies defendants’ obvious right to know precisely what is under challenge. And defendants have failed to explain why the asserted barrier of an absolute educational requirement, regardless of experience and ability, is somehow shielded from attack. With respect to the alleged failure to demonstrate that discrimination was the Union’s “standard operating procedure,” plaintiffs’ statistical proof, reaching the requisite showing of approximately two standard deviations, supplies an inference of discriminatory intent for the reasons set forth in Part IV.A.l. of this opinion. Defendants counter that there has been no showing of a causal connection between the challenged practice and the alleged discriminatory result. According to defendants, “[t]he Court cannot indulge in the assumption that Defendants caused people to drop out of the apprentice or training programs." Brief of Local 201 at 96. Defendants argue that no evidence suggested, much less demonstrated, that the higher minority dropout rate from the two programs was attributable to any of their actions. Here too, defendants have missed a basic point, which we have already elucidated: in a case based on statistical proof, direct evidence of discrimination is unnecessary. See Hazelwood School Dist. v. United States, 488 U.S. 299, 807-08, 97 S.Ct. 2736, 2741, 53 L.Ed.2d 768 (1977); Segar v. Smith, 738 F.2d 1249, 1277-79 (D.C.Cir.1984), cert. denied, 471 U.S. 1115, 105 S.Ct. 2357, 86 L.Ed.2d 258 (1985). Indeed, the entire notion of employing statistical proof is to eliminate non-discriminatory causes as likely explanations for racial disparities. In the present context, it seems to us that as a matter of logic there are four possible explanations to account for the failure of blacks to gain Union membership in equal proportions to whites: a) pure chance; b) blacks did not desire to join the Union in roughly equal numbers as whites; c) blacks failed the exam in greater percentages than whites; and d) the educational prerequisite discouraged blacks from taking the exam in greater proportion than whites. Possibility (a) was, of course, eliminated as part of plaintiffs’ prima facie statistical case. Possibility (b) can readily be eliminated since, as discussed above, there is no reason to assume that blacks in general had less desire than whites to join the Union. (Indeed, as we saw above, the evidence as to the Open Period suggests exactly the opposite). Possibility (c) is untrue as a matter of fact, as the parties agree. That being so, plaintiffs by their statistical proof established a prima facie case that possibility (d) provides the true explanation. Under settled principles, the burden thereupon shifted to defendants to demonstrate to the contrary. Defendants are therefore off the mark in arguing that plaintiffs have not shown causation. Contrary to defendants’ protestations, the burden fell to them to show that some racially neutral explanation existed for why blacks were failing, disproportionately, to become Union members. (d) The Experienced Pool Analysis Would Require the Union to Maximize its Membership Ranks to Achieve a Racial Balance Defendants argue that the experienced pool analysis operates so as to penalize the Union for not maximizing its membership ranks to achieve racial balance. See Brief of Local 201 at 103-05. According to defendants, the largest portion of the experienced pool consisted of workers over age 30; this group also happened to be disproportionately black. Defendants reason that because the Training Program was an affirmative action program, and whites over age 30 therefore had less effective avenues to Union membership than did blacks, plaintiffs’ complaint boils down to a quibble that greater numbers of blacks over age 30 were not admitted to Union ranks. And, defendants continue, the only reason for admitting these more experienced workers would be to achieve a desired racial balance. According to defendants, Title VII does not require them to achieve a racially balanced membership. To be sure, defendants are correct that unions have a legitimate, racially neutral interest in limiting the size of their memberships in accord with market demands. No one in this case has quarreled with the proposition that, at least insofar as the civil rights laws are concerned, Local 201 should be permitted to continue to “determine[] its number of apprentice and trainee indentures based on available employment.” Brief of Local 201 at 105. But it is equally clear that Unions are not permitted to limit their rolls in a discriminatory manner. Plaintiffs do not claim, and we emphatically do not hold, that the Union’s vice was in failing to open its doors to all experienced blacks over the age of 30. Rather, we hold today only that the means chosen by the Union to achieve its legitimate end of limiting membership were, in the face of plaintiffs’ proof, impermissibly discriminatory. The Union remains at liberty to employ a more stringent exam, or any other job-related system of qualifying workers for Union membership, so long as the result is not violative of the civil rights laws. (e) The Experienced Pool Analysis Did Not Account For Minimum Objective Qualifications Defendants’ final challenge to the plaintiffs’ prima facie case is fundamental in nature. It is well established that if a statistical model does not take into account the legitimate, objective qualifications for the jobs being analyzed, then it fails sufficiently to focus on an appropriate labor pool, and is therefore deficient in establishing a prima facie case. See Palmer v. Shultz, 815 F.2d 84, 91 n. 6 (D.C.Cir.1987); Segar v. Smith, 738 F.2d 1249, 1274 (D.C.Cir.1984), cert. denied, 471 U.S. 1115, 105 S.Ct. 2357, 86 L.Ed.2d 258 (1985). Accord