This is an appeal from those portions of a judgment rendered in favor of the employer in a class action brought under Title VII, 42 U.S.C. § 2000e et seq. Named plaintiffs Placido Rivera, Ernest Ceballos, Ernest Castillo, Pete Arthur Gonzales, Richard Gonzales, and David Maldonado are former employees of, or rejected applicants for employment with, the City of Wichita Falls, Texas. They represent a class defined as all Spanish surnamed Americans of Mexican descent who were discriminated against by the defendants, the City of Wichita Falls, Texas and certain of its officials, 1 through the establishment and application of various employment policies and practices. The claims asserted on behalf of the class challenged the full range of the City’s employment practices. After trial without a jury, however, the district court found discrimination only in the recruitment practices of the City’s Fire Department. 2 It rejected the plaintiffs’ contentions that screening procedures used by the City’s Police Department had a discriminatory impact on members of the class which could not be justified as job-related, that plaintiffs were unlawfully excluded from employment in the upper three levels of general City government, and that plaintiffs were discriminatorily denied promotions and training in the various divisions and levels of the City government. The plaintiffs appeal these rulings and the district court’s award of only $4,000 in attorney’s fees. We affirm the judgment of the district court.
I.
The plaintiffs’ attack was, at inception, a broad one. On behalf of the class they charged that the City was engaged in a pattern and practice of discrimination against Mexican-Americans which manifested itself in the entire range of the City’s employment practices, throughout all divisions of the City government. In addition, individual relief was pursued by three named plaintiffs and three intervenors who claimed to have been victimized by specific acts of discrimination. The action as now presented, however, is of considerably narrower dimensions. Through a series of waivers and self-initiated constrictions in the scope of the challenges pressed, 3 the *534 plaintiffs have limited the issues on appeal to these:
1. Did the district court err in holding that the City had successfully shown that the one element of its police recruit selection process which the court found to have a discriminatory impact is necessary for the performance of the job?
2. Did named plaintiff Ernest Ceballos present a prima facie case of discrimination by the City in its refusal to accept him in its police recruit training program?
3. Did the plaintiffs present a prima facie case of discrimination in hiring in regard to the upper three levels of the City government?
4. Did the plaintiffs present a prima facie case of discrimination in the training and promotion of class members employed by the City?
The class-wide claims charged the City with engaging in a “pattern and practice” of employment discrimination in violation of section 707(a) of the Civil Rights Act of 1964, 42 U.S.C. § 2000e-6(a)
4
The pattern or practice branch of employment discrimination law is designed to reach and to remedy pervasive resistance to the principles of equality of treatment and opportunity in the workplace articulated in Title VII. Accordingly, the threshold inquiry in a pattern or practice discrimination action is whether there is evidence of a correlation between employment decisions and the,
inter alia,
race, sex, or national origin of employees or job applicants. The plaintiffs’ initial responsibility of establishing a prima facie case of pattern and practice discrimination in hiring or promotion may, then, satisfied by a showing that “there is a significant statistical disparity between the racial, sexual, or ethnic balance and composition of an employer’s work force and that of the community from which the workers are hired,”
United States v. City of Alexandria,
The degree of the observed statistical disparity is, however, of critical importance. An absolute disparity between these percentages of representation is not enough. To be legally cognizable, the pat
*535
tern revealed must be at least “significantly discriminatory,”
Dothard v. Rawlinson,
Only upon the plaintiffs’ establishment of a prima facie case of pattern or practice discrimination would the defendants have either to discredit the plaintiff’s case, or to show that, despite the adverse impact, the challenged practices are necessary for the performance of the job.
Griggs,
District court determinations of the adequacy of these showings are reviewed under the variable standard appropriate to Title VII eases. Under this standard, as in the usual case, the district court’s findings of
*536
subsidiary facts would be overturned only if found to be clearly erroneous. The ultimate factual question, however, of whether the subsidiary findings of fact point to the defendants’ unlawful discrimination against the plaintiffs, is subject to an independent redetermination on appeal.
Danner
v.
United States Civil Service Commission,
II.
A. Police Recruit Selection Procedures
1. The Class Claims
In the district court, the plaintiffs challenged the entirety of the City’s police recruit selection procedures. The police recruit selection process employed by the City of Wichita Falls consists of four successive screening devices. First, applicants take the Basic Occupational Language for Police Officers (BOLPO) test. BOLPO, a test of the ability of an applicant to comprehend the type of reading material encountered in a police training academy, is used to predict the likelihood of the applicant’s successful completion of the police academy curriculum. An applicant passing BOLPO next takes a physical agility test. This test focuses on the applicant’s ability to perform those feats typically required of Wichita Falls police officers. Applicants clearing the first two hurdles are then the subjects of background investigations. A background investigation consists primarily of an inquiry into the applicant’s work history, a determination of whether he has a criminal record, and a check into whether he habitually associates with “police characters.” This step is considered by the City to be crucial in the selection of people suited to the highly sensitive, demanding and public position of a police officer. The final review of the applicant’s qualifications is made at an assessment center, where applicants, under the direction of police officers trained as “assessors,” engage in role playing, participate in simulated work experiences, and answer standard questions designed to measure their abilities to communicate effectively. Applicants passing all four tests are ranked by their cumulative scores and are offered positions with the police department in accordance with their rankings as openings occur.
Each of the four selection procedures were challenged by the plaintiffs as having a disparate impact on Spanish surnamed Americans of Mexican descent. The district court found, however, that the plaintiffs succeeded in adducing a prima facie case only in support of their challenge to the BOLPO test. The plaintiffs’ statistical proof on the comparative failure rates of whites and Mexican-Americans on the BOL-PO test is indeed adequate to support a prima facie case of BOLPO’s discriminatory impact on Mexican-Americans. In the relevant period of December 1974 to May 1978, 6 5.4 percent of all white applicants, twenty-one out of 387, failed BOLPO. During the same period, 40 percent of all Mexican-Americans, fourteen out of thirty-five, failed BOLPO. This substantially higher failure rate for class members is clearly statistically significant. 7
*537
The plaintiffs’ success was, however, short-lived. The district court held that the City had proved the test to be properly validated under the validation standards of the Uniform Guidelines on Employee Selection Procedures, 29 C.F.R. § 1607.
See Albemarle,
A test's degree of content validity is defined as the degree to which the subject matter of the test relates to those significant parts of the curriculum for which the test is intended to predict the applicants’ capability. To put it more clearly, an Early American history test which has content validity might well cover the Boston Tea Party, but would not cover the bombing of Pearl Harbor. The content validity of BOLPO was established in conformity with an August 1972 discussion draft of the Uniform Guidelines on Employee Selection Procedures by the Equal Employment Opportunity Coordinating Council. As the guidelines suggested, a “comprehensive job analysis” was performed to identify the cognitive skills required of police recruits. That analysis arrived at the unremarkable con-elusion that the ability to comprehend and effectively use materials such as those presented in training was a critical requirement for successful performance in the basic course at the academy. Accordingly, the test consisted of reading selections excerpted directly from standard training materials used in police academies in Texas.
After the test was designed, a “reliability estimate” was computed to determine the reliability of BOLPO test scores. Testing a representative sample of thirty incumbent police officers in Wichita Falls, 8 a reliability coefficient of .92 was obtained. The Civil Service Commission’s report concluded that this coefficient comfortably exceeds the minimum requirements which a test must satisfy in order to be considered to be content validated.
In addition to content validation of the test, the BOLPO test was measured for “criterion-related validity.” Criterion-related validity establishes a correlation between test scores and an independently determined measure of successful job performance. A “predictive” validity study, j.e., using knowledge of test performance to predict future performance at the police academy, was conducted with a group of trainee police candidates chosen from two academies in Texas prior to training. 9 *538 There was a strong correlation of .66 between the BOLPO test scores and the final grades at the training academies. 10
The strength of the correlations established by the content validity and predictive criterion-related validity studies amply supports the district court’s conclusion that BOLPO was properly validated. Further inquiry would be necessary only if the plaintiffs offered an alternative to BOLPO which has a less discriminatory effect on the class. They did not. The district court’s judgment that the City’s use of BOLPO in its selection of police recruits does not violate Title VII is accordingly affirmed.
The district court determined that plaintiffs failed to make a prima facie showing that any of the other three elements of the selection process, that is, the physical agility test, the background investigation and the job performance assessment, had a discriminatory impact. The plaintiffs do not argue that the subsidiary factual findings supporting these determinations are clearly erroneous, nor do they argue that the district court erred in failing to draw an inference of an absence of discrimination from the subsidiary findings.
11
Rather, they contend that these tests must nonetheless be properly validated. The duty to validate, however, arises only when the test has a disparate impact.
Albemarle,
Finally, plaintiffs argue that the district court should not have considered separately the individual tests for the selection of police recruits. Instead, they assert, the entire selection process should have been analyzed as a whole. Noting that 17.62 percent of white applicants completed the selection process while only 3.33 percent of class applicants did so, the plaintiffs argue that the process as a whole had a discriminatory effect on the class, and that therefore the process in its entirety must be validated.
*539 Such an approach might be appropriate if the effect of the individual elements in the selection process cannot be isolated, but a disparate impact can be shown to arise from the procedures’ completion. Applied, however, to situations in which the source of the disparate impact can be identified, the plaintiffs’ approach would allow the disparate impact of one element to require validation of other elements having no adverse effects. The burden of determining the validity of a screening procedure, weighing not only on the employer but also on the limited resources of the district court, will not be imposed where proof of an absence of discriminatory effect attributable to the procedure shows it to be unwarranted. The plaintiffs’ invitation to adopt this approach must be declined.
2. The Individual Claim
Ernest Ceballos, one of the named plaintiffs, appeals the district court’s holding that the City’s refusal to admit him to the police academy was for reasons other than his national origin. Ceballos originally complained that the City’s claim that he failed the background investigation was a subterfuge. 12 At trial, however, he testified:
Q. Don’t you think that it’s fair to say that your employment history at that time would cause considerable concern to someone who was considering you as a police officer?
A. Yes.
Then, of course, Officer Winn told you that that’s the reason that you weren’t being considered; isn’t that right? Q.
A. Right. . . .
Q. In fact, you testified in connection with this proceeding that whatever the reasons may have been for you not being employed as a police officer, you didn’t think it had anything to do with the fact that you were a Mexican American person; isn’t that right? . . .
A. Yes.
The district court is affirmed.
B. The City’s General Workforce
1. Administrative, Professional and Technical Employment
The plaintiffs’ efforts at trial concentrated on their allegation that the City has engaged in a pattern and practice of discrimination against class members in filling its upper level positions. To establish their prima facie case, the plaintiffs relied on evidence of an “underrepresentation” of class members in the City’s top three job categories and an “overrepresentation” of class members in the City’s lower four job categories. 13 They contended, first, that the lower four job categories constituted the labor pool from which the City selected people to fill positions in its upper three job categories, and that, therefore, the disparity between Mexican-Ameriean representation in the upper and lower levels indicated that the City discriminated against class members in training and promotions. 14 In the *540 alternative, they argued that the frequency of appearance of Mexican-Amerieans in the top three categories was significantly lower than that which would be expected in consideration of the number of Mexican-Amer-ieans in the relevant external labor pools, indicating discrimination by the City against Mexican-Americans in hiring. The district court concluded that comparison of the percentage of class members employed in the upper three job categories with the percentage of class members employed in the lower four job categories was inappropriate, and that the underrepresentation of class members in the upper three job categories was, when compared to an external labor pool, statistically insignificant. This determination led it to hold that the plaintiffs had failed to establish a prima facie case of pattern or practice discrimination.
A review of the statistical evidence found wanting by the district court must begin with a definition of the labor pools against whose composition the adequacy of class members’ representation in the City’s ranks is measured. There is perhaps no determination more critical to the ultimate resolution of a pattern and practice discrimination action than definition of that labor pool from which positions allegedly discriminato-rily assigned would properly be filled. See Wilkins at 395-99. Only in rare instances can an exact definition of the size and characteristics of the relevant labor market be attained. Id. at 398 n.13. However, a heightened sensitivity to the often-decisive effect of labor pool definition has led the courts to scrutinize carefully the definitions offered, and to require a reasonable semblance of accuracy in the definition ultimately accepted.
Of the many factors considered in definition of relevant labor pools, few have received more judicial attention than that of applicant qualifications. Although there are several variations on the approach to incorporation of applicant qualifications in labor pool definitions, the fundamental premise uniting all of the variations was articulated in
Hazelwood School District v. United
States,
One of the more significant refinements to the
Hazelwood
rule requires identification not only of the population having the requisite qualifications for the positions in question, but also identification of the source of qualified applicants to which the employer normally looks.
15
The source is almost always defined in part in geographical terms,
Hazelwood,
The plaintiffs have vigorously contended that their statistical evidence should be evaluated by the “internal comparison” approach. Pointing to undisputed evidence of a small percentage of class members in the City’s upper three job categories, a large percentage of class members in the City’s lower four job categories, and to the City’s ready concession that its policies favor filling positions by promotion, plaintiffs maintain that the labor pool from which candidates for upper level jobs are drawn is the City’s existing, lower level workforce. There is, admittedly, some superficial appeal to this argument. Evidence adduced at trial indicated that the percentages of Spanish-surnamed Mexican-Americans employed in the City’s various job categories in 1976 and 1977 were: 17
I II III IV v VI VII VIII
1976 0 1.87 1.56 2.47 30.76 10.53 10.90 20.44
1977 4.54 1.87 1.45 2.43 31.25 11.94 12.24 20.44
*542 The discrepancy is obvious. 18 That a discrepancy is obvious, however, does not necessarily lead to the conclusion that something is amiss. In this case, a closer scrutiny of the data leads only to the conclusion that no conclusions can be drawn from it.
The eight job categories listed above are titled:
I Official/Administrative
II Professionals
III Technicians
IV Protective Service (including fire and police)
V Para-Professional
VI Office/Clerical
VII Skilled Craft
VIII Service/Maintenance
The plaintiffs’ contention that Categories V through VIII provide the standard against which representation of class members in Categories I through III should be measured necessarily assumes, as discussed above, that employees presently classified in the former categories are eligible for promotion into the latter categories. As such an assumption would imply, the City does arrange its workforce into a hierarchical structure. It is first divided into departments, whose directors report to the City Manager, and then into divisions, whose superintendents report to department directors. Further distinctions exist within the divisions as is appropriate to the type of work the division performs. In the divisions of Utility Systems and Street Maintenance, on which the plaintiffs’ evidence focused, the bulk of the workforce is stratified as supervisors, foremen, crew leaders, senior laborers, and laborers.
The City’s eight job categories do not, however, represent sequential steps in that hierarchy. Category I, including the City Manager, department directors, and division superintendents, may properly be considered to contain “higher level” jobs. Categories II, III, V, VI, VII, and VIII, however, as their titles indicate, are distinguished from each other by the kind and function of jobs which they include, rather than by degree of responsibility within a particular function. Positions of greater responsibility within a particular function, up to and including what would commonly be considered middle management positions, are classified within that category which includes the entry level position subject to the higher level employees’ supervision. The Utilities Systems Division provides an example. Five of its six levels, from system worker through supervisor, are classified as category VIII jobs. Only the sixth level, Utility Systems Division Superintendent, is found in a “higher” category, i. e., Category I.
Against this background, the plaintiffs’ contention that the considerable proportion of Mexican-Americans observed in the lower categories of V through VIII should, then, be reflected in Category I, must be rejected. This argument fails to recognize that Category I positions, as the top rungs on the several ladders beginning in Categories V through VIII, would not be available to all employees classified in Categories V through VIII. Rather, they would be available only to those employees in Categories V through VIII who had reached within their respective divisions the highest level of progression encompassed within those “lower” categories. The entire employment of Categories V through VIII does not define the labor pool from which all Category I positions might be filled. At most, only a subset of Category V through VIII positions furnishes candidates for Category I jobs.
The plaintiffs produced no. evidence which defined the number of jobs within this subset of positions immediately below Category I, they did not identify the number of Mexican-Americans in these positions, and they did not identify the number of openings which have occurred in Category I superintendent positions since 1972, when the City became subject to Title VII. They provided only evidence of the total numbers and percentages of class members within Categories V through VIII. Compare Wilkins at 398 n.13. In light of the *543 plaintiffs’ failure to produce evidence properly identifying this subset, the district court’s conclusion that the City’s lower job categories do not provide the basis against which class members’ representation in Category I should be evaluated, must be affirmed.
The plaintiffs similarly urge that the adequacy of class members’ representation in Categories II and III is properly ascertained by an internal comparison to their‘rate of employment in Categories V through VIII. However, there is no support in the record for a finding that employees of Categories V through VIII would, in a normal course of progression, move into Categories II and III. Category II, Professionals, is primarily composed of data processing systems analysts, city planners, attorneys, public health nurses, sanitarians, and nutritionists. Category III, Technicians, includes data processing programmers, coders and system operators, engineers, surveyors, construction inspectors, and laboratory technicians. These are, as the district court observed, “positions which require special training for acquiring skills which persons in the general labor market do not possess or cannot readily acquire through minimal on-the-job training.”
Categories VI through VIII, by comparison, are primarily composed of, respectively, records clerks and typists, mechanics, and heavy equipment operators, and maintenance and construction workers. Category V, Para-Professionals, does appear to include employees whose skills are in the same areas as those of some Category II and III employees: it is composed of health aides, engineering aides and legal investigators. The plaintiffs, however, introduced no evidence whatsoever bearing on the qualifications required of, or actually held by, the City’s para-professionals, technicians and professionals. Id. There is in the record no basis for a determination of whether Category V employees would be eligible for Category II or III positions.
The plaintiffs, in an attempt to bridge the gap between the level of skills required of employees in Category II and III positions and of employees in Categories V through VIII, have adduced evidence that the City does make available some employee training. Company training programs, coupled with a corporate policy favoring promotion over recruitment, have in several cases made appropriate a comparison of minority representation in the organization’s higher echelons to minority employment among its rank and file.
See Fisher
at 544;
James
at 341; and
Robinson
at 659. In those cases, however, it was the employer’s policy and practice to hire unskilled workers into entry-level positions and to provide them with all training necessary for advancement into higher-level positions. As stated in
Fisher,
“[Wjhere skills are
commensurate
with the company training, we will approve statistical comparisons between [minority] make-up in key positions and [minority] composition in the total work force,”
The City readily concedes that its policy favors promotion over recruitment of new employees, and that it supports this policy by offering both opportunities for formal training and on-the-job training to its employees. Such facts do not, however, support the inference that plaintiffs would have us draw, that positions in Categories II and III would, by the City’s policy and design, be filled by employees in the lower four job categories. All evidence of the nature of the training made available by the City indicates that its purpose is to improve the employee’s skill at the type of job in which he is presently engaged, and not to train him in a different skill or qualify him for certification in a professional field. City-offered training teaches street maintenance workers to operate graders, and utility systems workers to read maps so that they may become utility systems foremen. It does not, nor is it required to, teach a utility systems foreman to be a surveyor or civil engineer. In the absence of the latter type of training, it cannot be said that professional and technical employees are drawn from the para-professional, clerical, mechanical, and maintenance workforces. The district court’s conclusion that an internal comparison would be here, too, inappropriate, is affirmed.
*544 The conclusion that an internal comparison is inappropriate does not, however, resolve the ultimate issue of whether discrimination exists in the City’s upper ranks. Such a conclusion means only that another, external labor market must be identified as the relevant comparative basis. Accordingly, the plaintiffs offered expert statistical testimony, in which the defendants concurred, that the external labor pools from which employees qualified for the first three job categories would be drawn could be either state, regional, or national pools.
Neither party, however, offered evidence of the number and minority compositions of persons in the regional or state labor pools which would supply applicants for the top three job categories. The court, therefore, on the concerted recommendation of both parties’ experts, used 1970 United States census data to derive the percentage of Spanish surnamed persons in the Wichita Falls Standard Metropolitan Statistical Area (SMSA) labor pool and the nationwide labor pool qualified for positions encompassed in the top three job categories. 19 Limitations inhering in the 1970 census data, however, required that the City’s job categories II and III, Professional and Technical employees, be analyzed in combination: the census reports cumulated census data on professionals and technicians into a single category. The court’s analysis was further limited by the plaintiffs’ failure to adduce, in any of the voluminous data and extended testimony which plaintiffs presented at trial, data on the total number of people and the number of Spanish sur-named Americans of Mexican descent hired by the City in the first eight months of 1978, in all of 1973, and in the last four months of 1972. This omission effectively withdrew from the court’s consideration the issue of the City’s hiring performance in two of the six years which it had determined the action properly to encompass. 20
Working within these constraints, the district court made the following findings upon comparison of employment in the City’s upper three categories with the relevant qualified labor pools:
36. The following table sets forth (a) the percentage of Spanish surnamed persons employed by the City in 1977 in EEOC job categories 1, 2, and 3; (b) the percentage of Spanish surnamed persons employed in the Wichita Falls SMSA in 1970 in EEOC job categories 1, 2, and 3; and, (c) the percentage of Spanish surnamed persons employed nationwide in 1970 in EEOC job categories 1, 2, and 3:
Job (a) City work (b) Wichita Falls Category force SMSA (rt Nationwide
1 4.5% 3.0% 2.0% 2 1.9% ) rt not ) 2 not S 1.4% ) 2M ) 2M
38. The following table sets forth the expected and observed numbers of Spanish surnamed persons hired by the City from *545 1974 through 1977 in EEOC job categories 1, 2, and 3. The expected numbers and the standard deviations are computed using 3% as the percentage of Spanish surnamed persons in the labor pool for EEOC job category 1 and 2% as the percentage of Spanish surnamed persons in the labor pool for EEOC job categories 2 and 3 combined.
Job Total Observed Expected Standard
Category Hires Sp.-sur. Hires Sp.-sur. Hires Deviation
1 14 0 .42 .64
2 & 3 146 1 2.92 1.69
On this basis, the district court concluded that plaintiffs failed to make out a prima facie case of discrimination.
The plaintiffs attack these findings with the single argument that the court erred in utilizing standard deviation analysis in its determination of the significance in the variation between the expected and the observed numbers of Spanish surnamed Americans of Mexican descent employed in the City’s top three job categories. The plaintiffs vigorously contend that any discrepancy unfavorable to the class emerging from a simple comparison of the percentages is evidence of discrimination adequate to support a prima facie ease. This argument flies in the face of rudimentary principles of sound statistical analysis. It is axiomatic that statistical significance may be attributed to an observed discrepancy only upon reduction to an acceptable level of the possibility that such a variation would, in the normal course of events, reasonably be expected to occur simply by random chance. Sullivan, Zimmer & Richards, supra n.7 at 71-80. Standard deviation analysis quantifies the likelihood of a benign explanation for an observed discrepancy; its use is essential where, as here, the discrepancy apparent upon a simple comparison of expected and observed occurrences is slight. 21
Nor does the plaintiffs’ argument find a basis in the case law defining the standard by which the adequacy of statistical proof of discrimination is to be measured. Judicial insistence on proper qualification of statistical proof is by no means new. The Supreme Court’s sophisticated statistical analysis in
Hazelwood,
*546
The district court’s conclusion that the plaintiffs failed to establish a prima facie case of discrimination against class members in the City’s upper three job categories is supported by the evidence. Only statistical proof was offered, and comparison between the expected and the observed numbers of class members in those positions cannot be said to evidence a “marked disproportion,”
Griggs
2. Promotion and Training
The plaintiffs alleged that the City’s job qualifications and employee assessment practices operated discriminatorily to deny promotions to class members. They produced, however, no statistical evidence on the progression — or lack thereof — of Spanish surnamed Americans of Mexican descent through the hierarchies of the divisions into which these class members were hired. They offered no evidence whatsoever, either statistical or testimonial, regarding the promotion of office and clerical workers, virtually all craft workers, and most service/maintenance employees. The only personal testimony produced regarding the City’s promotion practices related to the progression of Utilities Systems and Street Maintenance employees; with a single exception, this testimony was to the effect that Mexican-Americans received no fewer promotions than did any other class of employees. That exception was the testimony by .Raul Vasquez, one of two Mexican-American utility systems supervisors, 23 to his belief that the City looked more favorably on promoting minorities after Placido Rivera filed charges of discrimination with the Equal Opportunity Employment Commission in early 1973. Vasquez did not elaborate on his beliefs, nor did the plaintiffs pursue this avenue by either presenting evidence of specific instances of failure to promote, or introducing statistical evidence supporting an inference of an increased rate of promotion of class members after the EEOC charge was filed. Vasquez’ single bit of testimony, standing alone, is not sufficient to make out a prima facie case of failure to promote. The district court’s finding of an absence of evidence that the City’s promotion practices had a discriminatory impact is affirmed.
The same failure to present evidence dooms the plaintiffs’ attack on the district court’s finding of an absence of discrimination by the City in training its employees. What evidence there was of City-provided training indicated that on-the-job training was given to laborers, mechanics, equipment operators, and clerical employees; that technical, professional, and administrative employees were offered the opportunity to attend training seminars relating to the fields in which they worked; that the City made a college tuition subsidy program available to all employees; and that the City encouraged all utilities systems workers to obtain various certifications, but did not itself offer or subsidize related courses. All testimony, including that by the plaintiffs’ witnesses, indicated that this training was made equally available to all employees who by virtue of their position were qualified for it.
The plaintiffs did offer documentary evidence which indicated that only three percent of all City employees participating in seminars offered from the time of the City’s inclusion under Title VII in 1972 until trial in August 1978 were class members. The plaintiffs failed, however, to put this information into context. They did not show to which fields of employment the seminars were relevant, for which levels of employees the seminars were designed, or the proportion of class members in these fields and at the appropriate levels. Standing alone, this statistic has no probative value.
The plaintiffs contend that the City discriminates by offering seminars to its professional and technical employees while offering only on-the-job training to laborers, equipment operators, and clerical employees. There is, however, no requirement *547 that the City offer any particular form of training — or, for that matter any training at all — to its employees. It is required only to offer what training it chooses to make available on an even-handed, nondiscriminatory basis. This, it appears, it does. There is no basis on which a finding of discrimination in training could be rested. The district court is affirmed.
III.
The district court concluded that the only claim upon which plaintiffs prevailed was their class action claim that the City Fire Department discriminated in hiring class members. After analyzing and balancing the factors identified as guidelines in
Johnson v. Georgia Highway Express, Inc.,
IV.
We add in closing an echo of an observation well taken by a recent panel of this Court. Private enforcement of the employment discrimination laws is an essential component in the collective effort to rid our society of the debilitating effects of racial, ethnic, and sexual prejudice. When used in conjunction with a class action, the power of the laws is magnified many times over— but therein lies the danger, as well as the good. These actions determine the rights of untold numbers; frequently, those affected are wholly unaware of even the existence of the proceeding. The burden borne by those who would assert these interests has grown heavier in recent years, as the forms in which discrimination appears become more varied, and more subtly interrelated. The days — if they ever existed — of casual renditions of “broad conclusions from crude and incomplete statistics,” Wilkins at 410, are indeed long since past. The successful prosecution of class-wide employment discrimination claims demands considered and refined statistical analysis by counsel as well versed in the learning of mathematicians as in their own. The legal principles we apply today are familiar ones. Their invocation, particularly where the sought-after adjudication determines, for better or for worse, the interests of an undetermined count of unidentified people, must be with care and caution, unless the citizen’s sword against discrimination in the workplace would be turned unwittingly against the citizen.
The judgment of the district court is affirmed.
AFFIRMED.
Notes
. The City officials named as defendants are the City Manager, Mayor, Board of Aldermen, Chief of Police, and Personnel Director. All individual defendants are sued in their official capacity. The City Manager, Gerald G. Fox, and the Chief of Police, Curtis Harrelson, are sued as well in their individual capacity. Plaintiffs, however, seek only injunctive relief against them individually. All claims for exemplary damages, and all claims for back pay except those against the City itself, were waived before trial.
. The district court found that the Fire Department’s hiring practices had a discriminatory impact on Spanish surnamed Americans of Mexican descent, in that they discouraged class members’ applications for positions with the Fire Department. The plaintiffs did not challenge the actual qualifications which the City required of its firemen. The court ordered the City to develop a program for recruitment of Mexican-American applicants for the Fire Department and to publicize in the newspaper, on radio, and on television a notice to all Spanish surnamed persons in the Wichita Falls area that they may be
entitled to a cash award and/or employment based on your successfully making the following showing:
1. That you would have applied for a job as a fireman with the Wichita Falls Fire Department since September 10, 1972, but for its discrimination against Spanish-surnamed persons;
2. That you were qualified for a job as a fireman with the Wichita Falls Fire Department at the time you would have applied.
The City fulfilled this notice requirement. No class member applied for relief.
. The class claims originally included charges of harassment and retaliation by the City against class members who filed EEOC claims against it, and of discriminatory termination. These claims were abandoned, leaving as the class-wide issues discrimination in the City’s general recruiting, hiring, training, and promotion practices, and in its hiring practices for supervisory and administrative positions.
Of the six original individual plaintiffs and intervenors, only one actually tried his claim before the district court. Plaintiff-intervenors Rivera, Castillo, Pete Gonzales, and Maldonado *534 through their attorneys “waived” all claims, and Ceballos and Richard Gonzales “waived” their claims for monetary relief during the course of the trial. These “waivers” reduced Richard Gonzales’ claim for relief to a demand for reinstatement, and Ceballos’ claim for relief to a demand for admission to the police academy and seniority. Gonzales later stated, however, under questioning by his attorney, that he preferred his current job to his old job with the City. The court thereupon granted his request for a “waiver” of his claim to reinstatement. He thus effectively “waived” the entirety of his claim. Only Ceballos’ claim was tried.
. The plaintiffs also alleged violations of § 1981 of the Civil Rights Act of 1866, 42 U.S.C. § 1981, and § 1983 of the Civil Rights Act of 1871, 42 U.S.C. § 1983. Consideration of these alternative remedies for employment discrimination is necessary only if their violation can be made out on grounds different from those available under Title VII. The plaintiffs have not argued that such distinctions exist. Moreover, the case law controlling in this Circuit indicates that the elements of substantive claims of employment discrimination brought under § 1981 and § 1983 parallel those of claims brought under Title VII.
In
Williams v. DeKalb County,
Neither in this instance would § 1983 appear to provide an avenue for the prosecution of discrimination claims different from that available under Title VII. This Court has held that when § 1983 is used as a parallel remedy with Title VII in a discrimination suit, the elements of the substantive cause of action are the same under both statutes.
Whiting v. Jackson State University,
. A showing of “marked disproportion” between the representation of the allegedly disfavored group in the employer’s workforce and its representation in the labor market from which the employer hires suffices to establish a prima facie case of “disparate impact” discrimination.
Dothard,
Throughout this action, the plaintiffs have insisted that they allege, as well, a theory of “disparate treatment” arising from the City’s challenged patterns and practices. “Disparate treatment” denotes, in many contexts, an alternative approach to proof of employment discrimination. It may, but need not be, used in conjunction with allegations of pattern or practice discrimination, id. at 1854 n.15 (noting in the context of a “pattern or practice” discrimination case that either theory, of disparate impact or disparate treatment, may be applicable to a particular set of facts). Where it is not, it may be used to reach “isolated ... or sporadic discriminatory acts,” id. at 1854 n.15, 1855, 1855 n.16. The crux of it in all cases, however, is that the discriminatory act was purposeful.
When it is used in an action alleging pattern or practice discrimination, it reflects the plaintiff’s contention that the employer intended that its employment practices have the observed discriminatory effect. Allegations under a “disparate treatment” theory thus serve either to require direct proof of the employer’s intent through testimony to vignettes of discriminatory acts,
see Teamsters,
In this case, the plaintiffs have relied wholly on statistical proof. Accordingly, in determining whether they have made out a prima facie case of discrimination, it is appropriate in the first instance to review the evidence for the lesser degree of statistical disparity adequate to support a disparate impact action. Statistical proof failing to show a “marked disproportion,”
Griggs,
. The City began using BOLPO in conjunction with the other three elements of its police recruit selection procedure in December 1974. The plaintiffs’ statistical evidence of the class members’ comparative performance on the several elements of the selection process extended through May 1978, rather than through the time of trial. The district court properly considered May 1978 to define the close of the period under examination.
. In the evaluation of a testing procedure, the absence of discriminatory impact is defined as the occurrence of a uniform failure rate across all racial and ethnic groups. If the BOLPO failure rate experienced by white applicants, 5.4 percent, is assumed to be the designed failure rate and is, accordingly, applied as the norm, 1.89 of 35 Mexican-American applicants, with a standard deviation of 1.345, would be expected to fail BOLPO. The difference between the expected value of 1.89 failures and the observed value of 14 failures is approximately nine standard deviations. A difference of more than two or three standard deviations is generally considered to raise a compelling inference of discrimination.
Casteneda v. Par-tida,
We recognize that the application of probability theories to samples as limited as 35 is, of itself, problematic.
Cf. Wilkins v. University of
*537
Houston,
. Because only one class member both completed the selection process and accepted employment as a police recruit, the sample of applicants with which to revalidate BOLPO was too small up to the time of trial. While lack of an adequate applicant pool in Wichita Falls at the time of validation may justify using incumbent police officers when validating BOL-PO for purposes of its adoption, once an adequate applicant pool has developed, revalidation should be undertaken for purposes of continued use of BOLPO. See 18 C.F.R. § 1607.-5(K).
. The report also reviewed a “concurrent” criterion-related validity study which attempted to measure the correlation of success on the tests with success as a police officer. The criterion against which BOLPO scores were here compared was a ranking of police officers by their immediate supervisor from best qualified to least qualified. The correlation between the supervisory ranking and test scores was modest, only .38, but still statistically significant. The report concluded on the strength of this correlation that BOLPO was a useful measure of police performance.
This study was conducted with a sample of incumbent police officers from Wichita Falls. A differential validity study, in which the results of tests compared with job performance are separately reported for minority and non-minority groups, was not feasible and therefore not undertaken because there was not a sufficient sample size for protected classes.
Plaintiffs attack this aspect of the “criterion-related validity” study because it gives no indication if any guidelines were given for ranking
*538
by the supervisors, if the supervisors had the test scores or had access to them, or what if any controls were used to protect the accuracy of the evaluation process. This raises the danger of “criterion contamination”: the criterion measure is not independent of, but is influenced by, the test score or supervisor bias. See
Albemarle,
. Plaintiffs argue that the correlation established at other academies cannot be used to validate the test for use at the Wichita Falls Police Academy. However, validity studies conducted by one test user will justify use of that test by others when the jobs of the two users are substantially similar and the test was clearly validated by the first user. 29 C.F.R. § 1607.7.
. The physical agility test had no disparate adverse impact on class members. By plaintiffs’ own reckoning 2.85% of class member applicants failed the physical agility test from December 1974 to May 1978, while 19.9% of white applicants failed the agility test during the same period.
The background investigation resulted in the elimination of 21.8% of the white applicants from December 1974 to May 1978. During that same period, 25% of the class member applicants were eliminated on the basis of the background investigation. We cannot say that the district court erred in holding that the investigation did not have a disparate impact on employment discrimination of class members as police recruits.
The assessment center evaluation resulted in the rejection of 15% of the white applicants, and 20% of the class member applicants between December 1974 and May 1978. The district court held that this did not have a disparate impact on the employment of class members as police recruits. We affirm.
. Ceballos was twice refused admission to the police academy. Both times he filed charges with the EEOC. The first charge, filed in March 1974, alleged that the department’s height requirement, under which Ceballos had been rejected, had a discriminatory impact on Mexican-Americans. In due course, a right to sue letter issued on this charge. However, the charge became moot before it was tried, because when the City lifted the height requirement in February 1974, Ceballos reapplied, and was again rejected. In April 1974 he filed a second charge with the EEOC, alleging that his second rejection was attributable to a discriminatory impact which the City’s screening devices, particularly the BOLPO test and the background investigation, had on Mexican-Americans.
. The plaintiffs characterized the- rate of employment of class members in the City’s lower job categories as “overrepresentative” by comparison of these rates not only to the rate of employment in the higher job categories but also to the percentage of class members residing in the City of Wichita Falls. The district court found that 5.7% of the population in the Wichita Falls Standard Metropolitan Statistical Area is Spanish surnamed.
. In making these comparisons the plaintiffs disregarded Category IV, Protective Services. Their reasons for doing so were not articulated *540 in either their brief on appeal or in the record below. It may be that they believed that their particularized attacks on the recruiting and screening processes of Category IV’s component parts, the Police and Fire Departments, adequately aired all mechanisms of discriminatory practices operating in that category. Or they may have recognized that the general employment of Category IV bears, at best, a very limited relationship to employment in Category I, and no apparent relationship at all to employment in Categories II and III. If the latter is the case, the plaintiffs’ reasons for omission of Category IV from their “internal comparison” presaged our own analysis of the relationship of Categories V through VIII to Categories I, II, and III.
. An employer cannot, of course, limit the number of minorities in his employ simply by recruiting and hiring from predominantly white areas. In such a case, the relevant labor market would be considered to be that from which the employer, in the absence of discriminatory bias, would be expected to draw.
Markey v. Tenneco Oil Co.,
. A prima facie case may, of course, be shown without evidence of qualifications where the inference of discrimination is supported by a compelling level of minority underrepresentation in a sizeable workforce. In such a case, the burden of proving lack of qualifications is on the company.
United States v. Hayes International Corp.,
. The number of Spanish-surnamed Americans of Mexican descent in each job category, as compared to the total number of employees in each job category, for the two years in question was:
I II III IV V VI VII VIII
1976 0/22 2/107 1/64 5/213 4/13 14/133 12/110 58/282
1977 1/22 2/107 1/69 5/206 5/16 16/134 12/98 ■ 56/274
The plaintiffs presented data on the number of class members employed in each job category and the total number of employees in each job category for each of the years from 1973 through 1977. The latter two years are representative of the pattern of Mexican-American employment seen in the City’s workforce throughout the five-year period.
. The change in the first category was occasioned by the employment of one class member in that category, apparently by promotion, out of a total employment of 22 persons in that category.
. The plaintiffs did not advocate without reservation the use of the 1970 census data. Their statistical expert testified to several weaknesses in the data. One was that separate statistics on Mexican-Americans were not cumulated in the 1970 census; rather, Mexican-Americans, Cubans, Puerto Ricans, and all others with Spanish surnames were counted together as “Hispanics.” Another was that Government investigations into the accuracy of the 1970 census indicated that Mexican-Americans, like most minorities, were substantially under-counted. Yet another was that the census statistics analyzing the population by job skill included in each skill category only people actually employed in those skill categories. People qualified for, but not employed in, such positions were omitted from the statistics quantifying the proportion of the population eligible for the type of employment in question.
These are not insignificant defects. The plaintiffs did not, however, produce any other, more reliable, statistics on the number and qualifications of Mexican-Americans in either the state or national labor markets. Particularly in the absence of their suggestion of a reasonable alternative, they will not be heard to disparage the probative value of evidence which they presented in their case-in-chief, and which was offered for precisely the purpose to which it was put by the court.
. The district court certified the named plaintiffs to represent the class for the period beginning “180 days prior to the date of the earliest filing of a named Plaintiffs EEOC charge (March 9, 1973) to the present [time of trial].” So defined, the period relevant to determination of the class-wide claims ran from September 10, 1972 through trial in mid-August of 1978.
. Of course, where the disparity apparent upon simple comparison of percentages is compelling, c.
f. Hazelwood,
.
Castaneda
and
Hazelwood
were both cases in which the plaintiffs sought to establish through statistical evidence that they had been the victims of purposeful discrimination.
Castaneda
arose directly under the equal protection clause , of the fourteenth amendment;
Ha-zelwood
alleged purposeful discrimination in violation of Title VII. These cases indicated that an inference of purposeful discrimination fairly could be drawn “[a]s a general rule for such a large sample, if the difference between the expected value and the observed number is greater than two or three standard deviations,”
Hazelwood,
Where the issue is of the existence of a disparate impact, without regard to whether the adverse effect was actively sought, it is possible that a lesser disparity may lend support to an inference that the challenged practice tends to exclude the defined class of persons at a rate higher than the norm. Statistics such as those here presented, exhibiting a disparity of less than, and slightly more than, one standard deviation, do not, however, exhibit a “marked disproportion,”
Griggs
. Of five Utility Systems Supervisors employed by the City at time of trial, two were Mexican-American, one was American Indian, and two were white.