
    Jon F. BARNES, Homer L. Gibson, Jean Z. Gipson, Clarence P. Kennedy, Willy P. Kulhanek, Norman J. Muth, Febo C. Spagnuolo (89-3104), Plaintiffs-Appellants, v. GenCORP INC., Defendant-Appellee. Theophilos A. MILLIS (89-3192), Richard S. Novitsky (89-3193), Milan A. Rolik (89-3194), Norman S. Trommer (89-3195), Plaintiffs-Appellants, v. DIVERSITECH GENERAL, INC., Defendant-Appellee.
    Nos. 89-3104, and 89-3192 to 3195.
    United States Court of Appeals, Sixth Circuit.
    Argued Dec. 7, 1989.
    Decided Feb. 22, 1990.
    As Amended April 30, 1990.
    Rehearing and Rehearing En Banc Denied In Nos. 89-3104 and 89-3193 April 30, 1990.
    
      Richard V. Levin (argued), Akron, Ohio, Laurie F. Starr, William J. Novak, Sindell, Rubenstein, Einbund, Pavlik, Novak & Ce-lebrezze, Cleveland, Ohio, for plaintiffs-appellants.
    Lee J. Hutton, David J. Somrak, Thomas H. Barnard (argued), Duvin, Cahn & Barnard, Cleveland, Ohio, William Gorenc, Jr., GenCorp, Fairlawn, Ohio, for defendants-appellees.
    Before KEITH and KENNEDY, Circuit Judges; and PECK, Senior Circuit Judge.
   KENNEDY, Circuit Judge.

In this consolidated appeal, plaintiffs-appellants challenge the District Court’s grant of summary judgment for the defendants-appellees, their employers, in these age discrimination cases. The cases arose from a corporate restructuring in which a number of employees, including the plaintiffs, were permanently discharged. The plaintiffs assert that they created a jury question by establishing a prima facie case of age discrimination under McDonnell Douglas Corp. v. Green, 411 U.S. 792, 93 S.Ct. 1817, 36 L.Ed.2d 668 (1973), and by setting forth facts to rebut the defendants’ alleged nondiscriminatory reasons for the discharges. The defendants argue that the District Court correctly found that no pri-ma facie case was established and, in the alternative, argue that the plaintiffs have not rebutted the proffered nondiscriminatory reasons for the discharges. We AFFIRM in part, REVERSE in part, and REMAND for further proceedings.

I. BACKGROUND

A. General Information

In March 1987, GenCorp was the target of a hostile tender offer by General Acquisition, Inc. GenCorp defended against this takeover attempt by instituting a $1.4 billion stock repurchase program financed by the sale of two GenCorp subsidiaries — General Tire, Inc., a tire manufacturing operation and GenCorp’s largest subsidiary, and RKO Bottling, a soft drink bottling operation.

Prior to the attempted hostile takeover, GenCorp maintained a Research Division headquartered in Akron, Ohio where seven of the plaintiffs were employed. The Research Division conducted research in three areas: (1) tire technology; (2) projects associated with GenCorp’s automotive and polymer-based products; and (3) exploratory and support research. Following the divestiture of its largest subsidiary, GenCorp determined that it should curtail research activity and reduce the size of its Research Division staff. As a result, seven of the plaintiffs, as well as thirty-one other employees, were discharged. The specific decisions on which Research Division employees were to be eliminated were made by upper management employees in the Research Division, including Russell Livigni, Director of the Research Division.

GenCorp’s other subsidiaries, including DiversiTech, were also affected. Diversi-Tech instituted numerous cost-cutting measures including a hiring freeze, suspension of all training seminars, a 25% reduction in travel expenses, a 50% reduction in overtime, and the termination of all temporary help. Despite these measures, management was still required to reduce the payroll. The four DiversiTech plaintiffs and nine other employees were discharged after DiversiTech management determined that the positions occupied by the plaintiffs could be eliminated with the least disruption to company operations. The specific decisions were apparently made by Polymers Division President Howard Wheeler, who relied in part on recommendations made by upper-level management employees.

At both GenCorp and DiversiTech, no discharged employee whose position was eliminated was allowed to displace any other employee regardless of his or her relative seniority. The plaintiffs describe the so-called no-bump policy as an eleventh-hour change of position designed to enable the company to complete its discriminatory scheme. Both defendants deny that they had a bumping policy prior to the restructuring, but the plaintiffs have produced at least some evidence from which a jury could conclude that a policy existed. Gen-Corp cites the testimony of Russell Livigni, Director of the Research Division, who stated that the Division had never applied a bumping policy during his three-year tenure. Fred A. Deaner, GenCorp’s Director of Staffing, Development, and EEO, stated in his deposition that GenCorp did not have a “published policy” to allow bumping but did have a “practice” of allowing more senior employees to bump less senior employees. According to Deaner, the practice was eliminated “just prior” to the restructuring. While Deaner stated later in his deposition that he believed the policy was eliminated much earlier, this only creates a jury issue.

Similarly, there is some support for the existence and withdrawal of a bumping policy at DiversiTech. Richard D. First, DiversiTech’s Director of Personnel, testified that while DiversiTech had no bumping policy, it did circulate a document to company personnel shortly before the reorganization stating that no bumping would be allowed. He explained that this step was taken because bumping had been allowed during a factory closure in 1981 or 1982.

B. Statistical Proof

The plaintiffs presented two statistical reports — one for the GenCorp plaintiffs and one for the DiversiTech plaintiffs — prepared by Harvey S. Rosen and John F. Burke, Jr. The defendants do not here challenge their credentials as expert statisticians. The two reports analyzed the termination patterns of the employees who were discharged during the force reduction. Both reports utilized information provided by the defendants.

1. GenCorp

In the GenCorp cases, Rosen and Burke calculated the termination rates in Gen-Corp’s Research Division for employees age 48-and-over, 52-and-over, and 55- and-over, and for employees under ages 48, 52, and 55. The data included the number and percentage of employees in each age group in the Research Division prior to the work force reduction. The actual termination rates for each group were compared to the expected termination rates had employees been selected randomly for termination.

Data based on the 153 total staff members in the Research Division, 38 of whom were fired, revealed that the 48-and-over age group comprised 43.1% of the total employee population but 63.2% of those discharged. Twenty-four of the 66 employees, 36.4%, age 48-and-over were discharged. But only 14 of the 87 employees, 16.1%, under age 48 were discharged. The 52-and-over age group comprised 28.8% of the total employee population but 50% of those discharged. Nineteen of the 44 employees, 43.2%, age 52-and-older were eliminated. Only 19 of the 109 employees, 17.4%, under age 52 were discharged. The 55-and-over age group comprised 19.6% of the total employee population but 42.1% of those discharged. Sixteen of the 30 employees, 53.3%, age 55-and-over were eliminated. Only 22 of the 123 employees, 17.9%, under age 55 were discharged. This data was presented to the District Court in a series of charts divided by age group. The significance of the data was explained in a cover letter to the plaintiffs’ lawyers, a copy of which was submitted to the court.

In all but the age-48-and-over grouping, Rosen and Burke stated that “the Chi-Square statistic fell beyond three standard deviations from the hypothesized random result, indicating significantly higher-than-expected termination levels in the older age groups.” The age-48-and-over grouping “produced a Chi-Square sufficiently close to three standard deviations that it yielded a probability or confidence level of 99.6 percent, compared with 99.7 percent for three standard deviations.” Rosen and Burke concluded their report by stating: “Based upon the foregoing analysis, I conclude that, to a virtual statistical certainty, age was a determining factor in the termination of the employees in each of the indicated groupings.”

2. DiversiTech

Similar statistics were prepared in the DiversiTech cases for the 165 Polymers Division employees age 55-and-over, age 60-and-over, and those under ages 55 and 60. A separate analysis was prepared for the 109 employees who worked in departments of the Polymers Division in which terminations occurred. This latter analysis presented less of a disparity between the termination rates of the older and younger groupings. Data based on the smaller 109 employee grouping revealed that the 55- and-over age group comprised 22% of the total employee population but 76.9% of those discharged. Ten of the 24 employees, 41.7%, age 55-and-over were discharged. Only 3 of the 85 employees, 3.5%, under age 55 were discharged. The 60- and-over age group comprised 7.3% of the total employee population but 53.8% of those discharged. Seven of the 8 employees, 87.5%, age 60-and-over were discharged. Only 6 of the 101 employees under age 60, 5.9% were discharged.

In a report accompanying the statistics, a Chi-Square statistic and a binomial distribution were obtained for the groups analyzed. Rosen and Burke concluded, “[i]n all of the calculations, the Chi-Square statistic reached virtually the 100% level of significance, indicating significantly higher than expected termination levels in the older age groups.” Because the small number of employees age 60-and-over decreased the reliability of these statistics, Rosen and Burke confirmed the results of every group using a binomial analysis and found that “[i]n all four instances, the actual number of terminations in the stated categories deviated by more than four standard deviations from the expected terminations.” Rosen and Burke again concluded their report by stating: “Based upon the foregoing analysis, I conclude that, to a virtual statistical certainty, age was a determining factor in the termination of the employees in each of the indicated groupings.”

C. Facts Related to Individual Cases

The facts related to each individual plaintiffs discharge are presented in Section III(C), infra.

II. PRIMA FACIE CASE

A. Standard

The plaintiffs argue that they have established a prima facie case of age discrimination under the standard established in McDonnell Douglas, 411 U.S. at 792, 93 S.Ct. at 1819. In a broad sense, McDonnell Douglas set forth an analytical method to examine every intentional discrimination claim, whether it proceeds through an indirect method of proof or through direct, circumstantial, or statistical evidence.

First, the plaintiff has the burden of proving by the preponderance of the evidence a prima facie case of discrimination. Second, if the plaintiff succeeds in proving the prima facie case, the burden shifts to the defendant “to articulate some legitimate, nondiscriminatory reason for the employee’s rejection.” Third, should the defendant carry this burden, the plaintiff must then have the opportunity to prove by a preponderance of the evidence that the legitimate reasons offered by the defendant were not its true reasons, but were a pretext for discrimination.

Texas Dept. of Community Affairs v. Burdine, 450 U.S. 248, 252-53, 101 S.Ct. 1089, 1093, 67 L.Ed.2d 207 (1981) (citations omitted).

In a narrower sense, however, the importance of the McDonnell Douglas “test” is its discussion of the elements a plaintiff must prove to establish a prima facie case of discrimination absent direct, circumstantial, or statistical evidence of discrimination. In McDonnell Douglas, the Supreme Court held that a plaintiff alleging racially motivated refusal to hire could establish a prima facie case by showing, inter alia, that he was qualified for the position and that it remained open after the employer refused to hire him. The rationale for not requiring direct, circumstantial, or statistical evidence of discrimination has been explored in later cases:

Although the McDonnell Douglas formula does not require direct proof of discrimination, it does demand that the alleged discriminatee demonstrate at least that his rejection did not result from the two most common legitimate reasons on which an employer might rely to reject a job applicant: an absolute or relative lack of qualifications or the absence of a vacancy in the job sought. Elimination of these reasons for the refusal to hire is sufficient, absent other explanation, to create an inference that the decision was a discriminatory one.

International Brotherhood of Teamsters v. United States, 431 U.S. 324, 358 n. 44, 97 S.Ct. 1843, 1866 n. 44, 52 L.Ed.2d 396 (1977).

We do not believe the same rationale applies to a work force reduction where the plaintiff has done no more than show the elements of the McDonnell Douglas formula that relate to his or her situation: (1) that he or she was age forty or over; (2) that he or she was qualified to perform the job; and (3) that he or she was discharged. When work force reductions by the employer are a factor in the decision, “the most common legitimate reasons” for the discharge are the work force reductions. By showing the other elements of a McDonnell Douglas ease, a plaintiff has not presented any evidence indicating that the work force reductions are not the reasons for the discharge and therefore does not make out a prima facie case absent additional direct, circumstantial, or statistical evidence tending to indicate that the employer singled out the plaintiff for discharge for impermissible reasons. LaGrant v. Gulf and Western Mfg. Co., 748 F.2d 1087, 1090 (6th Cir.1984) (“[t]he mere termination of a competent employee when an employer is making cutbacks due to economic necessity is insufficient to establish a prima facie case of age discrimination”); McMahon v. Libbey-Owens-Ford Co., 870 F.2d 1073 (6th Cir.1989); Sahadi v. Reynolds Chemical, 636 F.2d 1116, 1118 (6th Cir.1980).

Our conclusion would not change even if a plaintiff additionally demonstrated that younger persons were retained in other jobs which the plaintiff was qualified to perform. A different result would allow every person age 40-and-over to establish a prima facie case of age discrimination if he or she was discharged as part of a work force reduction. Sahadi, 636 F.2d at 1118.

It is important to clarify what constitutes a true work force reduction case. A work force reduction situation oo-curs when business considerations cause an employer to eliminate one or more positions within the company. An employee is not eliminated as part of a work force reduction when he or she is replaced after his or her discharge. However, a person is not replaced when another employee is assigned to perform the plaintiffs duties in addition to other duties, or when the work is redistributed among other existing employees already performing related work. A person is replaced only when another employee is hired or reassigned to perform the plaintiffs duties. See Sahadi, 636 F.2d at 1117 (“[pjlaintiff was not replaced; his former duties were assumed by Alexander, who performed them in addition to his other functions”).

We do not believe this standard creates an undue hardship for age discrimination plaintiffs who are eliminated as part of a work force reduction. If the plaintiff was truly singled out for discharge because of age he or she should be able to develop enough evidence through the discovery process or otherwise to establish a prima facie case. For example, a plaintiff could establish a prima facie case by showing that he or she possessed qualifications superior to those of a younger co-worker working in the same position as the plaintiff. Alternatively, a plaintiff could show that the employer made statements indicative of a discriminatory motive as in Laugesen, 510 F.2d at 313. The guiding principle is that the evidence must be sufficiently probative to allow a factfinder to believe that the employer intentionally discriminated against the plaintiff because of age.

We turn then to the question of whether any or all of these plaintiffs have set forth sufficient direct, circumstantial, and/or statistical evidence to establish a prima facie case.

B. Statistical Proof

The plaintiffs initially argue that the statistical evidence they have presented is sufficient to create an inference of discrimination as to all of them. We agree. Appropriate statistical data showing an employer’s pattern of conduct toward a protected class as a group can, if unrebutted, create an inference that a defendant discriminated against individual members of the class. Laugesen, 510 F.2d at 317. See also McDonnell Douglas, 411 U.S. at 804-05, 93 S.Ct. at 1825 (defendant’s general policy and practice with respect to minority employment may be relevant to showing of pretext). To do so, the statistics must show a significant disparity and eliminate the most common nondiscriminatory explanations for the disparity. 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). Since all of the discharged employees are arguably qualified in work force reductions, the most obvious explanations for the discharge of any one employee are lower proficiency and/or random chance. Since it is reasonable to presume at this stage of the case that skill is distributed randomly over any given age group and since the plaintiffs have shown that the results depart significantly from those that chance alone would predict, we believe that the statistics, on their face, establish a prima facie case of discrimination. See, e.g., Castaneda v. Partida, 430 U.S. 482, 497 n. 17, 97 S.Ct. 1272, 1281 n. 17, 51 L.Ed.2d 498 (1977) (“[a]s a general rule for such large samples, if the difference between the expected value and the observed number is greater than two or three standard deviations, then the hypothesis that the jury drawing was random would be suspect to a social scientist”); NAACP v. City of Mansfield, 866 F.2d 162, 168 (6th Cir.1989). See also Simpson v. Midland-Ross Corp., 823 F.2d 937, 944 (6th Cir.1987) (“ ‘both the methodology and the explanatory power of the statistical analysis must be sufficient to permit an inference of discrimination’ ”) (citation omitted).

The defendants attack the plaintiffs’ statistics on several grounds. First, the defendants argue that the plaintiffs have artificially manipulated the statistics by choosing meaningless age categories— 48, 52, and 55-and-over versus those under 48, 52, and 55 for the GenCorp plaintiffs and 55 and 60-and-over versus those under 55 and 60 for the DiversiTech plaintiffs. The defendants argue that the only valid statistics would necessarily divide the employees into groups age 40-and-over and those under 40. We disagree. An employer violates ADEA when preference is given to a younger employee even if the younger employee is within the protected class of persons age 40-and-over. See, e.g., McCorstin v. U.S. Steel Corp., 621 F.2d 749, 754 (5th Cir.1980) (requiring a plaintiff to show that he or she was replaced by someone outside of the 40-and-over protected group “fails to take the reality of the working place into account. Because of the value of experience rarely are sixty-year-olds replaced by those under forty”); Moore v. Sears, Roebuck and Co., 464 F.Supp. 357, 366 (N.D.Ga.1979) (“[bjecause age is a relative rather than absolute status when taken as a basis for discrimination, it need not follow that all persons protected by the act be grouped together for purposes of delineating the extent of their protection”). We believe that the plaintiffs are correct when they assert that the age groups are relevant on their face. If employees younger than a specific plaintiff have a statistically significant lower discharge rate, then it is obvious that the statistics — absent another explanation for the discharge — are probative of discrimination.

Second, the defendants assert that the sample size is too small for the statistics to provide reliable results. They argue that a small change in the underlying data would produce a dramatic change in the outcome. The defendants have provided no evidence to support this assertion and, in any case, we believe this assertion, at least on its face, is incorrect. Further, the plaintiffs’ expert has asserted that the statistical pool is sufficient in size to render the results statistically reliable. At best the defendants’ unsubstantiated assertion raises a question that cannot be resolved on this record. See, e.g., Capaci v. Katz & Besthoff, Inc., 711 F.2d 647, 653-54 (5th Cir.1983) (“[t]he defendant must do more than raise theoretical objections to the data or statistical approach taken; instead, the defendant should demonstrate how the errors affect the results”).

Finally, the defendants argue that the assumptions used in framing the statistics remain unexplained. More specifically, the defendants argue that: (1) neither group of plaintiffs explains why statistics were prepared for the age groupings chosen; (2) the plaintiffs do not explain why the GenCorp set of statistics provide data excluding top management yet the DiversiTech data always includes top management; (3) the DiversiTech plaintiffs do not explain why the whole company, with the exception of a Texas plant, is used rather than a subset of the company; and (4) the GenCorp plaintiffs do not explain why the statistics were limited to the Research Center rather than the company as a whole. Since the plaintiffs have failed to answer these questions, the defendants argue, this Court should presume that the statistics were manipulated to produce “a preordained and partisan result.”

Most of these questions can be answered from the facts of the case themselves. The first question is merely a restatement of a question already raised and answered above. The answer to the second question is largely irrelevant since the statistics are also calculated including the management employees. As to the third question, the Texas plant was not included because a decision had already been made to transfer the plant from DiversiTech to another subsidiary of GenCorp, its parent company. Employees at this plant were apparently not considered in the restructuring.

The fourth question and the remainder of the third question challenge the relevance of including only the Polymers Division employees in the statistics for the Diversi-Tech cases and only the Research Division employees in the statistics in the GenCorp cases. Again the relevance of these statistics is apparent. As to the DiversiTech plaintiffs, all worked in the Polymers Division located in the Akron area. Apparently the decisions as to which employees were to be cut were made by the top management of the Polymers Division. Similarly, all of the GenCorp plaintiffs worked in the Research Division of GenCorp. The decisions as to which employees were to be cut by the Research Division were made by top management employees of this Division. In both cases we cannot say that it is illogical to include all of the employees who could have been considered by the management personnel involved in the decision.

The only serious problem with the statistics in this case — a problem not clearly raised by the defendants — is that all of the employees in each division are not interchangeable. Only a few employees occupy positions comparable to those occupied by each of the plaintiffs. While this diminishes the probative value of the evidence, we do not believe that it vitiates it. The discharge decisions were made by persons who had control over all of the division employees, not merely the employees who worked in each plaintiffs group. And while the statistical disparity for any one group would not be significant because of the sample size, it remains significant that the older employees were discharged in each of several groups.

III. PRETEXT

The defendants have set forth legitimate, nondiscriminatory reasons for discharging each of the eleven plaintiffs. Each individual plaintiff was chosen for discharge because his or her position was judged to be less essential to the company’s continued operations and because other individuals in the same position, if any, were judged to be better qualified. These reasons are discussed with greater specificity as to each plaintiff in Section III(C), infra, along with a discussion of the individual rebuttal evidence that individual plaintiffs have presented.

The plaintiffs allege that the asserted reasons for the discharges are pretextual. In the alternative, the plaintiffs argue that, even if the nondiscriminatory reasons are true, the defendants would not have acted absent an additional discriminatory motive. Thus the plaintiffs argue that the case could be considered one of mixed motives. The plaintiffs do not question the fact that the defendants discharged a large number of employees because of the reorganization. They merely assert that other employees would have been chosen for dismissal but for the age of each of the plaintiffs.

A. Statistical Rebuttal

The plaintiffs argue that the statistics used to establish a prima facie case would allow a jury to find that the defendants’ proffered reasons for all of the discharges are pretextual. The defendants counter that statistics can never be used to show pretext. We are unwilling to hold that statistics could never form the basis for a rebuttal, but we do not believe that the statistics presented in this case provide an adequate response to the defendants’ explanations.

When a plaintiff’s statistics indicate a disproportionate discharge rate for a protected group there are three possible explanations for the discrepancy: the operation of legitimate selection criteria, chance, or the defendant’s bias. See D. Baldus and J. Cole, Statistical Proof of Discrimination 291 (1980); see also Palmer v. Schultz, 815 F.2d 84, 90-91 (D.C.Cir.1987). When a plaintiff demonstrates a significant statistical disparity in the discharge rate, he or she has provided strong evidence that chance alone is not the cause of the discharge pattern. The statistics do not and cannot determine whether the more likely cause is the defendant’s bias or a legitimate selection criterion. We allow a plaintiff to establish a prima facie case without providing additional evidence that bias is the more likely cause of the discharge than legitimate explanations because we assume, absent some reason to believe the opposite is true, that job skills will be equally distributed across all age groups.

There are several methods to attack a prima facie case established by statistics. A defendant may establish that the statistical method is faulty and show that chance is in fact a likely explanation for the apparent discrepancy. A defendant might also attack the nonstatistical presumption that a nondiscriminatory explanation is unlikely. One method of doing this is to show that job skills are not equally distributed over the particular employees in question. In a discriminatory hiring case, for example, a defendant might show that older applicants in the relevant hiring pool in fact had less training than their younger counterparts.

A third possibility is the one actually chosen by the defendants in this case. They have chosen to demonstrate that even if the plaintiffs’ statistics and the court's assumption tend to indicate that bias could have played a role in some of the decisions, that bias did not play a role in the particular decision to discharge each of the plaintiffs. By presenting evidence that each plaintiff was less qualified than others occupying comparable positions, the defendants have undercut the importance of the plaintiffs’ statistical proof. This method of attacking a statistical prima facie case cannot be rebutted by reference to the statistics already presented since the statistics here do not tend to establish that age played a factor in any particular decision. Unless the plaintiffs can show that the defendants’ explanations are inherently suspect or can present other direct or circumstantial evidence suggesting that the proffered reasons are not true, then the defendants are entitled to summary judgment.

B. The Bumping Policy

The plaintiffs also argue that the defendants’ alleged elimination of the bumping rights shortly before the layoffs provides some evidence of discriminatory motives. This Circuit has clearly established that an employer has no duty under ADEA to permit an employee to transfer to another position or to displace workers with less seniority when the employee’s position is eliminated as part of a work force reduction. See, e.g., Ridenour v. Lawson Co., 791 F.2d 52, 57 (6th Cir.1986) (“[wjhere an employer reduces his workforce for economic reasons, it incurs no duty to transfer an employee to another position within the company”); Simpson, 823 F.2d at 942 n. 6. The plaintiffs seek to distinguish these cases by asserting that unlike the prior cases, the defendants here actually had a bumping policy in place but eliminated it in anticipation of the reorganization.

The plaintiffs’ argument might have merit if only a few employees had been discharged during the reorganization or if employees other than the plaintiffs had been allowed to bump those with less seniority. It is also noteworthy that the plaintiffs’ evidence does not show that a clear policy existed. Instead the policy — if it ever existed — was informal. While application of an informal bumping policy may not have caused serious disruptions in the past, the large scale reorganization that occurred when the plaintiffs were discharged presented a different situation. In a large reorganization such as this, we are unwilling to require an employer to further disrupt the business by forcing the employer to continue an informal bumping policy.

C. Individual Factors

While the plaintiffs as a group cannot demonstrate facts sufficient to allow a finding of pretext, some individual plaintiffs are able to make such a showing. The defendants urge that as to each individual plaintiff it has established a legitimate, nondiscriminatory reason why that individual was discharged. The plaintiffs argue that even if the proffered explanations are legitimate, then they presented enough evidence to allow a fact-finder to find the reason to be pretextual. We have already rejected the plaintiffs’ argument that they could demonstrate pretext on a group-wide basis. We now address the argument that individual plaintiffs have presented enough evidence to allow a finding of pretext.

1. GenCorp Plaintiffs

a. Jon F. Barnes

Barnes was employed as a nontechnical research technician in the Polymers Research Section and was forty-eight-years-old at the time of his discharge. Barnes’ duties consisted of laboratory characterization testing of polymers under the supervision of technical personnel.

Barnes was one of four nontechnical research technicians involved in polymer characterization. Two of these technicians were younger than Barnes, K.L. Hall and A.J. Kirschner. The third, J.W. Lea, was older than Barnes. Because GenCorp had sold its General Tire Division, it determined that only one nontechnical employee was necessary to continue polymer testing.

GenCorp selected Lea to remain as the sole nontechnical research technician in the Polymers Research Section. GenCorp claims that Lea was selected because of his superior skills; Lea apparently held the highest nontechnical classification in the lab. It is also noteworthy that Lea was the oldest of the four employees. Hall was also retained, but not as a nontechnical employee. At the time of the layoffs Hall was one month short of receiving his B.S. in Chemistry. Once he received his B.S. he would become a technical employee in the lab.

A reasonable jury could not find that Barnes was discriminated against because of his age. Barnes does not challenge Gen-Corp’s decision to retain only one nontechnical employee in the Polymers Research Section. Instead he argues that because of his experience and the quality of his past work, he should have been allowed to transfer to a different section of the Research Division. ' We have already rejected this contention. Only two of the persons occupying comparable positions in his department were retained. One of these two employees was younger than Barnes but was to receive his B.S. in Chemistry the following month and be promoted to another position. While the company retained Lea in a comparable position, Lea was older than Barnes and had a higher skills rating.

b. Homer L. Gibson

Gibson was employed as a research machinist in the machine shop and was fifty-five-years-old at the time of his discharge. Gibson provided machine services and other support to research activities and assisted in plant maintenance.

Gibson was one of three research machinists employed by GenCorp. R.G. Boyd was just over two years younger than Gibson and G.E. Dodson was six months older. Both had roughly the same experience and seniority as Gibson. Management determined that because of the decrease in research activities only one of the three machinists was required after the restructuring. GenCorp argues that Boyd, the youngest of the three, was retained because his performance reviews were consistently higher than those of Gibson or Dodson and because he was considered the best machinist of the three by the Department Supervisor, W. Miller.

A reasonable jury could not find that Gibson was discriminated against because of his age. Gibson does not challenge the decision to eliminate two of three machinist positions, nor does he argue that he was more qualified than Boyd. Instead, he asserts that GenCorp should have allowed him to transfer to another position in the company, presumably by displacing another employee. We have already rejected this position. The younger machinist who was retained had higher performance reviews and was considered a superior machinist than Gibson.

c. Jean Z. Gipson

Gipson was employed as a library technician in the Technical Information Center and was fifty-seven-years-old at the time of her discharge. Gipson’s duties included providing information resources to technical and professional personnel.

Gipson was one of two library technicians employed before the layoff. The other library technician was Norma K. Smith who was six and one-half years younger than Gipson. Smith had worked intermittently for GenCorp between 1956 and 1969. Smith had been rehired as a library technician approximately eight months before the reorganization. GenCorp determined that only one of the two library technician positions would be needed after the reorganization.

GenCorp asserts that it retained Smith because it believed Smith was better qualified than Gipson because she possessed a B.S. in Chemistry and had in the past worked for GenCorp as a chemist. With her chemistry degree background she could act as a backup to the head of the technical information center in his absence. Gipson did not have a college degree and had worked in clerical and operative positions prior to becoming a librarian in 1979.

A reasonable jury could not find that Gipson was discriminated against because of her age. Gipson asserts that a college degree was not required for the position, that she had received high marks in her performance reviews, and that she helped train Smith after Smith was hired. She also asserts that Smith had not worked in the field of chemistry in over seventeen years. While Gipson did have more seniority than the person retained, she did not possess a college degree in chemistry, a useful skill for a librarian who works with chemists and chemical engineers and essential to acting as a backup to the head of the center. While Gipson asserts that a college degree was not required for the job, we do not believe that the age discrimination laws require an employer to retain an older employee over a younger employee merely because the older employee meets the minimum requirements. An employer is free to retain the younger, less senior employee if the employer reasonably believes he or she is better qualified for the position.

Gipson also alleges that she was not notified of three secretarial openings within the company for which she was qualified. An employer is not required to inform former employees of all openings which the former employee might be qualified to perform in order to avoid liability for an age discrimination charge. Gipson has not shown that younger employees were notified of other openings, nor has she alleged that she ever inquired about other positions with the company. Cf. Ayala 831 F.2d at 1320 (upholding plaintiffs’ verdicts where plaintiffs showed they were not informed of other openings with the company even after inquiry).

d. Clarence P. Kennedy

Kennedy was employed as a nontechnical research assistant to a group leader, P.T. Suman, and was fifty-nine-years-old at the time of his discharge. Suman, who was fifty at the time, and Kennedy were the only two employees in their group. Both Kennedy and Suman were involved in analyzing and synthesizing rubber compounds for use in tire rubber. Kennedy generally performed tests under Suman’s supervision.

Because GenCorp sold its General Tire Division, management determined that it would eliminate one of the two employees in this tire rubber research group. Suman was retained because he performed more of the technical functions in the lab and, unlike Kennedy, possessed a B.S. in Chemistry.

A reasonable jury could not find that Kennedy was discriminated against because of his age. Kennedy’s only argument is that he should have been allowed to transfer to another position and bump a younger employee. We have already rejected this argument.

e. Willy P. Kulhanek

Kulhanek was employed as a nontechnical research assistant in the Colloid and Surface Science Section and was fifty-two-years-old at the time of his discharge. Kulhanek worked in colloid/latex research, although the parties dispute the exact nature of his work. GenCorp asserts that Kulhanek spent most of his time working on colloid/latex characterization, testing the colloidal properties of latex compounds. Kulhanek admits that he performed testing but also claims he engaged in other development work.

Kulhanek was one of two nontechnical research assistants working in the Colloid Section. The other assistant, R. Simmons, was seventeen years younger than Kulha-nek and had worked in the Colloid Section for only two years. Kulhanek trained Simmons in his job duties when Simmons transferred to the colloid group. GenCorp determined that it could eliminate one of the two positions at the time it restructured the company.

GenCorp argues that it chose to retain Simmons because he had “some training” in college-level chemistry that made him better qualified than Kulhanek and because he was better suited to the increased emphasis on new product development because he had done this type of work in the past.

Kulhanek survives summary judgment. Kulhanek does not challenge the decision to eliminate one of the two assistant positions in his department, but does allege that he was equally or better qualified. While GenCorp asserts that Simmons was better qualified than Kulhanek to work on new product development, the facts supporting this assertion are in dispute. Kul-hanek alleges that he trained Simmons. While GenCorp alleges that Kulhanek did not have duties in product development, Kulhanek disputes this. Finally, while GenCorp does assert that Simmons had some college chemistry, GenCorp does not allege that he was degreed and does not specify what additional skills he gained that made him more qualified. Absent a degree or a showing that the retained individual gained specific skills we are unwilling to say that the defendant’s assertion that Simmons had “some background” establishes greater expertise. Finally, Kul-hanek argues that even if Simmons was better qualified, GenCorp should have allowed him to transfer to another position. We have already rejected this position.

f. Norman J. Muth

Muth was employed as a Supervisor in the Research Center Services Section of the Research Division and was fifty-six-years-old at the time of his discharge. Pri- or to his discharge he supervised the work of four clerical employees in the Services Section. GenCorp eliminated two of the clerical positions and determined that the remaining two employees should report directly to Muth’s boss, W.L. Kollar.

A reasonable jury could not find that Muth was discriminated against because of his age. The facts as alleged by both parties establish that Muth’s supervisory position was eliminated and that Muth’s nonsupervisory duties were divided between two of Muth’s subordinates. As noted above, we do not believe that GenCorp was required to allow Muth to displace employees performing different duties merely because Muth was qualified to take over their positions and had more seniority with the company.

Muth also argues that he was not notified of two other openings for which he was qualified. Muth has not shown that younger employees were notified of other openings, nor has he alleged that he ever inquired about other positions with the company. Absent some inquiry about these other positions, we do not believe that failure to notify a discharged employee of positions with the company that opened after their discharge is evidence of discrimination.

g. Febo C. Spagnuolo

Spagnuolo was employed as a nonprofessional Laboratory Supervisor in the Polymer Physics and Physical Testing Section and was fifty-eight-years-old at the time of his discharge. Spagnuolo supervised the work of five research assistants who were engaged in the physical testing of tires and other rubber products.

Because of the decrease in physical testing, GenCorp eliminated an unknown number—two according to the plaintiff, four according to the defendant—of the assistants Spagnuolo supervised. GenCorp felt that the remaining assistants could be supervised directly by the professional staff. Accordingly, the supervisory position Spag-nuolo occupied was eliminated.

A reasonable jury could not find that Spagnuolo was discriminated against because of his age. While several of the employees Spagnuolo supervised were retained, we do not believe that this indicated that GenCorp discriminated against Spag-nuolo. As already discussed, GenCorp was not required to allow its employees to bump employees performing different duties even if they had less seniority.

2. DiversiTech Plaintiffs

a. Theophilos A. Millis

Millis was employed as a Chief Chemist at the Mogadore plant and was sixty-two-years-old at the time of his discharge. Millis tested raw materials and new chemicals for use in manufacturing processes.

At the time of the restructuring, Diversi-Tech asserts that it was engaged in a program of single-sourcing and hoped to switch the testing of raw materials to the single source vendor’s facilities. Diversi-Tech asserts that the program had already reduced the amount of testing performed by Millis, thus his position could easily be eliminated. Millis’ younger subordinate, G.E. Miller, was primarily responsible for inprocess material testing which was not affected by this shift in raw material testing. Thus Miller was retained, continuing his in-process testing and assuming some of Millis’ duties in addition to his own testing work.

Millis survives summary judgment. While DiversiTech asserts that Millis was fired because of a change in the company’s raw material testing, there are facts on which a jury could rely to reject this conclusion. The Polymers Division President, Wheeler, stated in his deposition that the company did not have a policy of single-sourcing and had not reduced the amount of in-house testing of raw materials.

DiversiTech argues that this Court should not rely on Wheeler’s deposition statements since Wheeler was not Millis’ supervisor and did not make the termination decision. Instead DiversiTech asserts that the Plant Manager’s opinion is most relevant. H.E. Jewett, the Plant Manager, supported DiversiTech’s assertions on single-sourcing and raw materials testing. While DiversiTech argues that Wheeler did not make the decision to eliminate Millis, Wheeler did state that all proposed position reductions were cleared by him before they were implemented. Further, even if Wheeler was not involved in the decision, his statements still contradict the explanation given by Millis’ plant supervisor. Additionally, Millis asserts that he should have been given the opportunity to replace Miller, a position we have already rejected.

b. Richard S. Novitsky

Novitsky was employed as the Manager of Ecology and Government Compliance at the Mogadore plant and was fifty-six-years-old at the time of his discharge. Novitsky had performed these duties since the position was created in November 1985. Novitsky had no subordinates and reported directly to the Latex Manufacturing Manager at the plant, H.E. Jewett. Jewett also had a background in environmental work.

DiversiTech determined that many of No-vitsky’s duties were redundant of functions performed or partially performed by other personnel. The position was eliminated and the nonredundant duties divided among four other persons occupying different positions: (1) Latex Manufacturing Manager, H.E. Jewett; (2) Personnel Administrator, OSHA/EEO, L.E. Gulka; (3) Safety Inspector, T.C. Stokes; and (4) Senior Process Engineer, R.A. Newgent.

A reasonable jury could not find that Novitsky was discriminated against because of his age. Novitsky’s central argument is that he should have been allowed to fill other positions with the company, including the position occupied by Stokes. As already stated, we do not believe that the company was required to allow bumping. Novitsky also asserts that Polymers Division President Wheeler contradicted DiversiTech’s stated reasons for discharging him. When Wheeler was asked “were there other individuals that were performing the same work as [Novitsky]” prior to No-vitsky’s discharge, Wheeler responded that there were not. We do not believe that this undermines DiversiTech’s explanation, however. Consistant with the DiversiTech’s explanation, Wheeler stated that Novitsky’s job functions were divided between several individuals and that Novitsky’s position was not recreated after his discharge. When asked whether another individual was doing “the same ” job as Novitsky before his discharge, Wheeler gave the correct answer. No one was doing “the same” job, even though other persons were performing related or overlapping work. We do not believe that the plaintiffs can rely on the answer to an isolated and ambiguous question to create an issue of fact for trial.

c. Milan A. Rolik

At the time of his discharge, Rolik was employed as a Senior Chemist in the Analytical Laboratory and was sixty-years-old. Rolik’s duties involved training, developmental work, and microscopy work.

DiversiTech alleges that it eliminated Ro-lik’s position because his duties could be easily assumed by the other professional laboratory employees, all of whom were younger than Rolik. The two other senior chemists occupying similar positions were chosen over Rolik because Rolik’s last performance report showed basic deficiencies in his analytical chemistry skills. Another of Rolik’s supervisors, S.J. Loinek, described Rolik’s skills as “outdated.”

Rolik not only challenges the accuracy of his last evaluation, he also alleges that his firing was merely the last act in a long series of discriminatory acts by Diversi-Tech intended to force him to leave the company. He points out that at the time of his December 1985 evaluation he had been performing the work of a Senior Chemist for less than two weeks. In the two months before this he had been filling in for a hospitalized technician. More importantly, however, he alleges that it was highly unusual to have given him a performance review at that time. According to Rolik, performance reviews were usually given on a yearly basis. Finally, he was transferred to the analytical group only under protest, feeling that his training and experience were not suited to this type of work. Thus, according to Rolik, his transfer to Analytical Chemistry was part of a discriminatory scheme to force him to leave the company.

Rolik alleges that the pattern of discrimination against him began in 1984 when his position as Pilot Plant Manager was eliminated. He filed a complaint with the EEOC at this time. The Company did not fire him, however. It kept him on at a series of temporary jobs until placing him in the Analytical Laboratory—over his objections—in October 1985. Rolik alleges that he applied for several other permanent positions but was always rejected in favor of younger applicants. Many of the temporary assignments he was given were allegedly dirty and dangerous, including scraping old tools and parts and disposing of outdated and unlabeled chemicals. In March 1986, after his transfer to Analytical Chemistry, he was informed that his salary would be reduced by more than $20,000 to $29,000. Only after writing numerous letters objecting to his pay cut—and voicing his concern that he might be the victim of age discrimination—was the pay cut rescinded.

Rolik survives summary judgment. Ro-lik does not claim that he was more qualified than the other persons occupying his position. Instead he relies on his lack of qualifications to challenge the decision to transfer him to the Analytical Group. Ro-lik alleges that this action was merely one of a series of actions intended to force him to resign.

While the issue is a close one, we believe that the series of actions taken by Diversi-Tech over the prior months could be seen as discriminatory. Rolik alleges that he applied for and was denied several jobs given to younger applicants. He also alleges that he was given a series of dirty and dangerous assignments and that the company attempted to drastically cut his salary. While DiversiTech has justified its decision to discharge Rolik from his last position, it has not presented any justification for its prior actions.

The other DiversiTech plaintiffs allege that they can rely on Rolik’s poor treatment to rebut DiversiTech’s alleged nondiscriminatory reasons for discharging them. We disagree. The basic question is whether the plaintiffs were as qualified as other younger individuals in comparable positions. Any discrimination directed at Rolik has nothing to do with the relative qualifications of the other plaintiffs. Even if Rolik’s treatment was discriminatory, this evidence alone would not be sufficient to entitle another plaintiff to a jury verdict where DiversiTech has shown the plaintiff to be less qualified than others in the position. Further, the plaintiffs have not shown that the same upper level management was involved in the decisions to eliminate their positions.

d. Norman S. Trommer

Trommer was employed as a Senior Process Engineer and was sixty-one-years-old at the time of his discharge. Trommer performed various duties related to the tire cord adhesive and carpeting textile markets.

Trommer was one of three Senior Process Engineers, all of whom reported to the Chief Process Engineer, M.J. Wise. The other two, V.A. Farone and R.A. Newgent, were younger than Trommer and had less seniority. Trommer had a chemistry background while both Farone and Newgent had chemical engineering backgrounds. Farone was primarily responsible for the paper and latex markets. Newgent worked with the tire cord adhesive and new product markets.

DiversiTech asserts that Trommer’s position was eliminated because of the product lines he served. DiversiTech felt that the textile markets served by Trommer were mature markets in a state of decline, and therefore the company decided to begin withdrawing from them. It asserts that Trommer’s position was eliminated along with other technical and sales positions in the textile area.

Trommer survives summary judgment. While DiversiTech asserts that he was selected for discharge over his peers because the textile market he served was in decline, this position is undercut by Wheeler’s deposition testimony. Wheeler stated that the textile market he served was in decline when the layoffs occurred, that no decision had been made to withdraw from the market, and that textiles continued to constitute a large portion of DiversiTech’s business. Trommer also asserts that he had more responsibility in the tire cord market than did Newgent. While Trommer’s supervisor may have given testimony to the contrary, we believe that Wheeler’s testimony and Trommer’s testimony created a fact question which cannot be resolved on this record.

IV. CONCLUSION

For all the reasons stated above, the judgment of dismissal is AFFIRMED as to plaintiffs-appellants Jon F. Barnes, Homer L. Gibson, Jean Z. Gipson, Clarence R. Kennedy, Norman J. Muth, Febo C. Spagnuolo, and Richard S. Novitsky. It is REVERSED as to plaintiffs-appellants Willy P. Kulhanek, Theophilos A. Millis, Milan A. Rolik, and Norman S. Trommer, and REMANDED for further proceedings consistent with this opinion. 
      
      . The cases were brought under the Age Discrimination in Employment Act of 1967 (ADEA), as amended, 29 U.S.C. §§ 621, et seq., and Ohio Rev.Code Ann. § 4101.17 (Anderson, 1980).
     
      
      . There is some confusion as to whether General Tire was sold before or after the reorganization. In any case, it appears the decision to sell the subsidiary was made prior to the reorganization that resulted in the staff cuts at issue in this case. Thus it is irrelevant for the purposes of this appeal.
     
      
      . Deaner did state, however, that he was not sure how the practice worked.
     
      
      . A separate analysis was done excluding the 10 upper-level employees at the Research Division, none of whom were fired. This analysis yielded a slightly greater disparity between the termination rates of the older and younger groups.
     
      
      . All calculations performed on groups excluding the senior management produced results more than three standard deviations from the expected random result.
     
      
      . The McDonnell Douglas test requires a plaintiff to show the following in order to establish a prima facie case: (1) that he is a member of a protected class; (2) that he applied for a job and was rejected; (3) that he was qualified for the job; and (4) that the employer continued to seek job applicants after the plaintiff was rejected.
     
      
      . When a plaintiff has established a prima facie case, he or she has established a "legally mandatory, rebuttable presumption” that the defendant discriminated. Burdine, 450 U.S. at 254 n. 7, 101 S.Ct. at 1094 n. 7. A defendant escapes this presumption by articulating a legitimate reason for the action.
      In saying that the presumption drops from the case, we do not imply that the trier of fact no longer may consider evidence previously introduced by the plaintiff to establish a prima facie case. A satisfactory explanation by the defendant destroys the legally mandatory inference of discrimination arising from the plaintiff's initial evidence. Nonetheless, this evidence and inferences properly drawn therefrom may be considered by the trier of fact on the issue of whether the defendant’s explanation is pretextual. Indeed, there may be some cases where the plaintiff's initial evidence, combined with effective cross-examination of the defendant, will suffice to discredit the defendant's explanation.
      
        Id. at 255 n. 10, 101 S.Ct. at 1095 n. 10.
     
      
      . The Court has also explained:
      A prima facie case under McDonnell Douglas raises an inference of discrimination only because we presume these acts, if otherwise unexplained, are more likely than not based on the consideration of impermissible factors. And we are willing to presume this largely because we know from our experience that more often than not people do not act in a totally arbitrary manner, without any underlying reasons, especially in a business setting. Thus, when all legitimate reasons for rejecting an applicant have been eliminated as possible reasons for the employer's actions, it is more likely than not the employer, who we generally assume acts only with some reason, based his decision on an impermissible consideration such as race.
      
        Furnco Constr. Corp. v. Waters, 438 U.S. 567, 577, 98 S.Ct. 2943, 2949-50, 57 L.Ed.2d 957 (1978) (citation omitted) (emphasis in original).
     
      
      . This Circuit’s decisions have not been entirely clear on when a plaintiff who establishes a prima facie case of age discrimination under McDonnell Douglas without providing other evidence of discrimination can escape summary judgment. In adapting the McDonnell Douglas test to a discharge case in which the plaintiff alleges age discrimination, a plaintiff would be required to demonstrate that:
      (1) he was a member of the protected class [i.e, age-40-and-over];
      (2) he was discharged;
      (3) he was qualified for the position;
      (4) he was replaced by a younger person.
      
        Ackerman v. Diamond Shamrock Corp., 670 F.2d 66 (6th Cir.1982) (citing Marshall v. Goodyear Tire and Rubber Co., 554 F.2d 730, 735 (5th Cir.1977)).
      We have stated that the McDonnell Douglas test is not to be applied mechanically, instead opting for a case-by-case approach that focuses on whether age was in fact a determining factor in the employment decision. See Merkel v. Sco-vill, Inc., 787 F.2d 174, 177 (6th Cir.), cert. denied, 479 U.S. 990, 107 S.Ct. 585, 93 L.Ed.2d 587 (1986); Blackwell v. Sun Elec. Corp., 696 F.2d 1176, 1179 (6th Cir.1983) (holding that a plaintiff who could not demonstrate every element of the McDonnell Douglas test could nonetheless demonstrate a prima facie case); Ackerman, 670 F.2d at 70; Sahadi, 636 F.2d at 1118 (refusing to find a prima facie case where the plaintiffs position was eliminated but younger employees were retained in positions plaintiff was capable of performing and willing to relocate to perform); Laugesen v. Anaconda Co., 510 F.2d 307, 312-13 n. 4 (6th Cir.1975).
     
      
      . Of course an employer could not avoid liability by changing the job title or by making minor changes to a job indicative of an attempt to avoid liability. Further, a plaintiff could attempt to show that the force reduction was itself pretextual.
     
      
      . Statistics for the 48-and-over age group into which Jon Barnes falls fell just less than three standard deviations. We need not decide whether Barnes established a prima facie case since we reject his claim in the pretext justification phase.
     
      
      . Such sub-group analysis may not apply to discriminatory impact cases, however. See Lowe v. Commack Union Free School Dist., 886 F.2d 1364 (2d Cir.1989).
     
      
      . Of course the closer in a age a retained employee is to a plaintiff, the less plausible it is that the statistics show the plaintiff was discharged because of age. For example, if the defendants showed that most of the nondis-charged employees were 47-years-old, then the discharge statistics would likely not be probative in the case of a 48-year-old plaintiff. Significantly, however, the defendants have not shown or alleged that the statistical results would change if employees only a few years younger than a plaintiff were included in the statistical class with that plaintiff. With respect to the GenCorp plaintiffs age 50-and-over and the DiversiTech plaintiffs age 60-and-over, the plaintiffs' statistics would specifically undercut such an assertion. At the summary judgment stage, a plaintiff is not required to anticipate and rebut every possible weakness in his otherwise valid statistical evidence.
     
      
      . For example, in the GenCorp cases if three persons in the 48-and-over age group had been retained and three persons in the under-48 group discharged then 31.8% of the older group would still have been discharged as compared to 19.5% of those younger than 48. The results would still have differed by more than a standard deviation from the expected random result.
     
      
      . There is arguably other evidence to support the establishment of a prima facie case. We need not address this evidence, however, since we believe the statistics alone present enough evidence to establish a prima facie case. Recognizing this possibility, the defendants have continually advanced legitimate, nondiscriminatory reasons for their actions.
     
      
      . For example, statistics could very well form the basis for a rebuttal if the defendants rebutted the plaintiffs’ statistical evidence with statistical evidence of their own.
     
      
      . Similarly, statistical proof alone cannot establish the existence of a mixed motive case in individual discrimination cases. For a true mixed motive case to arise a plaintiff must present either direct or circumstantial evidence of the presence of a discriminatory motive.
     
      
      . In a work force reduction case, unlike a hiring case, we can assume that all of the employees are qualified to perform their job since they were actually performing it at the time of their discharge.
     
      
      . The possibilities are, of course, not exhausted by the illustrative list presented here.
     
      
      . By presenting evidence that a large number of the discharged older employees were less qualified than the employees who were retained, the defendant may in fact undercut the statistical significance of the plaintiffs’ statistics requiring a reconsideration of their probative value. The District Court is free to consider the continued value of the statistics on remand.
     
      
      . While the plaintiffs’ expert concluded that the statistics demonstrate "to a virtual statistical certainty [that] age was a determining factor in the termination of the employees in each of the indicated groupings,” we do not believe that this conclusion is warranted. As the D.C. Circuit pointed out in Palmer, 815 F.2d at 91 (emphasis in original):
      A statistical analysis of a disparity in selection rates can reveal the probability that the disparity is merely a random deviation from perfectly equal selection rates. Statistics, however, cannot entirely rule out the possibility that chance caused the disparity. Nor can statistics determine, if chance is an unlikely explanation, whether the more probable cause was intentional discrimination or a legitimate nondiscriminatory factor in the selection process.
     
      
      . It is important to note that we do not believe the plaintiffs have shown that the defendants deviated from an otherwise established seniority system. In this case the defendants merely made clear that an informal bumping policy that had sometimes been applied in the past would not be used during the mass reorganization. Cf. Ayala v. Mayfair Molded Prods. Corp., 831 F.2d 1314 (7th Cir.1987) (deviations from an otherwise observed seniority system when combined with other circumstantial evidence was enough to support jury verdict for the plaintiffs in work force reduction case).
     