Mathematics and Education

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The Tyranny of the Mean

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In the appendix footnote 3 of a forthcoming 2015 article, Ceci and Williams state:

Many commentators have opined that female scientists are superior to their male counterparts, and therefore the fact that they are hired at the same rate as men obscures the fact that they should be hired at even higher rates, if merit was the basis for hiring.

Ceci and Williams give three quotes intended as examples of this argument. One comes from my blog post The Pipeline and The Trough. However, as I stated in a later post, I did not give this argument.

So why do Ceci and Williams think I did? The answer may lie in “the tyranny of the mean”—the assumption that the mean for a set, e.g., “female scientists,” is the same as the mean of a subset, e.g., “female applicants for a given job.” In this case, the population is “female scientists” and a sample is “female applicants for a given job.”

In particular, these may be applicants for jobs at Research I universities between 2001 and 2003. Statistics for applicants and hires are given in the National Research Council report Gender Differences at Critical Transitions in the Careers of Science, Engineering, and Mathematics Faculty. Ceci and Williams cite these statistics in articles published in the Proceedings of the National Academy of Sciences (2011 and 2015) and Psychological Science in the Public Interest (2014). In these articles, they note the report’s finding that “If women applied for positions at RI institutions, they had a better chance of being interviewed and receiving offers than male job candidates.”

I happen to think the explanation might be that the female applicants were, on average, superior to their male counterparts. However—contrary to the tyranny of the mean—this does not imply that the same was true in the labor pool from which applicants were drawn. In other words, female applicants for a given job may, on average, be superior to male applicants for the same job but that does not necessarily imply that female scientists in general (or recent female PhDs, or the females in some other labor pool) are, on average, superior to their male counterparts.

To see how the average for an applicant pool might differ from the average for a labor pool, consider the following scenario:

Suppose in discipline D one third of female recent PhDs are superior, one third of female PhDs are average, and one third are inferior—and the same holds for their male counterparts: one third are superior, one third are average, and one third are inferior. Thus, the labor pool looks like this:

Tyranny Table 1

For a given tenure-track RI job in discipline D, applicants are a subset of this labor pool. For example, the applicant pool might look like this:

Tyranny Table 2

In this case, the female applicants, on average, are superior to the male applicants. The means for females and males in the applicant pool differ from the means for females and males in the labor pool.

Ceci et al. (2014) say:

In the blogosphere, it is frequently suggested that female applicants are of higher quality than males. . . . Thus, the argument is that if the pool of female applicants is of higher quality than the male pool, then the high proportion of female PhDs hired may mask bias that prevented an even higher proportion from being hired.

Later, we review evidence and an argument that run counter to this claim, showing that, when taken together, objective measures of productivity (publications, grant dollars, citations per article) do not indicate that women in the applicant pool are stronger than men—publication measures favor men, as do total citations to their work; grant success is similar for both sexes; and citations per article tilt in favor of women—but on the whole, there is no evidence for the superiority of either gender applying for tenure-track jobs. (p. 102, emphasis added)

The “objective measures of productivity” are given for groups such as “assistant professors,” “associate professors,” “full professors,” “academics,” and “PhDs” (p. 104). Thus, in my opinion, the term “applicant pool” in the second paragraph of the quote above should be replaced by “labor pool.” The evidence given does not counter the possibility that the females in an applicant pool may be, on average, of higher quality than the males in the same applicant pool.

The quote at the beginning of this post also suggests that labor pool is not being distinguished from applicant pool. The first sentence is similar to that of Ceci et al. (2014), but “female scientists” replaces “female applicants”:

Many commentators have opined that female scientists are superior to their male counterparts, and therefore the fact that they are hired at the same rate as men obscures the fact that they should be hired at even higher rates, if merit was the basis for hiring.

According to the tyranny of the mean, it is acceptable to conflate labor pool and applicant pool because the means for the two are the same. However, there seems to be no reason to make this assumption.

 

* Linn, M. C. (1994). The tyranny of the mean: Gender and expectations. Notices of the American Mathematical Society, 41(7), 766-769.

 

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Written by CK

October 19, 2015 at 4:22 pm

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