In the mid-1970s, the University of California, Berkeley, found itself in some hot legal water over graduate school admissions. It turns out that the university accepted almost 2.5 times as many men as women to its graduate programs in the fall of 1973, which led to some accusations of gender discrimination. Actually, it wasn't just that so many more men were accepted --- after all, there were many more male applicants in the first place, so you'd also expect the majority of approved applicants to be male --- but that men were accepted at a much higher rate than their female peers. In fact, the university admitted approximately 44% of male applicants, but only about 35% of female applicants, enough of a discrepancy that it was unlikely to be due to chance alone.
Graduate Admissions Data (source)
At this point, things weren't looking so great for Berkeley. The likelihood of getting accepted to graduate school seemed to depend on an applicant's gender, which is generally otherwise known as "against the law." The university, being appropriately concerned, decided to take a closer look.
It's important to note that, as with most graduate schools, admissions decisions at the university weren't made by the graduate school, per se, but by each individual department. For instance, if you applied to the Ph.D. program in mathematics, then the math department reviewed your application. When the university looked at the admissions rates for the individual programs, they noticed that the gender bias seemed to disappear. In fact, in four of the six largest departments, women were actually admitted at a higher rate than men! How could that be?
The table below shows the number of applicants and acceptance rates by gender for the largest departments in the graduate school. The bold numbers highlight which gender had the higher acceptance rate.
Admissions Data by Department (source)
This case is a famous example of a phenomenon called Simpson's Paradox, where a particular trend in data is reversed when it is either aggregated or disaggregated. While calling it a paradox makes it sound waaaaay cooler, it's a little misleading, because there is no actual contradiction involved. We can usually trace the mystery back to either a faulty assumption or some hidden explanatory variable. So what accounts for the strangeness in the Berkeley case?
Things start to become clearer when we look at which departments men and women applied to. Notice that the overwhelming majority of women applied to departments that accepted well under half of all applicants, whereas the majority of men applied to less selective programs, which took well over half of all applicants.
In Department B, for example, almost 96% of applicants were men, so even though women were accepted at a higher rate, a lot more men were accepted to the graduate school on the whole. Same with Department A. Things are more balanced in the most selective programs, so the disproportionate number of male applicants in the least selective departments was enough to skew the overall percentages in favor of the men. The "discrimination" was more or less self-imposed; if more women had applied to Departments A and B, there would likely never have been any controversy.1
While there is usually a straightforward way to resolve Simpson's Paradox, it helps highlight the danger of attributing too much explanatory power to a single variable. When we look at data, we need to be mindful that the way in which it's organized can affect the types of conclusions we're likely to draw from it, and that there may be important relationships we haven't yet considered.
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1. Of course, there is a very reasonable argument to be made that certain discriminatory societal pressures led women to apply to particular departments and stay away from others, but that doesn't bear upon the question of whether U.C. Berkeley was intentionally denying admittance to women because of their gender.