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## Can You Identify the Discrimination?

“Study Finds that Smaller Percentage of Women are Admitted to Graduate School than Men: Government Moves to Address this Discrimination.”

Is the data evidence of discrimination? Does the policy proscription follow? Let’s put some hypothetical numbers to it. Suppose that for your university, 30% of male applicants are accepted to graduate programs while only 25.5% of female applicants are accepted to graduate programs – it would seem that something “bad” is happening to women. Suppose further that there are 4 different graduate programs at the university: English, Psychology, Mathematics and Physics.

Would you believe it if I told you that within each of the four graduate programs women have a larger acceptance rate than men? That might sound ridiculous. After all, how can only 25.5% of women overall be admitted while 30% of men, but that within each of the departments women have a better chance of getting in? I must be playing tricks!

But I am not. Suppose that there are 50 male applicants each to English and Psychology programs and 100 male applicants each to Physics and Math.  Suppose that 200 women apply to English and Psychology programs while only 20 women apply to Math and Physics programs.  Given the popularity of English and Psychology, those departments are highly selective. They take a much smaller share of their applicants than do their counterparts in Math and Physics. For example, in English and Psychology, suppose only 5 men out of 50 are admitted (10%) whole only 40 women out of 200 are admitted (20%). In this case, the English and Psych department favors women applicants by a factor of 2-1, while it has an overall acceptance rate of 18%.

Now, suppose that Math and Physics have an acceptance rate that is higher than  in English, but that it also favors women by a ratio of 2 to 1. In this case, suppose 40% of the men are admitted (40 of 100) while 80% of the women are admitted (16 of 20). In this case, the Math and Physics department would have an overall acceptance rate of 46.7%.

So what do we have now? In each department, we see that women have twice as good a chance of being accepted than men. But let’s aggregate what happens across all departments.

• There are a total of 150 male applicants to graduate school, and 45 of them are admitted, for an aggregate acceptance rate of 30.0%.
• There are a total of 220 female applicants to graduate school, and 56 are admitted, for an aggregate acceptance rate of 25.5%.

Within each department, women applicants are twice as likely to be accepted as their male counterparts, but when we aggregate the data we see that men at this university have a 20% higher chance of being admitted to the school at all. How the heck can this be? It is simple. Men are applying in disproportionately large numbers to programs with high acceptance rates while women are applying in disproportionately large numbers to programs with low acceptance rates.

What is “bad” is that such a headline would use aggregate data to make claims that cannot be supported from the aggregate data alone. Of course, it is POSSIBLE that women are being unfairly and that within each department they have a harder time being admitted, but it does not follow from the raw data itself.

On a related note, here is Mark Perry on females and graduate school admissions.

1. Harry says:

QED.