Simpson’s Paradox

According to Wikipedia, Simpson’s paradox is an apparent paradox in which the successes of groups seem reversed when the groups are combined. Often encountered in social and medical science statistics when frequency data are given causal interpretations.

That probably did as much for you as it did for me when I first read it. Let’s look at the example.

There are two departments (Department A and Department B). Each department receives an equal number of male and female applicants. Of the 510 male applicants, 251 men were accepted and of the 510 female applicants, 109 women were accepted. That means for 360 positions, men took almost 70% of the positions. Initially, this seems like a clear cut case of sexism in favor of men.

However, when the departments are analyzed individually we find that department A had 500 male applicants and only 10 female applicants. Of the male applicants, 50% were accepted while 90% of the female applicants were accepted. In department B, 10 males applied and 10% were accepted, while 500 females applied and 20% were accepted.

What is the result? Overall we see men taking 70% of the available jobs, yet department A favored women by 40% over men and department B favored women by 10% over men. Now it appears that we have sexism in favor of women.

 

	Men	Women
Department A	500 applicants (50% accepted)	10 applicants (90% accepted)
Department B	10 applicants (10% accepted)	500 applicants (20% accepted)
Totals	251 accepted	109 accepted

How is it that two departments, both favoring women, resulted in men taking more jobs than women? It is all about where the majority of each sex put their application. In the case of the male applicants, most applied to a department that had a much greater hiring rate.

There are some other great examples, one in particular where something quite similar to the above occurred.