During an online tutoring session I was hosting recently, the subject of Simpson's Paradox came up, and I realized that this beautiful example of counter intuitive statistics is not that well known yet, and even those who have heard of it have great trouble understanding it and resolving it in their own minds. And so I thought I would write a simple explanation that helped my students in the past to visualize how it works.

First an explanation of Simpson's Paradox. In general, this refers to a situation in which a statistical property present in several independent data sets is not present when the data sets are merged into one. Which in simpler terms means that statistical results can vary depending on how they are analyzed.

The most famous example of this is the University of California Berkeley discrimination case of 1973. The university was sued when it was discovered that their graduate schools accepted 44% of the men who applied, but only 35% of the women applicants. Clearly it was more difficult for women to get accepted than men, which was clearly biased. However when the university tried to find which departments were doing this, they found that no single department was discriminating against women. In fact, four of the top six departments at the university were actually biased in favour of the female applicants!

But how can this be? How can the entire graduate school favor men, but every department is either neutral or favoring women? That is Simpson's paradox.

It has a simple explanation, but unfortunately most statistics textbooks and courses get so focused on the subtleties of the mathematics that only statisticians can understand them. So let me present an extreme but simple example in lieu of an explanation.

Suppose there exists a university that is so small it has exactly two departments, called Department A and Department B. And suppose that Department A is very appealing to men, and so they receive 90 applications from males and only 10 from females. But they have funding to accept 50 students, and are completely fair, so that each applicant has a 50% chance of getting accepted. They accept 45 men and 5 women.

Over in Department B, they receive applications from 90 women and only 10 men. And because they are facing budget shortfalls, they can only accept 10% of their applicants. But they are also completely fair in their acceptance practices, so each applicant has the same 10% chance of being accepted, leading to the department accepting 9 women and 1 man.

Clearly neither department has discriminated based on gender, as men and women in each department have exactly the same chance of getting accepted.

Then the chair of the university decides to cut costs by merging the two departments into one. The applications have already been processed, so they cannot be changed. However this new department has effectively accepted 46 of the 100 applications from men, and only 14 of the 100 applications from women. In this new department, men are more than three times as likely as women to get accepted, which is blatant discrimination!

So which result is correct? Each department separately is fair, but together they are biased.  Often in statistics the result depends on how the data is handled, and unfortunately in situations like this one, it can be manipulated to give either result without actually cheating or falsifying data. The answer is simply that both results are correct.

And that is Simpson's Paradox in a nutshell.