Simpson’s paradox

Simpson’s paradox is a statistical paradox named after a British statistician. The paradox is that if we have two subjects, and one of them is more successful than the other in every observation, it can still happen that the other one is more successful overall.

Example

We have two different students at two different schools studying two different subjects. Let’s call them Amelia and Oliver. Each of them sits two tests in their subject during a term. Amelia has a success rate of 30% in the first and 100% in the second. Oliver has a success rate of 25% in the first and 75% in the second.

It seems that Amelia is the more successful student. But if we add the number of correctly answered questions, it may not look that way any more. The key point is that Amelia and Oliver sat different tests, because they went to different schools.

In the first test Amelia could have answered 3 out of 10 questions correctly (30% success rate) and then 2 out of 2 questions (100%). Altogether she answered 5 out of 12 questions correctly. Oliver could have answered 1 out of 4 (25%) and then 6 out of 8 questions (75%). Altogether he answered 7 out of 12 questions. From this point of view Oliver is the more successful one after all.

Simpson’s paradox is quite common and there is nothing incomprehensible about it. It is named after Edward H. Simpson, who was the first to describe the phenomenon properly – of course, the paradox had appeared before that.