What is Simpson's Paradox?

Simpson's paradox is a comparison that points one way in the combined data and the other way inside every group the data is made of. In 1973 the University of California, Berkeley admitted about 44% of the men who applied to graduate study and about 35% of the women, a gap of 9 points. Looked at department by department, the gap was gone: six departments admitted women at a higher rate than men, and four did the opposite. The combined figure flipped because women had applied mostly to departments that were hard to get into, and men mostly to departments that were easy. So the total was comparing which departments each sex had chosen, not how each sex was treated. That is the paradox in one line: the total can point the opposite way from every one of its parts. The fix is to compare like with like, group by group, whenever the two sides did not face the same mix of cases.

Simpson's paradox is a comparison that reverses when the data is split into groups: one side wins on the combined figure and loses inside every group. Two things have to be true at once: the groups differ in difficulty, and the two sides are spread unevenly across them, one mostly in the hard group and the other mostly in the easy one. Then the combined figure measures the spread, not the sides. Two phone repair shops each take 100 jobs. Shop A fixes 90 of every 100 cracked screens and 30 of every 100 water-damaged phones. Shop B fixes 80 of every 100 screens and 20 of every 100 water-damaged phones, so A is better at both. But A's 100 jobs are 20 screens and 80 water-damaged: 18 plus 24 is 42 fixed. B's are 80 screens and 20 water-damaged: 64 plus 4 is 68 fixed. B wins the total while losing both kinds of job, because B got the easy work. Berkeley's 1973 graduate admissions had the same structure. Overall, 44% of men were admitted against 35% of women, yet department by department more departments favoured women than men. Women had applied to the departments that were hard to get into. The department is a confounding variable, a hidden factor tied to both the thing compared and the outcome. Splitting is not automatically the honest view, though. It is right when the grouping genuinely sets the difficulty, as damage type and department do, and splitting by something irrelevant can manufacture a flip the other way. Nor is Simpson's paradox bad data: every number in it is correct, and the total still misleads. A skewed sample is selection bias, a separate problem. The question that catches it is always the same: did the two sides face the same mix of cases?

Simpson's paradox is a comparison that flips when the data is split into groups, and two pizza places on a delivery app are the same idea. The app says Place A averages 32 minutes to your door and Place B averages 26, so B looks faster. Split the orders by distance and A is faster on both: 18 minutes against B's 20 for nearby orders, and 40 against B's 45 for far ones. A sits on the edge of town, so most of its orders are far, and B sits in the centre, so most of its orders are near. The overall average is grading the map, not the kitchens. The nearby and far orders are the groups, A's faster time in each is the side that wins every group, the app's average is the total, and where each shop sits is the uneven spread that flips it. The match ends here: on the app you can see the map, while in research the thing that splits the groups has to be thought of and measured.