The school that passes more students in every subject

Question

Two schools report exam pass rates, split by subject.

Math (a hard exam): School P passes 30 of 100 (30%); School Q passes 12 of 50 (24%).
English (an easier exam): School P passes 45 of 50 (90%); School Q passes 88 of 100 (88%).

Overall, School P passes 75 of 150 (50%) while School Q passes 100 of 150 (about 67%), so Q looks better in total.

Which school performs better, and why does the overall rate favor the other one?

Accepted Answer

School P performs better. It has the higher pass rate in math (30% versus 24%) and in English (90% versus 88%). In each subject, P wins. The overall rate flips the ranking, a Simpson's paradox produced by a confounder. The confounder is which subject the students sat. English is the easier exam (high pass rates) and math is the harder one (low pass rates). The two schools sent very different numbers of students into each. Most of P's exam-takers sat the hard math exam (100 of 150), while most of Q's sat the easy English exam (100 of 150). So Q's overall rate is lifted by a favorable subject m…

The school that passes more students in every subject

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