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Simpson’s paradox — Data, 14–17 years

A comparison can point one way in separate groups and the opposite way when the groups are combined. The change is caused by how many cases each group contains.

Idea

Simpson’s paradox occurs when a pattern appears in several separate groups but reverses after the groups are combined. The combined result gives more weight to groups with more observations. So an overall percentage can hide what happens inside the groups that matter.

Why

This problem matters because people often compare totals before asking whether the groups have the same mix. The paradox became famous through examples in medicine, education and university admissions, where difficulty levels or group sizes differed. Splitting the data can reveal a fairer comparison.

Worked example

Treatment A succeeds for 90 of 100 mild cases (90%) and 1 of 10 severe cases (10%). Treatment B succeeds for 8 of 10 mild cases (80%) and 80 of 90 severe cases (88.9%). A is better in both groups, but overall A is 91/110 = 82.7%, while B is 88/100 = 88%. The larger number of severe B cases reverses the total.

Common trap

The tempting mistake is to trust the overall percentage because it uses all the data. That feels sensible: more data usually sounds more reliable. But totals can be unevenly weighted, so you should inspect important subgroups and ask whether their sizes or difficulty levels differ.

Use

Hospitals may compare treatments across patients with different levels of illness. Schools may compare results while separating subjects or year groups, and businesses may compare products across regions. The lesson is not to split data forever, but to choose comparisons that respect meaningful differences.

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