Simpson's Paradox

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Tap a term to see what it means, or .

Calculate

Example: in a study of kidney stones, treatment A cured 81 of 87 patients with small stones and 192 of 263 with large stones; treatment B, 234 of 270 and 55 of 80. A is better in both groups. And overall? Change the values to work out your own case.

Groups (one per row) In each row: successes of A, total of A, successes of B, total of B. Groups separated by semicolons.

Step by step

What each term means

Overall Rate: The success rate pooling every group, for one treatment. It is the one that can tell a story opposite to the groups'.
Sum over the Groups: Adds each group's contribution, such as small stones and large stones.
Group Weight (n_g/N): The share of that treatment's cases that fell into group g. When A and B are spread differently across the groups, the weights differ, and that is where the reversal comes from.
Rate in the Group (p_g): The treatment's success rate within group g. Comparing rates within each group neutralizes the group's influence.