Can it win each group yet lose?

One option can beat another in every subgroup and still come out behind once you pool the numbers. It isn't a trick of arithmetic — it's confoundingA lurking variable tied to both which option each case got and how hard that case was. Pool over it and the comparison is no longer like-for-like.: the option that looks worse per group was simply handed the easier cases. Edit the table and watch the pooled winner flip.

The two options
measuringrate
Overall
%✓
of
%✓
of
78%
of 350
%
of
%
of
82.9%✓
of 350
🔀 Simpson's paradoxOpen surgery has the higher success rate in both small stones and large stones — yet Keyhole wins once the groups are pooled. The pooled average is dragged by who got which cases, not by which option is better.
Where the cases went
Open surgery
25%
75%
Keyhole
77%
23%
Small stonesLarge stones

The pooled rate is a size-weighted average, so the mix above is the whole story. When the two options draw from the groups in different proportions — one loaded with the high-rate group, the other with the low — pooling stops comparing like with like, and the subgroup winner can lose. Keep the mixes equal and the paradox can't happen.

Simpson's paradox — default figures from a 1986 kidney-stone study