Simpson's Paradox — when every part says one thing and the whole says the opposite
Two treatments, A and B, are compared on two kinds of patients, group 1 and group 2.
In the initial data, treatment A has the higher success rate in each group.
Yet when the two groups are pooled, treatment B can have the higher success rate.
Data — successes / trials (editable)
Edit any count; rates update as you type.
Group 1
Group 2
Pooled (all patients)
Treatment A
/
/
Treatment B
/
/
Better in this column
Explore — shift the case mix
Move patients between the groups while each within-group rate stays fixed to within roughly percentage points (integer rounding).
Bar chart — success rates
treatment Atreatment B
Compare the two group columns with the pooled column. A reversal occurs when the same treatment has the higher rate in both groups but the other treatment has the higher pooled rate.
Vector picture — why pooling can flip
A: (trials, successes) in group 1, then group 2B: same
Each treatment is represented by two vectors (n, s), one for each group. The pooled success rate is the slope of their resultant vector, not the vertical position of its endpoint.