Simpson's paradox shows that the same data can tell opposite stories depending on how it is grouped.
It is not a trick of arithmetic but a sign that groups were mixed in unequal proportions.
The paradox matters because combined figures are what most reports show first.
A frequently cited case involved kidney stone treatments, where one did better for both small and large stones but worse overall.
It was used far more often on large stones, which are harder to treat, and that dragged its overall rate down.
The figures within each group are the fair comparison there, because stone size affected both the choice and the outcome.
Another frequently cited case is university admissions, where a lower overall acceptance rate came from applying to more competitive departments.
The general pattern is a third variable linked both to the grouping and to the outcome.
That third variable is often a confounder, and splitting by it removes the distortion.
Splitting is not always right, though: if the third variable is caused by the treatment, the combined figure is the honest one.
For example, if a treatment works partly by preventing severe cases, splitting by severity hides part of its effect.
So the paradox cannot be resolved from the table alone; it needs a view of what causes what.
Checking results within key segments, such as region or customer type, catches reversals before they reach a decision.
In practice, any surprising overall result is worth splitting by the main variables that differ between the groups.
The usual misreading is that one of the 2 answers must be a calculation error, when both are correct for different questions.