Simpson's Paradox

Definition

Simpson’s paradox is when a trend that appears in every subgroup reverses or disappears once the groups are combined. It happens when group sizes differ and the grouping variable is linked to both the choice and the outcome, so the combined figure hides it.

In plain terms

Imagine comparing two sellers’ success rates. Seller X does better than seller Y among easy products and also among hard products. Yet overall Y looks far better, because Y sells mostly easy products, which succeed more often, while X takes mostly hard ones. The overall rate mixes “who is better” with “who gets which products”.

The paradox is not an arithmetic trick but a reminder that a combined figure answers a different question from the subgroup figures. Which one answers yours depends on how the data came about, which is a causal question (see Confounding).

Example

Successful sales per attempt, split by product difficulty (synthetic):

X wins in both columns, yet Y wins overall by a wide margin: X’s work is mostly hard products. Ranking the sellers by the “Overall” column would reverse the verdict that holds within each kind of product.

Common misunderstandings

• “The overall number is the real one.” It answers a different question and can reverse the within-group story.
• “The subgroup numbers are always the real ones.” Which is right depends on the causal structure; sometimes the combined figure is correct.
• “It is a rare curiosity.” It appears whenever groups of unequal size and different base rates are merged.