Simpson's Paradox as an Interview Trap

The problem. A trader claims Strategy A beats Strategy B in small-cap names and beats it in large-cap names. Yet across the combined portfolio, Strategy B has the higher win rate overall. Both statements can be true at once — and it isn't a trick of small samples, it's arithmetic. This flip is Simpson's paradox, and interviewers love it because it exposes whether you actually check how groups are weighted before trusting an aggregate number.

Why it happens: weighting hides inside the average

An overall win rate is a weighted average of the group win rates, and the weights are the group sizes, not something visible in the win-rate numbers themselves. If Strategy A wins more of its trades in each size bucket but happens to concentrate its trades in the bucket where everyone does worse, its overall average gets dragged down by that concentration — even though bucket-by-bucket it's the stronger performer. The paradox isn't a probability quirk; it's what happens whenever you combine ratios without accounting for unequal denominators.

Worked example: trade win rates by bucket

	Small-cap trades	Small-cap win rate	Large-cap trades	Large-cap win rate
Strategy A	10 (of 100)	90% wins (9/10)	90 (of 100)	20% wins (18/90)
Strategy B	90 (of 100)	80% wins (72/90)	10 (of 100)	10% wins (1/10)

A beats B in small-caps (90% vs 80%) and beats B in large-caps (20% vs 10%). But look at the totals: A wins $9+18=27$ out of 100 trades overall (27%), while B wins $72+1=73$ out of 100 (73%). B crushes A overall despite losing every bucket comparison. The reason: A placed 90% of its trades in the low-win-rate large-cap bucket, while B placed 90% of its trades in the high-win-rate small-cap bucket. Each strategy's overall number is dominated by whichever bucket it traded most, and the two strategies traded completely different mixes.

The general technique: never trust an aggregate without the split

When someone hands you a single combined statistic — a win rate, a conversion rate, a survival rate — the reflexive question is: what are the underlying groups, and are the two things being compared weighted toward those groups the same way? If the weighting differs, the aggregate comparison can reverse the group-by-group comparison, and the aggregate is telling you as much about the mix as about the underlying quality. The fix is always the same: break the aggregate back into its groups, check both the rates and the group sizes, and only trust the combined number once you understand why the weights are what they are — sometimes the weighting itself is the real finding (e.g., A avoids trading small-caps for a good reason), and sometimes it's an artifact worth ignoring.

Drag the exponent on the curve above and notice how a relationship that looks consistent locally can bend once you view it across a wider combined range — the same visual intuition that makes Simpson's paradox surprising: a trend within each slice doesn't have to survive when the slices are stitched together.