Why can grouped and combined rates reverse?

An overall rate combines groups using their sizes as weights. It is not generally the simple average of the group percentages. When two records contain very different mixes of easier and harder tasks, their combined comparison can reverse the comparison within each group. This pattern is called Simpson's paradox.

The fictional puzzle records show system A completing 81 of 90 easy puzzles and 1 of 10 hard puzzles. Its within-group rates are 90% and 10%. System B completes 19 of 20 easy puzzles and 16 of 80 hard puzzles, giving 95% and 20%. B has the higher completion rate within both supplied difficulty groups.

Combining completions and attempts gives A 82 out of 100, or 82%, and B 35 out of 100, or 35%. A's combined rate is higher because 90% of its attempts are in the easy group, compared with only 20% of B's. Each overall rate weights its own easy and hard rates by those different attempt shares.

The reversal is an arithmetic consequence of the recorded group rates and weights. It does not make either set of calculations wrong, but it changes what each comparison summarizes. The grouped rates compare within difficulty labels; the combined rates describe the actual mixes in these records.

Neither view alone establishes a causal advantage or justifies recommending a system. These invented records are descriptive, not randomized assignments. Group labels and a weighted average cannot by themselves explain what caused the differences.