Simpson's paradox: the more reliable courier with the worse on-time rate

Question

Two delivery couriers are compared by on-time rate (on-time deliveries / total), split by traffic conditions:

	Off-peak	Rush hour	Overall
Blue	95/100 (95%)	540/900 (60%)	635/1000 (63.5%)
Red	846/900 (94%)	55/100 (55%)	901/1000 (90.1%)

Blue is more punctual off-peak (95% vs 94%) and in rush hour (60% vs 55%), yet Red's overall on-time rate is far higher (90.1% vs 63.5%).

Which courier is more reliable, and explain the reversal.

Accepted Answer

This is Simpson's paradox. The confounder is traffic: everyone is far more punctual off-peak (about 94 to 95 percent) than in rush hour (about 55 to 60 percent), and the two couriers work opposite shift mixes: - Blue runs 900 of 1000 deliveries in rush hour, the hard conditions. - Red runs 900 of 1000 off-peak, the easy conditions. Red's higher overall number reflects when it drives, not steadier driving. The overall rates are traffic-weighted averages: 	ext{Blue: } 	frac{100}{1000}(0.95) + 	frac{900}{1000}(0.60) = 0.635, \qquad 	ext{Red: } 	frac{900}{1000}(0.94) + 	frac{100}{1000}(0.55…

Simpson's paradox: the more reliable courier with the worse on-time rate

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