Simpson's paradox: the better hitter with the worse average

Question

Two hitters are compared by batting average (hits / at-bats), split by the handedness of the pitcher they faced:

	vs RHP	vs LHP	Season
Player X	35/100 (.350)	100/400 (.250)	135/500 (.270)
Player Y	136/400 (.340)	24/100 (.240)	160/500 (.320)

X hits better against right-handers (.350 vs .340) and against left-handers (.250 vs .240), yet Y's overall average is higher (.320 vs .270).

Who is the better hitter, and explain the reversal.

Accepted Answer

This is Simpson's paradox. The confounder is the mix of pitchers each hitter faced. Everyone hits better against right-handers (about .34 to .35) than against left-handers (about .24 to .25), and the two players faced opposite mixes: - X took 400 of 500 at-bats against left-handers, the tougher matchup. - Y took 400 of 500 at-bats against right-handers, the easier matchup. Y's higher season average reflects whom he faced, not superior hitting. The season figures are matchup-weighted averages: 	ext{X: } 	frac{100}{500}(.350) + 	frac{400}{500}(.250) = .270, \qquad 	ext{Y: } 	frac{400}{500}…

Simpson's paradox: the better hitter with the worse average

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