A Derivation of the Pythagorean Won-Loss Formula in Baseball

By Steven J. Miller. 2005

73 citations recorded by the source corpus.

It has been noted that in many professional sports leagues a good predictor of a team’s end of season won-loss percentage is Bill James’ Pythagorean Formula RSobs γ RSobs +RAobs γ , where RSobs (resp. RAobs) is the observed average number of runs scored (allowed) per game and γ is a constant for the league; for baseball the best agreement is when γ is about 1.82. This formula is often used in the middle of a season to determine if a team is performing above or below expectations, and estimate their future standings. We provide a theoretical justification for this formula and value of γ by modeling the number of runs scored and allowed in baseball games as independent random variables drawn from Weibull distributions with the same β and γ but different α; the probability density is f(x; α, β, γ) = ( γ α ((x− β)/α)γ−1 e−((x−β)/α)γ if x ≥ β 0 otherwise. This model leads to a predicted won-loss percentage of (RS−β)γ (RS−β)γ+(RA−β)γ ; here RS (resp. RA) is the mean of the Weibull random variable corresponding to runs scored (allowed), and RS−β (resp. RA−β) is an estimator of RSobs (resp. RAobs). An analysis of the 14 American League teams from the 2004 baseball season shows that (1) given that the runs scored and allowed in a game cannot be equal, the runs scored and allowed are statistically independent; (2) the best fit Weibull parameters attained from a least squares analysis and the method of maximum likelihood give good fits. Specifically, least squares yields a mean value of γ of 1.79 (with a standard deviation of .09) and maximum likelihood yields a mean value of γ of 1.74 (with a standard deviation of .06), which agree beautifully with the observed best value of 1.82 attained by fitting RSobs γ RSobs +RAobs γ to the observed winning percentages.

Subfields: Artificial Intelligence, Economics and Econometrics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9d82ad2bfbdb6e34bfa1605c9a1fa3cd02debe3d