A Derivation of the Pythagorean Won-loss Formula in Baseball

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