Incorporating the Effects of Designated Hitters in the Pythagorean Expectation
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The Pythagorean Expectation is widely used in the field of sabermetrics to estimate a baseball team’s overall season winning percentage based on the number of runs scored and allowed in its games thus far. Bill James devised the simplest version of the formula through empirical observation as Winning Percentage = (RS)^2 /[(RS)^2+(RA)^2], where RS and RA are runs scored and allowed, respectively. Statisticians later found 1.83 to be a more accurate exponent, estimating overall season wins within 3-4 games per season. Steven Miller provided a theoretical justification for the Pythagorean Expectation by modeling runs scored and allowed as independent continuous random variables drawn from Weibull distributions. This paper aims to first explain Miller’s methodology using recent data and then build upon Miller’s work by incorporating the effects of designated hitters, specifically on the distribution of runs scored by a team. Past studies have attempted to include other effects on run production such as ballpark factor, game state, and pitching power. The results indicate that incorporating information on designated hitters does not improve the error of the Pythagorean Expectation to better than 3-4 games per season.