The Pythagorean Theorem of Baseball and Alternative Models

摘要

The "Pythagorean Theorem of Baseball" uses a simple function of a team's runs scored and runs allowed to predict the team's winning percentage. Runs scored reflects a team's offensive ability, while runs allowed reflects defensive and pitching ability. Values from the formula are posted daily during the Major League Baseball (MLB) season on websites such as ESPN.com [Neyer 2004]. Baseball managers and fans use the statistic to see if their team is playing up to its predicted value ("its potential") [Schwarz 2004]. The "Theorem" has also been suggested as a criterion to evaluate the success of baseball managers themselves [Horowitz, 1994], spawning a debate on its appropriateness [Ruggiero et al. 1997; Horowitz 1997]. We use calculus to explore the mathematical characteristics of a class of predictive functions of which the Pythagorean Theorem of Baseball is a special case. We restrict our attention to models that predict the season-ending winning percentage as a function of runs scored and runs allowed over the course of a season. We use linear regression on the data at the ends of the 1969-2004 seasons to investigate the accuracy of the Pythagorean Theorem and other Pythagorean-type models that predict winning percentage based on the ratio of runs allowed to runs scored, as well as models based on a linear combination of runs scored and runs allowed. Although we considered beginning our analysis in 1973, the year when the designated-hitter rule went into effect, we selected 1969, the year when the pitcher's mound was lowered 5 inches. We also offer suggestions on how to extend the analyses for other sports and for other baseball questions.