{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-derivation-of-the-pythagorean-won-loss","title":"A Derivation of the Pythagorean Won-Loss Formula in Baseball","arxiv_id":"math/0509698","date":"2005-09-29","proceeding":null,"authors":["Steven J. Miller"],"abstract":"It has been noted that in many professional sports leagues a good predictor of a team's won-loss percentage is Bill James' Pythagorean Formula RSobs^c / (RSobs^c + RAobs^c), where RSobs (resp. RAobs) is the observed average number of runs scored (allowed) per game and c is a constant for the league; for baseball the best agreement is when c is about 1.82. We provide a theoretical justification for this formula and value of c by modelling the number of runs scored and allowed in baseball games as independent random variables drawn from Weibull distributions with the same b and c but different a; the probability density f(x;a,b,c) is 0 for x < b and is (c/a) ((x-b)/a)^{c-1} exp(-((x-b)/a)^c) otherwise. This model leads to a predicted won-loss percentage of (RS-b)^c / ((RS-b)^c + (RA-b)^c); here RS (resp. RA) is the mean of the random variable corresponding to runs scored (allowed), and RS - b (resp. RA - b) 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 or a maximum likelihood analysis give good fits; least squares gives a mean value of c of 1.79 with a standard deviation of .09, and maximum likelihood gives a mean value of c of 1.74 with a standard deviation of .06, which agree beautifully with the observed best value of 1.82 attained by fitting RSobs^c / (RSobs^c + RAobs^c) to the observed winning percentages.","url_abs":"https://arxiv.org/abs/math/0509698v4","url_pdf":"https://arxiv.org/pdf/math/0509698v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-derivation-of-the-pythagorean-won-loss","repo_url":"https://github.com/mindonly/sta631-project","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}