Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Quantitative Analysis in Sports

An official journal of the American Statistical Association

Editor-in-Chief: Steve Rigdon, PhD

4 Issues per year

CiteScore 2017: 0.67

SCImago Journal Rank (SJR) 2017: 0.290
Source Normalized Impact per Paper (SNIP) 2017: 0.853

See all formats and pricing
More options …
Volume 12, Issue 1


Volume 1 (2005)

Is there a Pythagorean theorem for winning in tennis?

Stephanie Ann Kovalchik
  • Corresponding author
  • Institute of Sport, Exercise & Active Living, Victoria University, Footscray Park, VIC, Australia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-02-17 | DOI: https://doi.org/10.1515/jqas-2015-0057


Bill James’ discovery of a Pythagorean formula for win expectation in baseball has been a useful resource to analysts and coaches for over 30 years. Extensions of the Pythagorean model have been developed for all of the major professional team sports but none of the individual sports. The present paper attempts to address this gap by deriving a Pythagorean model for win production in tennis. Using performance data for the top 100 male singles players between 2004 and 2014, this study shows that, among the most commonly reported performance statistics, a model of break points won provides the closest approximation to the Pythagorean formula, explaining 85% of variation in season wins and having the lowest cross-validation prediction error among the models considered. The mid-season projections of the break point model had performance that was comparable to an expanded model that included eight other serve and return statistics as well as player ranking. A simple match prediction algorithm based on a break point model with the previous 9 months of match history had a prediction accuracy of 67% when applied to 2015 match outcomes, whether using the least-squares or Pythagorean power coefficient. By demonstrating the striking similarity between the Pythagorean formula for baseball wins and the break point model for match wins in tennis, this paper has identified a potentially simple yet powerful analytic tool with a wide range of potential uses for player performance evaluation and match forecasting.

Keywords: performance evaluation; Pythagorean method; sports forecasting


  • Baumer, B. and A. Zimbalist. 2014. The Sabermetric Revolution: Assessing the Growth of Analytics in Baseball. Philadelphia, Pennsylvania: University of Pennsylvania Press.Google Scholar

  • Braunstein, A. 2010. “Consistency and Pythagoras.” Journal of Quantitative Analysis in Sports 6(1):1–16.Google Scholar

  • Caro, C. A. and R. Machtmes. 2013. “Testing the Utility of the Pythagorean Expectation Formula on Division One College Football: An Examination and Comparison to the Morey Model.” Journal of Business & Economics Research 11(12):537–542.Google Scholar

  • Cha, D. U., D. P. Glatt, and P. M. Sommers. 2007. “An Empirical Test of Bill James’s Pythagorean Formula.” Journal of Recreational Mathematics 35(2):117–130.Google Scholar

  • Cochran, J. J. and R. Blackstock. 2009. “Pythagoras and the National Hockey League.” Journal of Quantitative Analysis in Sports 5(2):1–13.Google Scholar

  • Davenport, C. and K. Woolner. 1999. “Revisiting the Pythagorean Theorem: Putting Bill James’ Pythagorean Theorem to the Test.” The Baseball Prospectus.Google Scholar

  • Faraway, J. J. 2005. Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models. Boca Raton, Florida: CRC Press.Google Scholar

  • Gilsdorf, K. F. and V. A. Sukhatme. 2008. “Testing Rosen’s Sequential Elimination Tournament Model Incentives and Player Performance in Professional Tennis.” Journal of Sports Economics 9(3):287–303.Web of ScienceCrossrefGoogle Scholar

  • Hamilton, H. H. 2011. “An Extension of the Pythagorean Expectation for Association Football.” Journal of Quantitative Analysis in Sports 7(2):1–18.Google Scholar

  • Hammond, C., W. P. Johnson, and S. J. Miller. 2015. “The James Function.” Mathematics Magazine 88(1):54–71.CrossrefGoogle Scholar

  • James, B. 1981. Baseball Abstract. Self-Published, Lawrence, KS.Google Scholar

  • Klaassen, F. J. and J. R. Magnus. 2001. “Are Points in Tennis Independent and Identically Distributed? Evidence from a Dynamic Binary Panel Data Model.” Journal of the American Statistical Association 96(454):500–509.CrossrefGoogle Scholar

  • Knottenbelt, W. J., D. Spanias, and A. M. Madurska. 2012. “A Common-opponent Stochastic Model for Predicting the Outcome of Professional Tennis Matches.” Computers & Mathematics with Applications 64(12):3820–3827.Web of ScienceCrossrefGoogle Scholar

  • McHale, I. and A. Morton. 2011. “A Bradley-Terry Type Model for Forecasting Tennis Match Results.” International Journal of Forecasting 27(2):619–630.CrossrefWeb of ScienceGoogle Scholar

  • Miller, S. J. 2007. “A Derivation of the Pythagorean Won-loss Formula in Baseball.” Chance 20(1):40–48.CrossrefGoogle Scholar

  • Miller, S. J., T. Corcoran, J. Gossels, V. Luo, and J. Porflio. 2014. “Pythagoras at the Bat.” in Social Networks and the Economics of Sports, 89–113. Springer.Google Scholar

  • Morris, C. 1977. “The Most Important Points in Tennis.” Optimal Strategies in Sports 5:131–140.Google Scholar

  • R Core Team. 2015. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.Google Scholar

  • Rosenfeld, J. W., J. I. Fisher, D. Adler, and C. Morris. 2010. “Predicting Overtime with the Pythagorean Formula.” Journal of Quantitative Analysis in Sports 6(2):1–19.Google Scholar

  • Stefani, R. T. 1997. “Survey of the Major World Sports Rating Systems.” Journal of Applied Statistics 24(6):635–646.CrossrefGoogle Scholar

  • Tibshirani, R. 1996. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B (Methodological) 267–288.Web of ScienceGoogle Scholar

  • Vollmayr-Lee, B. 2002. More than You Probably ever Wanted to Know about the “Pythagorean” Method. http://www.eg.bucknell.edu/bvoll-may/baseball/pythagoras.html.

  • Winston, W. L. (2012). Mathletics: How Gamblers, Managers, and Sports Enthusiasts use Mathematics in Baseball, Basketball, and Football. Princeton, New Jersey: Princeton University Press.Google Scholar

About the article

Corresponding author: Stephanie Ann Kovalchik, Institute of Sport, Exercise & Active Living, Victoria University, Footscray Park, VIC, Australia, Tel.: +61 450 509 098, e-mail:

Published Online: 2016-02-17

Published in Print: 2016-03-01

Citation Information: Journal of Quantitative Analysis in Sports, Volume 12, Issue 1, Pages 43–49, ISSN (Online) 1559-0410, ISSN (Print) 2194-6388, DOI: https://doi.org/10.1515/jqas-2015-0057.

Export Citation

©2016 by De Gruyter.Get Permission

Comments (0)

Please log in or register to comment.
Log in