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Licensed Unlicensed Requires Authentication Published by De Gruyter February 17, 2016

Is there a Pythagorean theorem for winning in tennis?

  • Stephanie Ann Kovalchik EMAIL logo


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.

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


I am grateful to the staff at the ATP and for making a vast amount of tennis data available to the public and the research presented in this paper possible.


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Published Online: 2016-2-17
Published in Print: 2016-3-1

©2016 by De Gruyter

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