A Bias-Corrected Estimator of Competitive Balance in Sports Leagues

Abstract

The ratio of the actual standard deviation (ASD) to the idealized standard deviation of win percentages (RSD) is the conventional measure of competitive balance (CB). RSD is designed to control for the effect of season length on the sample standard deviation of win percentages (ASD). Theoretically, the RSD should be greater than 1, but empirical values below 1 have been found in previous studies. This article employs a mathematical statistics approach to evaluate the statistical properties of RSD and ASD. In doing so, this study finds that RSD is constructed by an invalid normalization approach and that ASD is biased. It also presents a bias-corrected standard deviation (BCSD) as a new estimator of the standard deviation of true win probabilities. Results from the simulations confirm the following: (i) ASD is prone to underestimate CB levels when the number of games is small, (ii) the RSD values become unreasonably large when the number of games is large, and (iii) BCSD performs well with respect to mean bias and root mean squared errors. According to empirical analysis of the English Premier League (EPL) and the Korea Baseball Organization (KBO), BCSD shows that the KBO was more competitively balanced than the EPL between 2000 and 2015, while the RSD implies that the two leagues were roughly equal.