Universality, limits and predictability of gold-medal performances at the olympic games

Abstract

Inspired by the Games held in ancient Greece, modern Olympics represent the world's largest pageant of athletic skill and competitive spirit. Performances of athletes at the Olympic Games mirror, since 1896, human potentialities in sports, and thus provide an optimal source of information for studying the evolution of sport achievements and predicting the limits that athletes can reach. Unfortunately, the models introduced so far for the description of athlete performances at the Olympics are either sophisticated or unrealistic, and more importantly, do not provide a unified theory for sport performances. Here, we address this issue by showing that relative performance improvements of medal winners at the Olympics are normally distributed, implying that the evolution of performance values can be described in good approximation as an exponential approach to an a priori unknown limiting performance value. This law holds for all specialties in athletics-including running, jumping, and throwing-and swimming. We present a self-consistent method, based on normality hypothesis testing, able to predict limiting performance values in all specialties. We further quantify the most likely years in which athletes will breach challenging performance walls in running, jumping, throwing, and swimming events, as well as the probability that new world records will be established at the next edition of the Olympic Games.

Figure 1. Performances of male gold medalists in 400 meters sprint.

a. Best estimate of the asymptotic performance value. For each value of lower than the actual Olympic record, we evaluate the goodness of the fit of performance improvements with a normal distribution. is determined as the value of the asymptotic time that maximizes the statistical significance (-value). For men 400 meters sprint, our best estimate is seconds, where we find that relative performance improvements are normally distributed with a confidence of 98%. For this value of , the best empirical estimates of the average value and standard deviation are respectively and . b. The cumulative distribution function of the -scores obtained for (red curve) is compared with the standard normal cumulative distribution (black curve). c. Normal sample quantile are plotted against normal theoretical quantiles . The dashed line corresponds to the theoretically expected behavior in case of a perfect agreement between sample and theoretical distributions. d. -scores of relative performance improvements between consecutive editions of the Games.

Figure 2. Statistical properties of performance improvements in athletics.

In the main panels we show the determination of the best estimate of the asymptotic performance value, while in the insets we provide a graphical comparison between the sample cumulative distributions (red line) and the standard normal cumulative distribution (black line). a and b. We report the results obtained by the analysis of the performances of male athletes in marathon ( seconds, -value ) and female athletes in long jump ( meters, -value ). c and d. We show the outcome of our method for performances of men and women in 100 meters sprint (respectively, seconds and -value , seconds and -value ).

Figure 3. Scaling law between asymptotic time and running length, and prediction of performances at future editions of the Olympic Games.

a. Relation between the best estimates of the limiting performance value and the length of the race for men running events in athletics (red circles). We excluded from the analysis relay and hurdles events. We find that , and the best estimate of the power-law exponent is (black line). b. Probability density functions of the winning time for the men 400 meters sprint in future editions of the Games. The dashed line represents the winning time in the latest edition of the Olympics in Beijing 2008. This value is used as initial condition for the prediction of future performances. c. The probability density of the winning time in men 400 meters predicted by our model is compared to past performance data (black circles). The density plot is obtained by convoluting the various prediction curves derived from real data. d. Probability that athletes will breach challenging walls in various specialties of athletics as a function of time.