Statistical Curse of the Second Half Rank

TL;DR

This paper explains the 'Statistical Curse of the Second Half Rank' in competitions with many races, showing how scores tend to cluster near the center and how this affects final rankings, especially for those slightly below median scores.

Contribution

It provides a quantitative analysis of the effect using the Central Limit Theorem and derives exact distributions for small numbers of races, including variants with dropped worst results.

Findings

Scores follow a Gaussian distribution near the center for many races.

Final rank distribution becomes skewed, amplifying the disadvantage of slightly below-median scores.

Exact formulas are provided for two and three races, verified by simulations.

Abstract

In competitions involving many participants running many races the final rank is determined by the score of each participant, obtained by adding its ranks in each individual race. The "Statistical Curse of the Second Half Rank" is the observation that if the score of a participant is even modestly worse than the middle score, then its final rank will be much worse (that is, much further away from the middle rank) than might have been expected. We give an explanation of this effect for the case of a large number of races using the Central Limit Theorem. We present exact quantitative results in this limit and demonstrate that the score probability distribution will be gaussian with scores packing near the center. We also derive the final rank probability distribution for the case of two races and we present some exact formulae verified by numerical simulations for the case of three races. The variant in which the worst result of each boat is dropped from its final score is also analyzed and solved for the case of two races.