Anyone for chess? Analysing chess ratings above high thresholds

Suppose some cleverness score parameter is sufficiently interesting to be defined and then measured, perhaps for different strata of specialists or for the broader population. Such phenomena could have Gaussian distributions, when it comes to all players in a stratum, but when interest focuses on the very tails, for the top few percent, those above certain high thresholds, different models are called for, along with the need to analyse such based on the listed top scores only. In this note I develop such models and tools, and apply them to the top-100 and above 2100 points lists for regular chess ratings, for the currently active 14671 men and 753 women, as given by the FIDE, January 2026. It is argued that even when two or more distributions have close to identical expected values, or medians, even smaller differences in variance may explain gaps for the few very best ones.

💡 Research Summary

The paper introduces a statistical framework for analysing scores that lie far above a chosen threshold, with a particular focus on the upper tail of chess rating distributions. The author begins by recalling that, for a normally distributed population, the expected maximum of an i.i.d. sample of size n grows as ξ + σ√(2 log n). This relationship shows that, while the mean (ξ) and the standard deviation (σ) both affect the extreme values, the sample size only enters through a slowly growing logarithmic term. When individuals are positively correlated, the factor √(1 − ρ) reduces the effective variability, further emphasizing the role of dispersion in shaping extremes.

Motivated by the need to model only the highest‑scoring individuals, the author proposes a two‑parameter parametric family for scores above a fixed cutoff r₀ = 2100 (the FIDE “master” threshold). The density is

 f(x; a, θ) =