8.1 FIDE ratings of chess players

FIDE (the International Chess Federation) publishes ratings of chess players regularly and these are the ratings that will be discussed here. Like many sporting organisations they use a system based on Elo’s ideas (Elo (1978)). The main assumption is that the performance of a player is normally distributed about the player’s current strength. A further assumption is that players’ performances in a game are independent of one another. Ratings are updated based on game results. Ratings in the December 2020 dataset range from 1001 to 2862 (the then world champion, Magnus Carlsen).

Figure 8.1 gives one view of the distribution of player ratings.

Figure 8.1: Histogram of ratings of all chess players rated by FIDE in December 2020

There is certainly no sign of a normal distribution, but then that would not be expected. There is a sharp lower boundary, presumably because lower ratings are not included. The very best players are not visible, as the histogram bars where they would appear are too small. There is a suggestion of a one-sided peak at 2000. Perhaps players who reach this level stop playing to avoid the risk of falling below 2000. Making the binwidth of the chess ratings histogram narrower (10 rating points instead of the 25 points used in Figure 8.1), there are weak hints of one-sided peaks at 1250 and 1500, similar to the one at 2000.

The best performing players are of interest and they are not visible in Figure 8.1. A boxplot of the ratings data is shown in Figure 8.2 and now the best players stand out (individual outliers drawn in red). It is clear that there were some exceptional players with ratings over 2750. Going back to the data shows that there were 18 of them.

Figure 8.2: Boxplot of ratings of all FIDE rated chess players in December 2020