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Rating distributions are often not normal because some subset of players study the game and take it more seriously resulting in a bimodal distribution. See [0] for an example in Chess.

[0] https://chess.stackexchange.com/questions/2550/whats-the-ave...

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Even without the bimodality, you wouldn't expect a normal distribution of ratings.

1. Assume that chess ability is normally distributed in the population.

2. Assume that people who are terrible at chess are more likely to stop playing chess than people who are successful.

Then you've sampled the underlying normal distribution mostly from the top end, and that new, highly skewed distribution is what you'll see when you measure everyone's rating.

That's fascinating, thanks! It looks like you can model it as a mixture distribution made up of two underlying normal distributions.

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