# Statistical distribution of centipawn losses

I am wondering what function describes best the statistical distribution of centipawn losses in human chess games. (I do not mean the average centipawn loss for a completed game, but losses of single moves.)

Will it be more like an Erlang/Gamma distribution starting from a small value at low loss, increasing to its global maximum at finite loss and then decaying to zero with a long tail? Or rather more like an exponential distribution which is monotonically decaying for all loss values?

For sure, there will be parameters depending on the player's strength (Elo) and time control. I am asking for a rough model which would fit the data.

This is an interesting question.

We can plot the distribution using data from the Lichess database of games that have evaluations.

I have held constant:

1. Player rating
2. Time-Control (here I used Blitz games)
3. Evaluation before the move was made (perhaps people make worse moves if the position is far from equal)

The first thing to notice is that there is a discrete mass at 0. This is the probability that players find the best move (which always has 0 centipawn loss). Reasonably, 2000 rated players find the best move ~30% of the time, whereas 1000 rated players only find the best move ~20% of the time.

It is also reasonable to expect that the remainder of the mass is continuous (since all the moves which aren't the next best moves are going to be distributed along a continum.

Looking just at the distribution, the simple answer to OP's question is "more like an exponential distribution which is monotonicallly decaying for all loss values").

However, just by eye, this has fatter tails than an exponential distribution. Fitting a Pareto distribution seems to do a better job, although I'm not entirely convinced that's right either.

This result (discrete probability mass at 0, and then strictly descreasing pdf beyond that) is consistent with Regan's model that players are more likely to play better moves. With a better model of what the distribution of candidate-move evals is we could probably spell this out in more detail

• This is a very nice answer! Thank you for your effort. I have more detailed questions to it: which parameters of Pareto did you get? (power, shift?) For which ratings did you fit? Could you provide the extracted data (e.g. online) or script to obtain them? I am very interested in that topic. Could we chat more about it? E.g. on discord or so... (since no DM on SE possible) Apr 4 at 22:23
• Sure, simonm101 Apr 5 at 7:51