Answers.Body: <p>This is an interesting question.</p>
  <p>We can plot the distribution using data from the <a href="https://database.lichess.org/" rel="nofollow noreferrer">Lichess database</a> of games that have evaluations.</p>
  <p>I have held constant:</p>
  <ol>
  <li>Player rating</li>
  <li>Time-Control (here I used Blitz games)</li>
  <li>Evaluation before the move was made (perhaps people make worse moves if the position is far from equal)</li>
  </ol>
  <p><a href="https://i.sstatic.net/GI5qC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GI5qC.png" alt="Centipawn Loss (Blitz)" /></a></p>
  <p>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.</p>
  <p>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.</p>
  <p>Looking just at the distribution, the simple answer to OP's question is &quot;more like an exponential distribution which is monotonicallly decaying for all loss values&quot;).</p>
  <p><a href="https://i.sstatic.net/ZKV9r.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZKV9r.png" alt="Centipawn Loss Density" /></a></p>
  <p>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.</p>
  <p><a href="https://i.sstatic.net/taYyd.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/taYyd.png" alt="Log-density" /></a></p>
  <p>This result (discrete probability mass at 0, and then strictly descreasing pdf beyond that) is consistent with <a href="https://cse.buffalo.edu/%7Eregan/papers/pdf/ReHa11c.pdf" rel="nofollow noreferrer">Regan's model</a> 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</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-04-04T22:23:50.533Z
Answers.Text: 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)
Answers.UserDisplayName: Nikodem
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-04-05T07:51:44.833Z
Answers.Text: Sure, simonm101
Answers.UserDisplayName: Simon M
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2024-04-04T20:20:37.100Z
Answers.LastActivityDate: 2024-04-04T20:20:37.100Z
Answers.UserDisplayName: Simon M
Body: <p>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.)</p>
  <p>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?</p>
  <p>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.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-02T12:32:37.740Z
Comments.Text: (+1) For your analysis, how are you handling "missed forced mate" or when one blunders a forced mate?  Are these excluded?  Is this set to a certain value?  Seems this could affect the answers.
Comments.UserDisplayName: SecretAgentMan
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-02T12:35:52.077Z
Comments.Text: Related (not a dupe): [How is average centipawn loss calculated when a mate is missed?](https://chess.stackexchange.com/q/22087/26335)
Comments.UserDisplayName: SecretAgentMan
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-20T16:45:13.077Z
Comments.Text: Related:  [Centipawn Loss Distribution](https://chessenginelab.substack.com/p/centipawn-loss-distribution)
Comments.UserDisplayName: SecretAgentMan
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-04-21T18:59:10.370Z
Comments.Text: This is a very nice analysis, @SecretAgentMan! Could you enlarge it to other games of other players? (e.g. by using the evaluated database database.lichess.org mentioned below by @Simon). I am most interested in the large average distribution over games of many players having similar strength (rating).
Comments.UserDisplayName: Nikodem
Tags: <evaluation>
Title: Statistical distribution of centipawn losses
ThreadText: # Statistical distribution of centipawn losses
  
  (Asked by: username_0 on 2024-04-01)
  
  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.
  
  **Tags:** evaluation
  
  > **Comments:**
  > **username_1**: (+1) For your analysis, how are you handling "missed forced mate" or when one blunders a forced mate?  Are these excluded?  Is this set to a certain value?  Seems this could affect the answers.  
  > **username_1**: Related (not a dupe): [How is average centipawn loss calculated when a mate is missed?](https://chess.stackexchange.com/q/22087/26335)  
  > **username_1**: Related:  [Centipawn Loss Distribution](https://chessenginelab.substack.com/p/centipawn-loss-distribution)  
  > **username_0**: This is a very nice analysis, @username_1! Could you enlarge it to other games of other players? (e.g. by using the evaluated database database.lichess.org mentioned below by @username_2). I am most interested in the large average distribution over games of many players having similar strength (rating).  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2024-04-04. Score: 3
  
  This is an interesting question.
  
  We can plot the distribution using data from the [Lichess database](https://database.lichess.org/) 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)
  
  [![Centipawn Loss (Blitz)](https://i.sstatic.net/GI5qC.png)](https://i.sstatic.net/GI5qC.png)
  
  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").
  
  [![Centipawn Loss Density](https://i.sstatic.net/ZKV9r.png)](https://i.sstatic.net/ZKV9r.png)
  
  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.
  
  [![Log-density](https://i.sstatic.net/taYyd.png)](https://i.sstatic.net/taYyd.png)
  
  This result (discrete probability mass at 0, and then strictly descreasing pdf beyond that) is consistent with [Regan's model](https://cse.buffalo.edu/%7Eregan/papers/pdf/ReHa11c.pdf) 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
  
  > **Comments:**
  > **username_0**: 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)  
  > **username_2**: Sure, simonm101  
  
  ---
  
  