Body: <p>What is the <em>quantitative</em> predictive value of Elo rating for tournament performances? (Or, same, what is the natural strength variabitity of an average player, measured in Elo?)</p>
  <p>A quantitative version, so to say, of <a href="https://chess.stackexchange.com/questions/4884/how-well-does-elo-rating-estimate-a-players-strength">this question</a>, but the answers are unsuitable, although #3 points in the right direction.</p>
  <p>The question has some practical value: assume you get 4.0 points where your Elo predicted 3.0 (D=H-E=+1.0) - are these to read as, on some arbitrary confidence level, rather as 3.0 ± 2.0 (you were just lucky) or 3.0 ± 0.5 (you played good)?</p>
  <p>First of all, the Elo rating is a snapshot in time, averaged out a bit by past tournaments, and essentially assumed to be a constant measure of strength. We thus should expect that juniors and seniors in average have larger differences D.</p>
  <p>Let's look at a (somewhat) random tournament (pre-Corona by suggestion of Brian), <a href="https://www.schachbund.de/dem2015.html?file=files/dsb/spielbetrieb/2015/dem/Elo-Auswertung.pdf&amp;cid=36030" rel="nofollow noreferrer">German championship 2015</a>, and observe the last column, which measures the difference D of predicted and actual performance. Mean is ~0 (duh), standard deviation s=~13, interval [-32,+21]. I've seen many such columns and it looks pretty &quot;normal&quot; to me.</p>
  <p>Thus two natural questions:</p>
  <ol>
  <li>How large is s for an &quot;average&quot; tournament? (I.e. pick a large number of random tournaments, compute s for each, compute the average of that.) My own estimate is around 20.</li>
  <li>Of course there are no &quot;average&quot; tournaments anyway - for example, I expect that s will be far higher in a lower Elo tier. Is that correct?</li>
  </ol>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-08-01T20:18:37.730Z
Comments.Text: Are you asking for a confidence or a prediction interval (not the same)?  What you provide as an example is not a "variance" mathematically.
Comments.UserDisplayName: SecretAgentMan
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2024-06-01T09:56:50.633Z
Comments.Text: USCF makes use of this for rating bonuses when a person does significantly better in a tournament then predicted.   Their opponents are also related in an addational calculation phase using the fast improving players post tournament rating rather then published rating.
Comments.UserDisplayName: Ian Ringrose
Tags: <statistics><elo>
Title: "Variance" of Elo rating
ThreadText: # "Variance" of Elo rating
  
  (Asked by: username_0 on 2022-08-01)
  
  What is the *quantitative* predictive value of Elo rating for tournament performances? (Or, same, what is the natural strength variabitity of an average player, measured in Elo?)
  
  A quantitative version, so to say, of [this question](https://chess.stackexchange.com/questions/4884/how-well-does-elo-rating-estimate-a-players-strength), but the answers are unsuitable, although #3 points in the right direction.
  
  The question has some practical value: assume you get 4.0 points where your Elo predicted 3.0 (D=H-E=+1.0) - are these to read as, on some arbitrary confidence level, rather as 3.0 ± 2.0 (you were just lucky) or 3.0 ± 0.5 (you played good)?
  
  First of all, the Elo rating is a snapshot in time, averaged out a bit by past tournaments, and essentially assumed to be a constant measure of strength. We thus should expect that juniors and seniors in average have larger differences D.
  
  Let's look at a (somewhat) random tournament (pre-Corona by suggestion of Brian), [German championship 2015](https://www.schachbund.de/dem2015.html?file=files/dsb/spielbetrieb/2015/dem/Elo-Auswertung.pdf&cid=36030), and observe the last column, which measures the difference D of predicted and actual performance. Mean is ~0 (duh), standard deviation s=~13, interval [-32,+21]. I've seen many such columns and it looks pretty "normal" to me.
  
  Thus two natural questions:
  
  1. How large is s for an "average" tournament? (I.e. pick a large number of random tournaments, compute s for each, compute the average of that.) My own estimate is around 20.
  2. Of course there are no "average" tournaments anyway - for example, I expect that s will be far higher in a lower Elo tier. Is that correct?
  
  **Tags:** statistics, elo
  
  > **Comments:**
  > **username_1**: Are you asking for a confidence or a prediction interval (not the same)?  What you provide as an example is not a "variance" mathematically.  
  > **username_2**: USCF makes use of this for rating bonuses when a person does significantly better in a tournament then predicted.   Their opponents are also related in an addational calculation phase using the fast improving players post tournament rating rather then published rating.  
  
  ---
  
  ## 0 answers
  
  