# Computer Chess Club Archives

## Messages

### Subject: Re: A question about statistics...

Date: 11:03:05 01/06/04

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```On January 05, 2004 at 18:11:25, Peter Fendrich wrote:

>The rest is basic formulas for
>the standard deviation.

Actually, I am puzzled about these basic formulas for the standard deviation.
Somehow I don't get it (in German, ich vermute, ich sitze auf der Leitung).

Anyway, I made a few experiments. I did some Monte-Carlo simulation of matches
(its easier, than to figure out the mathematics - actully I can figure it out
with only wins and losses, but not with draws, too). When I assume, that white
wins/draws/losses and black wins/draws/losses for the player have the same
probabilties, and I fit the data m vs. p(m) with a Gauss-function I get exactly
sigma = s/sqrt(n) with your formula.

s=sqrt((W*(1-m)^2 + D*(0.5-m)^2 + L*(0-m)^2)/(n-1))

Not too surprising the function fits perfectly, with the deviations randomly
distributed and of an order of magnitude as one can expect from the Monte Carlo
method (proportional to 1/sqrt(N), N number of matches used in the simulation).

When the Monte-Carlo method simulates 1 million games, there are about 4
significant figures, that are the same from the fit and the formula.

However, the function m vs. p(m) looks different, when I assume different w/d/l
probabilities for white and black. So, we not only have W, D, L, but wW, wD, wL,
bW, bD, bL and say nb, nw, mb, mw (typically nb=nw). Can you also give a formula
for this scenario for s?

What do you think of the margins given by Elostat - something seems wrong there
in the case of the W=20, D=980, L=0.

Regards,
Dieter

```