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Subject: Re: A question about statistics...

Author: Peter Fendrich

Date: 16:03:58 01/06/04

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On January 06, 2004 at 14:03:05, Dieter Buerssner wrote:

>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

I'm sorry I've been quite busy today and will be for a while.
I must confess that I didn't study your examples carefully enough.
Anyway I'm not sure what you're after with the variables above. The variance in
the sample is what it is and is computed as I told before. In order to estimate
the standard deviation of the population and to use preknown distribution of
W/D/L you are into the Bayesian thing. Is this what you want?
Maybe we can continue by email - I will send you one when things are calmed down
here...

I did put your examples into my formulas and got slightly different Rating-Dif's
than you have. Maybe it's just decimal errors. I also used another formula than
you did.
I know nothing about Elostat and have never used it and can't really tell what's
inside. The "W=20, D=980, L=0" case seems to be wrong by Elostat. A lower bound
of 0 in the 95%-interval after 1000 games just can't be right.
I think that you should be able to check that with your simulations.

I will get in touch with you within a few days.
Regards
/Peter



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