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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