Author: Eelco de Groot
Date: 09:56:55 04/18/05
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On April 17, 2005 at 05:26:22, Tony Hedlund wrote: >On April 16, 2005 at 09:32:14, Eelco de Groot wrote: > >> Hi Anson >> >>For a first order approximation, I always use 2812 - (50%-45%)*7 =2777, which is >>increasingly inaccurate with greater score-difference. Precise formula involves >>calculating an integral, which you can do numerically. >> >>I once wrote a little program for this for a Casio calculator, but that will not >>help you much I'm afraid without an explanation. It took the first order >>approximation and went from there. But it is not hard to do if you have the >>formula from professor Elo, Excel can do it too, easily. I don't have the >>formula at hand right now. But it is a nice programming excercise! >> >> Eelco > >To my knowledge USCF uses Rp = Rc + 400(W-L)/N which I have in a Excel >spreedsheet. But it's not that hard to use pen and paper either. > >Tony > Hello Tony, Yes it may well be true that the USCF uses that formula. Professor Arpad Elo suggested this linear approximation in a time that calculations were commonly only done with slide-rulers or pen and paper. No calculators or computers to do the hard work for us! The actual graph of scoring-percentage versus ratingdifference assuming a Normal distribution (bell curves and all that) looks like an extended letter S, centered on 0 Elo, 50%. This formula Rp = Rc + 400(W-L)/N has the advantage that it can be used for larger ratingdifferences, first diving under the S, then over it. Instead of 7 elo per percent it uses 8. The formula I gave would be identical to Rp = 350(W-L)/N, it has the advantage that it is to my knowledge the actual tangent to the graph at 0 Elo, 50%. At least, that is what came out of my CASO FX 501P experiments, but I haven't proven it analytically! So it is a straight line following the S first very closely going from 0 Elo, 50% until the S starts bending away at the two tails. I think that the formula Kolls gave is the Logistic formula that Prof. Elo also treated in his book. I will have to look that up later. As far as I know the Integral of the Normal distribution has no analytical solution so it can only be approximated by doing a numerical integration, not by an exact formula. But the Logistic formula follows the bend of the S very closely, only at the tails of the S it curves away a bit more under it at the top end, over it at the lower end. This formula was used in other fields too for predicting results. What is the effect of the different formulas on the end calculation for overall Elo's for the whole pool of results and players, I honestly don't know precisely. I don't think that calculation uses the TPR formula, so it is not critical for the SSDF lists for instance. Regards, Eelco P.S. I don't have Prof. Elo's book with me, so I can't look this Logistic formula up! Later..
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