Author: Odd Gunnar Malin
Date: 00:27:48 04/19/05
Go up one level in this thread
On April 18, 2005 at 12:56:55, Eelco de Groot wrote: >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. > There was posted a link a few weeks ago to a page with a formula that gives a little better approx than the logarithm formula. I didn't save the link but here is what I put as comment into my program. // Alghorithm from Simon Schmitt's site on the net. // Die offiziellen Elo-Zahlen werden nach folgender Formel errechnet: // Erwartung = 0,5 + 1,4217 x 10^-3 x D // - 2,4336 x 10^-7 x D x |D| // - 2,5140 x 10^-9 x D x |D|^2 // + 1,9910 x 10^-12 x D x |D|^3 // wobei gilt: // D = Differenz = Elo(eigene, alt) - Elo(Gegner, alt) // |D| = Absolutbetrag von D Odd Gunnar
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