Author: Eelco de Groot
Date: 15:05:26 04/26/05
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On April 26, 2005 at 16:49:48, Eelco de Groot wrote: >On April 26, 2005 at 14:41:25, Dieter Buerssner wrote: > >>Eelco, did you compare the numbers calculated with the approximation formula to >>those given in the FIDE handbook? There was a larger difference than what you >>have given as the error, when I tried it. I used a very accurate version of >>erf() by Stephen Moshier (I think, 15 digits correct). I used the following >>formulas: >> >>p = 0.5+0.5*erf(dp/400) >> >>and >> >>p = 1/(1+10^(-dp/400)) >> >>p is the expected score, dp the rating difference. IIRC, USCF uses the later >>formula based on the logistic distribution. In the center (rating difference >>small), both formulas give practically identical values. In the tails, they >>differ (not much in absolute terms, but significantly taken into relation). The >>numbers given on the FIDE site where in between. >> >>Regards, >>Dieter > >Hi Dieter, > >Are we talking about the same approximation formula? The elo-numbers that Odd >Gunnar Malin computed for the Hastings formula came out pretty well compared to >what Excels own approximation of the normal distribution would give. > >http://www.talkchess.com/forums/1/message.html?422011 > >And Hastings best approximation could be found in the manuals for both Casio's >and Texas Instruments programmable calculutors. (I have the Casio manual that >came with my FX-501P calculator, bought that in 1980, it had no less than 128 >programmable steps!) This approxiation is probably around since the 1950's but >for calculator use (8 digits or thereabout) it should work well enough. > >Regards, Eelco These are some numbers that I just got with Hastings approximation in Excel '97 Visual Basic, compared with Excel's STAND.NORM.VERD: x phi(x)Hastings STAND.NORM.VERD Exact(*) 1 0,841344555 0,84134474 0 0,499999429 0,5 0,5 0,1 0,539827365 0,539827896 0,539827837277029 0,2 0,579259199 0,579259687 1,3333333 0,908788608 0,908788713 2 0,977249913 0,977249938 2,5 0,993790314 0,99379032 3 0,998650031 0,998650033 3,5 0,999767326 0,999767327 4 0,999968314 0,999968314 In Dutch we use a comma instead of a decimal point, Standaard Normaal Verdeling is Standard Normal Distribution. Curiously around zero the discrepancy between the twoseems largest, phi(0) should of course be 0.5 I put in the following macro: Function phi(x) As Double Dim sndf, t As Double p = 0.2316419 c1 = 0.31938153 c2 = -0.356563782 c3 = 1.78147937 c4 = -1.821255978 c5 = 1.330274429 Pi = 3.14159265358979 sndf = Exp(-(x * x) / 2) / Sqr(2 * Pi) t = 1 / (p * x + 1) phi = -((((c5 * t + c4) * t + c3) * t + c2) * t + c1) * t * sndf + 1 End Function (*)Exact value as given here http://math.uc.edu/~brycw/preprint/z-tail/z-tail.htm that also mentions Hastings approximation accredited to a TI manual. So, an erf() implementation with 15 decimal accuracy is undoubtedly better, but for the elo list this at least is better that what FIDE seems to think is the Normal Distribution, if that in fact is what they are using :) Regards, Eelco
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