Hello Fabio:
Fabio Gobbato wrote:To compute the ELO error margin I have found this formula:
700*sqrt(4*scoreratio*(1-scoreratio)-drawratio)/sqrt(ngames)
For example if I played 4000 games with these results:
W: 1534 D: 950 L: 1516
scoreratio = 0.502
drawratio = 0.238
The error margin is 700*sqrt(4*0.502*0.498-0.238)/sqrt(4000) = 9,66 ELO
Is this the most accurate formula to compute the error margin?
The error margin depends on the level of confidence you want: it is not the same 95% that 99%, for example. A higher level of confidence implies wider error margins with the used model, which is the same you found. I have written a lot on this subject in TalkChess, so you can use the search engine of the forum.
Dividing by
ngames or by
ngames - 1 is a question of taste because differences are very small unless
ngames is very low.
The number 700 is the key here. I explained it a little more in the following post:
Re: Critter 1.6 - Critter 1.4a, ponder ON/OFF.
I call s = sqrt{[score*(1 - score) - draw_ratio/4]/(
ngames - 1)} and then |error| = 1600·z·s/ln(10), where z is the z-score in a normal distribution in a two-tailed test (for example: |z| < 1.96 for confidence ~ 95%). In the formula you found, sigma' = sqrt[4*scoreratio*(1-scoreratio)-drawratio]/sqrt(ngames) ~ 2·s.
Then, in my case |error| ~ 694.87·z·s ~ 694.87·z·(sigma')/2, but I guess that z ~ 2 (circa 95.45 % confidence) and you get 694.87·sigma' ~ 700·sigma'. Using 700 means using z ~ 2.0148 or a confidence interval of 95.61% more less if I am not wrong.
Usual values of z are 1.96 (in reality 1.95996398...) and 2, but you can see that differencies are small. I hope you can follow the explanation of the formulæ and decide yourself. Please note that a draw ratio of 100% or near it would give very low values of standard deviation and this model cracks. The same with score < 15% and score > 85% (or other values near these ones, just say in Elo gaps over 300 Elo), where score*(1 - score) is also very low and the model cracks again.
Regards from Spain.
Ajedrecista.