To compute the ELO error margin I have found this formula:
700*sqrt(4*scoreratio*(1scoreratio)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*0.4980.238)/sqrt(4000) = 9,66 ELO
Is this the most accurate formula to compute the error margin?
ELO error margin
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 Posts: 217
 Joined: Fri Apr 11, 2014 10:45 am
 Full name: Fabio Gobbato

 Posts: 1988
 Joined: Wed Jul 13, 2011 9:04 pm
 Location: Madrid, Spain.
Re: Elo error margin.
Hello Fabio:
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 zscore in a normal distribution in a twotailed test (for example: z < 1.96 for confidence ~ 95%). In the formula you found, sigma' = sqrt[4*scoreratio*(1scoreratio)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.
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.Fabio Gobbato wrote:To compute the ELO error margin I have found this formula:
700*sqrt(4*scoreratio*(1scoreratio)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*0.4980.238)/sqrt(4000) = 9,66 ELO
Is this the most accurate formula to compute the error margin?
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 zscore in a normal distribution in a twotailed test (for example: z < 1.96 for confidence ~ 95%). In the formula you found, sigma' = sqrt[4*scoreratio*(1scoreratio)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.

 Posts: 217
 Joined: Fri Apr 11, 2014 10:45 am
 Full name: Fabio Gobbato
Re: Elo error margin.
Thank you so much.
So with 95% confidence z is ~ 1.96
error = 1362*sqrt((score*(1score)drawratio/4)/(ngames1))
So with 95% confidence z is ~ 1.96
error = 1362*sqrt((score*(1score)drawratio/4)/(ngames1))

 Posts: 1988
 Joined: Wed Jul 13, 2011 9:04 pm
 Location: Madrid, Spain.
Re: Elo error margin.
Hello again:
If you want to check your results with an external tool, I programmed a few years ago one that can serve your purpose. The link is in my profile and in the boton [www] below this post, and the tool is called LOS_and_Elo_uncertainties_calculator.
Error margins are calculated with this model (zscore can be read from a standard normal distribution table or computed with approximations once given a level confidence) and LOS is calculated in two ways: with an assumption of normal distribution using the zscore (I called this one 'LOS with draws') and with a division of two integrals like Rémi Coulom in his WhoIsBest programme (I called it 'LOS without draws'). I agree that these names are misleading but the formulæ are better explained in the source code, which could be a pain (Fortran 95 plus my nonprogrammer status).
Good luck with Pedone development! My best wishes for you.
Regards from Spain.
Ajedrecista.
That's it! The other formula you found gives Elo intervals that are around 2.79% bigger than this one.Fabio Gobbato wrote:Thank you so much.
So with 95% confidence z is ~ 1.96
error = 1362*sqrt((score*(1score)drawratio/4)/(ngames1))
If you want to check your results with an external tool, I programmed a few years ago one that can serve your purpose. The link is in my profile and in the boton [www] below this post, and the tool is called LOS_and_Elo_uncertainties_calculator.
Error margins are calculated with this model (zscore can be read from a standard normal distribution table or computed with approximations once given a level confidence) and LOS is calculated in two ways: with an assumption of normal distribution using the zscore (I called this one 'LOS with draws') and with a division of two integrals like Rémi Coulom in his WhoIsBest programme (I called it 'LOS without draws'). I agree that these names are misleading but the formulæ are better explained in the source code, which could be a pain (Fortran 95 plus my nonprogrammer status).
Good luck with Pedone development! My best wishes for you.
Regards from Spain.
Ajedrecista.

 Posts: 217
 Joined: Fri Apr 11, 2014 10:45 am
 Full name: Fabio Gobbato
Re: Elo error margin.
Thank you very much.
I'll have a look at your useful tools.
I'll have a look at your useful tools.