teamdonkey
05-03-2005, 08:25 PM
Irie's post set me to thinking, mostly about winrates and his guess that only 2% of online players beat the rake over the long haul (too low in my opinion, but that's a different post). This, combined with his luckiest 10% of the luckiest 10% theory, presents a real problem to the casual player: if you're winning, how can you tell if you're good or just lucky? The more you play, the more certain you can be, but how much do you have to play to be confident you're actually beating the game?
The answer is, there are statistical calculations that can help you out. These calculations give you a range, and your true win rate is somewhere in that range. The more you play, the tighter that range gets, and the more confident you can be in your win rate.
I play mostly small stakes ring games, and this (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=inet&Number=1342415&fpart= &PHPSESSID=) thread explains these calculations for those games. It looks imposing but once you get the formula into a spreadsheet it's simple to use.
I've never seen anyone use these type of calculations for SNG's, but it's really just a small change to Homer's formula. Instead of table hours use # of SNGs, and instead of EV/hr use your ROI. So it looks like this:
u = upper limit of your win rate
l = lower limit of your win rate
c = confidence interval
n = # of SNG's
ROI = ROI
sd = standard deviation
u=ROI-normsinv[(1-c)/1]*sd*[1/sqrt(n)]
l=ROI+normsinv[(1-c)/1]*sd*[1/sqrt(n)]
normsinv and sqrt are excel functions, not values you put in. The only thing you're missing is standard deviation. For ring games pokertracker calculates it for you, but for SNGs it's not as easy. If you have your results in a spreadsheet you can figure this yourself. It's not bad to use an approximation, everyone's will be similar after about 100 SNGs. I figured with a 16.3% ROI, with an equal distribution of 1-2-3 finishes, SD is 165 at 100 SNGs, and slowly drops to 164.2 after 10,000. With a 0% ROI, SD is 157.5 at 100 games, and drops to 156.7 after 10,000. Using any of these numbers will give you a very similar range. So if you're picky you can calculate your own SD, if you're lazy just use 160.
Now you plug your numbers into excel and can quickly see what your win range is.
For example: If i've played 100 SNG's with a 16.3% ROI, use 160 for SD, and 0.95 as the confidence interval, i can say this:
"i am 95% confident that my true ROI is between -15.1% and +47.7%."
As you can see, after 100 SNGs you can't tell crap. The range is huge, and you can't be at all sure you're even a winning player, even though you've made almost $1300. The range gets tighter the more games you play:
500: +2.3% to +30.3%
1000: 6.4 to 26.2
2500: 10 to 22.6
5000: 11.9 to 20.7
10000: 13.2 to 19.4
If you drop the confidence interval to 90% those numbers get tighter, but i think 95% is the most commonly used interval in these types of calculations. Obviously it takes a lot of tournaments to get a decent handle on your actual win rate.
*DISCLAIMER* - normsinv in excel is a statistical function that assumes the data falls into a normal distribution. With ring games this is probably a fair assumption, but with SNG ROI data it obviously isn't. I'm not sure how much affect that has on the resulting numbers. Someone with a better background in statistics might be able to provide a better solution, if it is in fact a problem.
The answer is, there are statistical calculations that can help you out. These calculations give you a range, and your true win rate is somewhere in that range. The more you play, the tighter that range gets, and the more confident you can be in your win rate.
I play mostly small stakes ring games, and this (http://forumserver.twoplustwo.com/showflat.php?Cat=&Board=inet&Number=1342415&fpart= &PHPSESSID=) thread explains these calculations for those games. It looks imposing but once you get the formula into a spreadsheet it's simple to use.
I've never seen anyone use these type of calculations for SNG's, but it's really just a small change to Homer's formula. Instead of table hours use # of SNGs, and instead of EV/hr use your ROI. So it looks like this:
u = upper limit of your win rate
l = lower limit of your win rate
c = confidence interval
n = # of SNG's
ROI = ROI
sd = standard deviation
u=ROI-normsinv[(1-c)/1]*sd*[1/sqrt(n)]
l=ROI+normsinv[(1-c)/1]*sd*[1/sqrt(n)]
normsinv and sqrt are excel functions, not values you put in. The only thing you're missing is standard deviation. For ring games pokertracker calculates it for you, but for SNGs it's not as easy. If you have your results in a spreadsheet you can figure this yourself. It's not bad to use an approximation, everyone's will be similar after about 100 SNGs. I figured with a 16.3% ROI, with an equal distribution of 1-2-3 finishes, SD is 165 at 100 SNGs, and slowly drops to 164.2 after 10,000. With a 0% ROI, SD is 157.5 at 100 games, and drops to 156.7 after 10,000. Using any of these numbers will give you a very similar range. So if you're picky you can calculate your own SD, if you're lazy just use 160.
Now you plug your numbers into excel and can quickly see what your win range is.
For example: If i've played 100 SNG's with a 16.3% ROI, use 160 for SD, and 0.95 as the confidence interval, i can say this:
"i am 95% confident that my true ROI is between -15.1% and +47.7%."
As you can see, after 100 SNGs you can't tell crap. The range is huge, and you can't be at all sure you're even a winning player, even though you've made almost $1300. The range gets tighter the more games you play:
500: +2.3% to +30.3%
1000: 6.4 to 26.2
2500: 10 to 22.6
5000: 11.9 to 20.7
10000: 13.2 to 19.4
If you drop the confidence interval to 90% those numbers get tighter, but i think 95% is the most commonly used interval in these types of calculations. Obviously it takes a lot of tournaments to get a decent handle on your actual win rate.
*DISCLAIMER* - normsinv in excel is a statistical function that assumes the data falls into a normal distribution. With ring games this is probably a fair assumption, but with SNG ROI data it obviously isn't. I'm not sure how much affect that has on the resulting numbers. Someone with a better background in statistics might be able to provide a better solution, if it is in fact a problem.