zachvac
Legend
Silver Level
Statistics term, defined by the square root of the variance (which is the average squared distance from the mean).
This may seem complicated but it really isn't, for example say you have a data set of 1, 3, 5, 7, 9. The mean is 5, squares of the distance are 4^2 + 2^2 + 0 + 2^2 + 4^2 = 32 + 8 = 40. divide by 5 and you get 8 as the variance, sqrt(8) is the standard deviation. Although actually I think for some reason you divide by n-1 so you'd divide by 4, get 10 and sqrt(10), but that's irrelevant for large data sets anyway (if you've played 50k hands, does it matter that you divide by 49999 or 50k?).
Anyway, what does this mean? Well basically the relevance of this is in a normal set of data. I'm sure most have seen the bell curve. Basically a large data set that is determined randomly will have an approximately normal distribution.
Anyway, if we assume a normal distribution (which we can't, since in a normal distribution any amount of loss or win would be possible although extremely improbable while in NLHE if you leave before you get too deep you cannot win 1k big blinds in a hand) we can then approximate the probability of any given result in any given amount of time.
First off to apply this you have to learn how to determine the standard deviation of a sample when given standard deviation/100 hands. The way this works is that when you combine data sets, you add the variances, which means that to get standard deviation you square the standard deviations, add together, then take the square root again. For example, say you have an 80 bb/100 standard deviation yet you have 1k hands.
You would end up with this expression:
sqrt((80^2)*10) ~= 253 BB/1k hands
ok but now we get to how we can use this. I can post one later, but look on the internet for a z-table (think that's what it's called). Basically it lists the amount of area under a bell curve for each standard deviation. As a rule of thumb, 1 standard deviation either way encapsulates 68%, 2 is 95%, and 3 is 98%. So let's say that in these 1k hands you have won 253 BBs, meaning you are up the exact standard deviation in that amount of time (and also winning at an amazing 25.3 BB/100).
The probability of you actually being a losing player is the amount that is less than 1 standard deviation. If the 1 covers 68%, 32% is outside it and half of that, 16% is below that. So if you are winning at 25.3 BB/100 after 10 hands with an 80 BB/100 standard deviation, you can say with 84% certainty you are a winning player. Now try the numbers yourselves with bigger samples and more reasonable winrates. You'll understand why it's so easy for people to have upswings and downswings lasting 10k hands, but how if you play hundreds of thousands of hands and have a good winrate, you can be pretty certain you are a winning player. Also we can use this with other numbers as well, not just 0. For example say you've been running at 6 PTBB/100 for the last 20k hands. You can figure out the probability that you are at least a 5 PTBB/100 winner.
I think I've pretty much covered everything, the one thing I'm interested in actually though is the normal distribution. I don't think we can treat hands as if they follow the normal distribution simply because we will have a lot of small losses and a few big wins if we are good. With a normal distribution the odds of winning an amount x units above our mean (winrate) and an amount x units below our mean are identical. There's a test you can use to test how normal a data set is and I think I may run it on my data. I'll let you know how it goes.
This may seem complicated but it really isn't, for example say you have a data set of 1, 3, 5, 7, 9. The mean is 5, squares of the distance are 4^2 + 2^2 + 0 + 2^2 + 4^2 = 32 + 8 = 40. divide by 5 and you get 8 as the variance, sqrt(8) is the standard deviation. Although actually I think for some reason you divide by n-1 so you'd divide by 4, get 10 and sqrt(10), but that's irrelevant for large data sets anyway (if you've played 50k hands, does it matter that you divide by 49999 or 50k?).
Anyway, what does this mean? Well basically the relevance of this is in a normal set of data. I'm sure most have seen the bell curve. Basically a large data set that is determined randomly will have an approximately normal distribution.
Anyway, if we assume a normal distribution (which we can't, since in a normal distribution any amount of loss or win would be possible although extremely improbable while in NLHE if you leave before you get too deep you cannot win 1k big blinds in a hand) we can then approximate the probability of any given result in any given amount of time.
First off to apply this you have to learn how to determine the standard deviation of a sample when given standard deviation/100 hands. The way this works is that when you combine data sets, you add the variances, which means that to get standard deviation you square the standard deviations, add together, then take the square root again. For example, say you have an 80 bb/100 standard deviation yet you have 1k hands.
You would end up with this expression:
sqrt((80^2)*10) ~= 253 BB/1k hands
ok but now we get to how we can use this. I can post one later, but look on the internet for a z-table (think that's what it's called). Basically it lists the amount of area under a bell curve for each standard deviation. As a rule of thumb, 1 standard deviation either way encapsulates 68%, 2 is 95%, and 3 is 98%. So let's say that in these 1k hands you have won 253 BBs, meaning you are up the exact standard deviation in that amount of time (and also winning at an amazing 25.3 BB/100).
The probability of you actually being a losing player is the amount that is less than 1 standard deviation. If the 1 covers 68%, 32% is outside it and half of that, 16% is below that. So if you are winning at 25.3 BB/100 after 10 hands with an 80 BB/100 standard deviation, you can say with 84% certainty you are a winning player. Now try the numbers yourselves with bigger samples and more reasonable winrates. You'll understand why it's so easy for people to have upswings and downswings lasting 10k hands, but how if you play hundreds of thousands of hands and have a good winrate, you can be pretty certain you are a winning player. Also we can use this with other numbers as well, not just 0. For example say you've been running at 6 PTBB/100 for the last 20k hands. You can figure out the probability that you are at least a 5 PTBB/100 winner.
I think I've pretty much covered everything, the one thing I'm interested in actually though is the normal distribution. I don't think we can treat hands as if they follow the normal distribution simply because we will have a lot of small losses and a few big wins if we are good. With a normal distribution the odds of winning an amount x units above our mean (winrate) and an amount x units below our mean are identical. There's a test you can use to test how normal a data set is and I think I may run it on my data. I'll let you know how it goes.
R
RazzleDazzle
Rock Star
Silver Level
The standard of deviation will tell you the reproducibility of the statistic which the standard of deviation is describing. Of course, with poker, that requires that you play just as well each time (unless of course the statistic is something like how often your getting AA)
If your statistics are X during one session, and you have standard of deviation Y, 68% of the time your statistics during your next session will fall in the range of X +/- Y, 95 % will be between X +/- 2Y, and 99+% will fall in the range of X +/- 3Y.
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BelgoSuisse
Legend
Silver Level
That's because what you describe, zach, is the estimator of the variance of the random process based on one of its realisations, not the variance of the process itself. And in order to get an unbiased estimator, you need to divide by n-1, not by n.
See Bias of an estimator - Wikipedia, the free encyclopedia for more. It's not trivial, and not really related to Chuck's question.
Chris_TC
Cardschat Elite
Silver Level
Here are some recent stats. I'm kinda curious how this compares. It's all 6-max.
50NL (32364 hands) = $47.67/100 (47.67 ptbb/100)
100NL (30408 hands) = $98.54/100 (49.27 ptbb/100)
200NL (24153 hands) = $202.05/100 (50.51 ptbb/100)
400NL (27146 hands) = $388.91/100 (48.61 ptbb/100)
600NL (27066 hands) = $623.70/100 (51.98 ptbb/100)
It seems like my standard deviation is almost exactly one stack per 100 hands. I pulled this data from Pokeroffice btw, not from Pokertracker. But I guess they're the same numbers.
50NL (32364 hands) = $47.67/100 (47.67 ptbb/100)
100NL (30408 hands) = $98.54/100 (49.27 ptbb/100)
200NL (24153 hands) = $202.05/100 (50.51 ptbb/100)
400NL (27146 hands) = $388.91/100 (48.61 ptbb/100)
600NL (27066 hands) = $623.70/100 (51.98 ptbb/100)
It seems like my standard deviation is almost exactly one stack per 100 hands. I pulled this data from Pokeroffice btw, not from Pokertracker. But I guess they're the same numbers.
Steveg1976
...
Silver Level
Standard deviation makes a lot more sense when it is graphed out so that you can the mean and what one standard deviation actually is. I will try to post a graphic explanation later when I get home if zach doesn't beat me to it, it may or may not help.
zachvac
Legend
Silver Level
This has a decent graphical explanation: Standard deviation - Wikipedia, the free encyclopedia
The entire article though I think may be tougher to understand than my post lol, but that section seems decent.
zachvac
Legend
Silver Level
Also from that article it appears I could be wrong, that we don't use the 68-98-95 rule that we use for normally distributed data. Looks like if we want a 95% (94.5's close enough) confidence that we are above a certain winrate we can use 3 standard deviations, unless somehow it comes out normal. Like I said I think I'm going to run a test on my data.
ChuckTs
Legend
Silver Level
Well yes, the swingier your style, the higher your SD, and the more likely you are to go bankrupt if you don't pad your bankroll sufficiently.
As for how to actually change (lower obv) your SD by adjusting your game, well I'm sure you've read about high and low-variance play before.
I personally wanted to learn about it because I want to compare just how swingy FR, 6max and HU are...so far I've only played a few thousand hands of 6max and I wasn't sure the variance was worth it.
As for how to actually change (lower obv) your SD by adjusting your game, well I'm sure you've read about high and low-variance play before.
I personally wanted to learn about it because I want to compare just how swingy FR, 6max and HU are...so far I've only played a few thousand hands of 6max and I wasn't sure the variance was worth it.
NineLions
Advanced beginner
Silver Level
So how are you finding 6max verses FR? Different standard deviation showing up?
My current project is learning short stack; only 2,035 hands yet but Std Dev of 19.07 BB, 37.44bb. Maybe an indication of the short stack strategy?
ChuckTs
Legend
Silver Level
Yeah I think the shorter-handed you are (ie the looser/more aggressive you/your opponents play), the higher the variance will be. I'm so far seeing 34BB/46BB for FR/6max, thought it would be a much bigger jump.
I have absolutely no idea how long it takes SD to converge, so I'm not sure how accurate those shortstacking stats are, but 19 sounds incredibly low.
I have absolutely no idea how long it takes SD to converge, so I'm not sure how accurate those shortstacking stats are, but 19 sounds incredibly low.
NineLions
Advanced beginner
Silver Level
If your FR and 6 max are that close, what would that indicate? That your style is not as aggressive/swingy at 6 max relative to the average players at 6 max as compared with your play verses FR?