A question about SD

[TheShootah]

Call me crazy or stupid. I was talking to Howard Lederer about Standard Deviation and I just don't get it. When I took college stats in High School, I of course learned the equation of Standard Deviation. Now that equation which is blah blah over I think (n-1) of course implies that the SD decreases with a bigger sample. But he told me he did alot of work with this stuff and claimed and that at the table SD increases as radical n. I don't get it. With more hands played, how does the deviation get bigger? Am I dumb or misinterpreting something here?

[PokrLikeItsProse]

You are asking about the standard deviation of what quantity?

I think that if you are talking about the standard deviation of stacks at the table, the average stack size remains the same, but the standard deviation among sizes of the stacks increases with the square root of time, here measured by number of hands.

[TheShootah]

That is must be what he was talking about, and I was talking about the SD of win rates....so when you play more hands, your deviation should get smaller eh?

[PokrLikeItsProse]

OK, I see what he was getting at now. He was probably talking about the size of your bankroll (assuming you funnel all money won into your 'roll). The longer you play, the greater the volatility; you can win more or you can lose more in a ten-hour session than in a two-hour session no matter your skill level and standard deviation is basically a measure of volatility. You are going to see a higher standard deviation in your win rates if you are comparing four-hour session to two-hour sessions. This increase in standard deviation is proportional to the square root of time (measure in number of hands).

So, the standard deviation of net profit is proportional to the square root of the number of hands, but you are asking about the standard deviation of hourly win rate, so divide both sides by n. Do some magic involving l'Hospital's rule and you find that as your number of hands approaches infinity, the standard deviation of your hourly win rate is proportional to 1 over the square root of n. So, you become more confident of what your hourly win rate is as you play more hands, but the range of what your bankroll might be explodes. For a winning player, the more hands you play, the more likely you are to go broke (but also, the more likely you are to have doubled or tripled your bankroll--neither of those are likely to occur in ten sessions, assuming you are properly bankrolled).

So, he was talking about how much you win and you were talking about how much you win per hour.

[TheShootah]

Dang it, I had a chance to talk to one of the best pros, and he and I were holding different conversations!

[dana33]

Your confusion probably comes from the fact that there are multiple standard deviations relevant to poker, and you need to be precise about which one you are talking about.

SD/100 is the standard deviation in your bankroll per 100 hands you play. The estimate of this SD gets more precise as you play more hands, but its value trends neither bigger nor smaller.

The SD/100 can be used to estimate the uncertainty in your computed win rate per 100 hands. This uncertainty is properly called the standard error in the mean (where the "mean" here is your mean win rate), but is sometimes loosely referred to as standard deviation. This value scales as 1/sqrt(n) for n hands played. I.e., the more hands you play, the more confident you are of your win rate.

The "SD over all hands played" (for lack of a better term) is a measure of the fluctuations in your bankroll over your entire playing lifetime (or some subset of this time). This SD scales as sqrt(n). I.e., the longer you play, the bigger the swings that you'll experience.

So, depending on which "SD" you are talking about, it may remain roughly constant, decrease, or increase with the number of hands you play.