How many hands for Std. Deviation to Converge?

[mmcd]

I just started a new PT database on a new computer and my standard deviation seems to be abnormally low (12.81 ~15k hands). IIRC in my old (100k+) PT database it was somewhere around 16. How many hands does it take to get an accurate read of standard deviation?

[AaronBrown]

This is not an easy question. There is a theoretical answer if you are drawing from a Normal distribution, but it is not useful in practice for this question. The key is something called "kurtosis," basically, the size and frequency of extreme variations.

If the real question is whether your standard deviation has decreased, the best way to answer it is to look at subsamples from your large database, say 5,000 hands each. If you frequently had runs with standard deviations below 12.81, then the current run is reasonably likely to be the result of chance. But if standard deviations of 12.81 were rare or non-existent in the older database, it seems likely that your standard deviation has really gone down (although not necessarily all the way to 12.81).

[yellowjack]

Sorry to threadjack (AaronBrown seems have answered the question already anyways) but can someone post their thoughts on standard deviation and how it is seemingly 5x the size of a winrate (i.e. 3BB/100, s.d. of +/- 15BB/100)?

From what I recall about s.d. in my early days with normal distributions, 68% of all possibilities lie within 1 s.d. of the mean. So in the 32%/2=16% of the best/worst possible outcomes, your winrate is -12BB/100 or worse OR +18BB/100 or better?

Is this correct, or oversimplified? Also, is there any reason not to assume a normal distribution? Even if a hand doesn't follow this, with the CLT I think that it should.

[AaronBrown]

The generalization that standard deviation is about 5 times win rate is an empirical observation from Poker, there's no math behind it. I'm not sure it's accurate, in my experience bad players have the highest standard deviations and they have negative win rates.

There are reasons to doubt the Normal distribution. The Central Limit Theorem applies only to independent observations. If there are any patterns in your play, it could affect the distribution even over long periods of time. You might play differently against different opponents, or at different limit structures, or after wins versus losses. Other people might play differently against you.

There is a certain kind of professional player, who plays mostly in cardrooms and casinos, that seems to have reasonably constant standard deviation, at least at a specific game and limit structure, and therefore reasonably Normal long-term results. However, this is based on anecdotal and self-reported statistics, so it's hard to be too sure. Most players have obvious patterns and non-Normality in their results.

Still, it's not a bad approximation to assume 1/6 of the time you will do worse than one standard deviation below the mean (-12 BB in your example), 1/3 you'll be below the mean but within one SD (-12 to 3), 1/3 you'll be above but within one SD (3 to 18) and 1/6 you'll be more than one SD above. You remembered your Normal theory right.

What might not be true, and in my experience is not true for most players, is that you can project these results over a year's play by multiplying by the square root of the number of 100-hand sessions played. Most players will have varying means and standard deviation over the year, and you can spot these in their results.

[mmcd]

Went -100bb then back to +10bb in a ~1200 hand session today, and that brought it right up to 14.3.