[jason1990]

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The formulas assumes a bias in that you might be playing better on one day than another or that there is some session to session difference.

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I assume you mean that the user of the formula is (implicitly) assuming a bias in that you might be playing better on one day than another or that there is some session to session difference. (Since a formula cannot assume anything. [img]/images/graemlins/smile.gif[/img])

Anyway, I disagree completely (with what I think you're trying to say). Any statistician who wanted to prove something about the accuracy of this formula, whether asymptotically or for specific sample sizes, would undoubtedly assume independence of the session outcomes, as well as some consistency in the probability distributions of the session results. They obviously couldn't be identically distributed since the session times are different, but a natural assumption would be that the variance of each session is proportional to the duration of that session, and that the constant of proportionality is the same for all sessions. Without these assumptions (which wouldn't be there if you assumed some sort of bias or session to session differences), you would be hard pressed to prove anything about the reliability of this estimator. This is precisely why you should recompute your standard deviation whenever game conditions change. You may not always know when they change, but when you do, you must correct for this phenomenon, because it is not part of the original formula.

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You're confusing hands with session-hours. You can't chop up the hands and create your own sessions. That's not how the formula works.

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Again, I completely disagree with the claim that I cannot chop up the hands and create my own sessions. Consider an online player who plays long hours at multiple tables. Suppose he plays 1000 hands per session on average and plays 30 sessions. He then applies the formula and gets a number, call it \sigma_1. Suppose he then chops up each session into 10 "artifical" sessions of 100 hands each. He now has 300 "sessions" and he applies the formula again, getting a new number \sigma_2. Now, I'm a probabilist, not a statistician, but I'm sure that even I could prove that \sigma_2 is a more reliable estimator of his true standard deviation than \sigma_1, in the sense that \sigma_2 has a smaller variance. In fact, it's no great leap to believe that the optimal estimator (in the sense of having the smallest variance) is obtained by using the formula with 30000 1-hand sessions. (Of course, there's a subtle difficulty with all of this, because the smaller the session length, the less it will behave like a Gaussian. But Gaussian assumptions are not strictly necessary to compute and compare the variances of different estimators.)

Obviously, the formula is most valuable for people who do not have hand-by-hand results. (For example, a B&M player who has a diary containing only the results of each night.) It is presented in a way that they can use, since they don't have the ability to chop up their sessions into single hands, let alone into sessions of equal length.

I don't know your mathematical background, so I may have assumed you have a higher degree of familiarity with probability lingo than you do. If so, I apologize. Anyway, my original question still stands:

"A good rule of thumb is to have at least 30 observations (playing sessions) for the estimate to be reasonably accurate." How long should these playing sessions be in small stakes Holdem?