Are Winrates Normally Distributed?

[stinkypete]

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You just made an argument for running the sims with 100 hand sets.

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i think his point was that the samples don't provide a realistic approximation of 100 hand blocks since you could easily have, for example, 15 big blind hands, which you almost never would in an actual 100 hand block, so the distributions wouldn't be entirely comparable.

i don't think it really makes much of a difference though.

[Justin A]

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You just made an argument for running the sims with 100 hand sets.

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i think his point was that the samples don't provide a realistic approximation of 100 hand blocks since you could easily have, for example, 15 big blind hands, which you almost never would in an actual 100 hand block, so the distributions wouldn't be entirely comparable.

i don't think it really makes much of a difference though.

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Oh I get it now. Yeah that complicates things a bit.

[MaxPower]

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You just made an argument for running the sims with 100 hand sets.

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i think his point was that the samples don't provide a realistic approximation of 100 hand blocks since you could easily have, for example, 15 big blind hands, which you almost never would in an actual 100 hand block, so the distributions wouldn't be entirely comparable.

i don't think it really makes much of a difference though.

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Picking hands randomly is a better way to to control for these extraneous factors.

I could run 100 hand blocks, but I'm playing poker right now and it will crash my machine. I'll do it at work.

I don't know why you guys are so hung up on 100 hands. It is just an arbitrary number.

[stinkypete]

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Picking hands randomly is a better way to to control for these extraneous factors.


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why do you say that? it's a better way to control for factors like game conditions and tilt, but i don't see how it's a better way to control for position/blinds.

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I could run 100 hand blocks, but I'm playing poker right now and it will crash my machine. I'll do it at work.

I don't know why you guys are so hung up on 100 hands. It is just an arbitrary number.

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blame it on pokertracker pat.

[Justin A]

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I don't know why you guys are so hung up on 100 hands. It is just an arbitrary number.

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PT does everything in BB/100, so I'd like to know the significance of this stat over certain sample sizes.

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It looks pretty close to normal here, but I think the way you did the sampling is not quite right.

You need to draw random samples from the total group of hands. Chopping them up into blocks is easier, but not appropriate. The way you have done it, we might find a different results using a different database.

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Exactly.

It looks like the effect that you're seeing is caused by the fact that playing badly costs you more than playing well earns you. Your $/hand should still be normally distributed, but not if you break it up into temperal blocks because you're more likely to see effects of tilt or playing poorly (no offense intended) or bad tables, etc.

[MaxPower]

OK, I did run 10,000 samples of 100 hands each last night and I can post the results. But before I do, I want to make a point. I do not think the results of my simulation have any practical implications.

Why do we keep track of BB/100 and SD/100? We use these to determine our bankroll needs, how much we can expect to win (confidence intervals), how long one can break even, etc.

I don't know if Mason Malmuth was the first to apply these concepts, but he certainly popularized it. I assume that BB/hour and SD/hour were used in order to simply record keeping and computation. It could have been done per hand, but then you would need to keep track of how many hands you played.

With the advent of internet multitabling, BB/hour was replaced by BB/100, but once again the choice of 100 hands was arbitrary.

We could keep track of win rate and SD on a per hand basis and it would work just as well.

Obviously, the win rate for individual hands are not normally distributed (since you win/lose zero for a majority of you hands), but that does not matter.

What matters is your total sample size, how you compute your test statistic is not important (as long as it is accurate and consistent). We could make it BB/hand, BB/10, BB/1000, or BB/134 and it wouldn't matter.

So if I play 20,000 hands and my BB/100 is 1.5, I know that the sampling distribution for samples of that size is normally distributed.

The fact that the sd is based on 100 hands is also irrelevant, because we use the standard error to compute confidence intervals and that takes the number of hands played into account.

I'm not certain about the bankroll formulas, but I'm pretty sure that it is the same concept.

It has been about 8 years since I studied stats and I am a little out of practice, so please correct me if I am wrong here.

I will post the results of the simulation for those that are interested.

[MaxPower]

OK, this is 10,000 random saples of 100 hands each drawn from 164,724 hands at 15/30 with a win rate of 1.13BB/100.

First, the Descriptive Statistics:

The skeweness is positive and the ratio of the skewness to the standard error of the skewness indicates that it it different from the normal distribution. The distribution is positively skewed - as you can see the mean is higher than the median.

The same is true for the kurtosis. The value are more closely clustered about the mean than in a normal distribution.


This is a test of normality. The significance value indicates that this distribution differs significantly from normal.





These are the extreme sample values:


This is a histogram of the distribution with a normal curve superimposed:



These are some additional fun plots for the geeks out there:

[DcifrThs]

excellent post...i was thinking of the "normality test" but couldn't think of the kolmogorov smirnov name...

PS- even in their self named tests, the russions represent smirnov [img]/images/graemlins/smile.gif[/img]

Barron

[edtost]

from looking at the q-q plot, it seems that the upper tail of the poker data is fatter than the gaussian, and the lower is thinner. shouldn't this result in large downswings happening less often than a normal assumption would predict?

i need to spend some more time thinking about this, the repetition of trials inherent in poker makes this more complicated than a standard VaR calculation.