Are Winrates Normally Distributed?

[sthief09]

one of the possible conclusions that justin a came up with is that maybe BB/100 is not the optimal measure for those who want to do tests on it. maybe we could get a more accurate standard deviation from BB/1000, though for most people, playing that many hands before getting a standard deviation would be infeasible and plain annoying

[felson]

I think PTjvs is dead on. Also, this effect is strongest when the blocks of hands are very small. If the block is just one hand in length, then (in a 10-handed game) around 90% of your sample points will be <= 0, and about 10% will be greater than zero. As the block size gets larger, the median block value tends towards the mean. You can see this reflected in the plots, which shift to the right as the block size gets larger.

[UprightCreature]

For the reason PTjvs states the winrate for one hand should not be normally distributed. There is an interesting fact though, that is the distribution of groups of a samples from a non-normal distribution aproach normal as the size of the group increases (Central Limit Theorem). Eg. winrate/1000 will be more normal than winrate /10.

[Derek123]

This would suggest that winrates are not normally distributed, which would mean you are more likely to run good but the bad runs will be worse.


This sentence seems backwards to me. If it is skewed to the left, there are more instances of bad, but the few big wins make up for it.

[Justin A]

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This would suggest that winrates are not normally distributed, which would mean you are more likely to run good but the bad runs will be worse.


This sentence seems backwards to me. If it is skewed to the left, there are more instances of bad, but the few big wins make up for it.

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Yeah you're right. I think Josh got it mixed up.

In an extreme sense, it's like we're usually treading water with a few really good runs in between that makes our results better.

[MarkD]

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In an extreme sense, it's like we're usually treading water with a few really good runs in between that makes our results better.

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LOL, this is funny because this quote seems to very accurately describe my experience at poker. Nice big bursts between periods of losing or breaking even.

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In an extreme sense, it's like we're usually treading water with a few really good runs in between that makes our results better.

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LOL, this is funny because this quote seems to very accurately describe my experience at poker. Nice big bursts between periods of losing or breaking even.

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This has been my experience as well. Since I moved up to 15/30 in May and later 20/40 in September, I have never had a losing month, but I did make about half my money in one 30 day span in which I ran insanely well, and as a result, played a ton of hands.

What the other poster said about taking random hands and combining them to make a sample is appropriate. As tilt proof as all of us think we are (or aren't), it is still perhaps not an accurate statement to call each group of 100 hands independent. Combining hands from different sessions to form samples would be a much better indicator of overall play in my opinion.

[disjunction]

http://forumserver.twoplustwo.com/showth...rue#Post3134879

Also there is no reason that I know of to believe that winrates are normally distributed. The normal distribution is a hammer but not every problem is a nail.

Edit: Also I forgot to say in the linked post that a bad table or a bad seat, rather than bad play, can be "Mr. Hyde".

Hi Josh,

I'll take a stab at addressing a few points.

The fundamental Random Variable in poker is the amount of money you win on one hand. This random variable has a distribution, which is certainly not Guassian.
First off, it's a discrete random variable. The mean is your winrate per hand. The max value values it can take are +12BB and -12BB (on Party Poker). The most probable event is 0, since you fold most hands.
Other frequently occurring values are -0.50BB and -0.25BB since these are the values you lose when you fold your blinds, and maybe -1.5BB since this is how much you lose when you raise pre-flop, completely blank the flop, bet the flop, and get raised.

So we get a sense of what the probability mass function of this random variable looks like: It's centered at your winrate (say .02bb) but its peak value is at 0. Then it has smaller peaks at popularly occuring values, such as -0.50BB, -0.25BB, etc. It is, obviously, not a normal distribution.

The Central Limit Theorem tells us that if we ADD together enough of these strange random variables, the sum, regarded as a random variable, must start looking more and more Guassian.

In your charts, when you group together a string of hands, you are adding all the random variable in each group, and this sum should starting looking Guassian the larger the group is (BB/1000 should look more Guassian than BB/10). With a 150k hand sample, I don't think you have enough hands to get a graph that shows this, since if you went to, say, BB/1000, you would only have 150 sample points. But I'm pretty sure that at some point, it would look like a nice bell-shaped curve.

Edit: You can start to see at BB/50 how the graph is looking more Guassian. Below BB/50 you have the nice feature that you have many smaple points. BUT each sample point is not yet being taken from a very Guassian distribution. Above BB/50 (BB/100 and up), you have the nice feature that the samples are being taken from a pretty Guassian distribution, BUT you don't have enough samples to draw the curve. If your DB was much larger, I think you would see the BB/100 look much closer to Guassian than the BB/50.

-v

[damaniac]

Isn't your theoretical max winrate (or win) for a hand 12BB x N(number of players)? You can only lose 12 bets but you can certainly win far more.