Are Winrates Normally Distributed?

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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.

[jetsonsdogcanfly]

There are two easy numbers that reflect the "normalness" of the distribution. Skewness measures the bias towards one side of the mean, and kurtosis measures the fatness of the tails of the distribution. In excel, you can easily get these numbers using the functions skew() and kurt(), with the arguments for the functions being simply the winrate series. Can you do that, and post or PM me the results, for each of the block sizes?

[B Dids]

I liked this better when you tried to explain it drunk off your ass at Craftsteak.

[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.

Yes, you are right. Max loss is -12BB and Max win is 9*12BB = 108BB - rake.

[Justin A]

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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

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Ok you seem to know a lot about stats. When I first started looking into this I did so because I was wondering if the confidence interval calcs we've done in the past are accurate when dealing with BB/100. So if you have a winrate of x bb/100 after 100k hands or whatever, then we do a calc for a 95% or 99% or whatever confidence level we choose to find out where are true winrate most likely falls. Likewise we can do the same for level of confidence that winrate > x.

So my question to you, is how accurate are these confidence intervals when dealing with a statistic that is not distributed normally?

[MaxPower]

If the distribution is not normal but not much different from the normal distribution, then in practical terms I don't think it would make much difference. Even if it were not strictly accurate, it would be good enough and probably not worth doing all the extra work to get an accurate confidence interval.

I am still not convinced that win rates are not normally distributed around the mean.

If you want to base you confidence interval on the actual distribution, you might look into bootstrapping.

[oreogod]

you pictures are huge, even on 1600*1200 in makes reading this thread no fun. If u can, resize them before u post next time.

[jason_t]

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you pictures are huge, even on 1600*1200 in makes reading this thread no fun. If u can, resize them before u post next time.

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He can resize them right now through photobucket.

[DcifrThs]

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If the distribution is not normal but not much different from the normal distribution, then in practical terms I don't think it would make much difference. Even if it were not strictly accurate, it would be good enough and probably not worth doing all the extra work to get an accurate confidence interval.

I am still not convinced that win rates are not normally distributed around the mean.

If you want to base you confidence interval on the actual distribution, you might look into bootstrapping.

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one thing thats important to consider is the nature of the process that drives the win rate of a given player or a pool of players.

in discrete time, its easier to deal with but when we move to continuous time, the driving force could be a set of stochastic processes which COULD nullify any inferences made from using the current normal distribution as a base for analysis.

basically, if random processes drive parts of winrate (one process could be how a given person does x,y, or z and have it based on randomness or even have real life like jumps-like the poker graphs show- by making those processes brownian motions that accumulate quadratic variation at rate 1 per unit time) then we wont see the distribution as normal or even a good approximation unless ALL processes meet a few criteria:

-they all have to individually be random and not deterministic (though they can change over time, so long as its random)

-their drift/diffusion (mean/variance) must be adapted to the SAME information that drives the whole system

-they are jointly normally distributed

NOTE: these are some seroiusly strict conditions ... especially the last one.

if these are met then the distribution of the results of the process may approximate a normal distrubution with some confidence.

either way, studies have shown that almost all biological/psychological phenomenon are normally distributed or very easily and readily approximated by a normal distribution. since the win rate is driven by largely biological phenomenon, it would seem as if on a large enough scale, the results of the win rate observations would converge to a normal distribution as well.

the whole thing is interesting and i like thinking about it but im not good enough at all types of higher level math to write out a proof of this...

well, its bedtime.

Barron