Normal Distribution -- Applicable to Winrates?

[75s]

No more than 1/k2 of the values are more than k standard deviations away from the mean

Equality holds exactly for any distribution that is a linear transformation of this one. Inequality holds for any distribution that is not a linear transformation of this one.

The theorem can be useful despite loose bounds because it applies to random variables of any distribution, and because these bounds can be calculated knowing no more about the distribution than the mean and variance.

Chebyshev's inequality is used for proving the weak law of large numbers.

[JavaNut]

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Javanut,

It doesn't matter whether we're talking about cash games, STTs, MTTs, or anything else. With a large enough sample size from ANY population with finite variance (here the "population" is all possible hands or tourneys), the average will have a limiting normal distribution, by the Central Limit Theorem.

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Ah, so you say that if we take X = the average win of say 100 tourneys then X will follow a normal distribution and if you get enough sets of 100 tourneys you can get an estimate of you average 100 tourneys winrate from the normal distribution.

Das 10 Pfennig ist gefallen. (AKA I got it!). [img]/images/graemlins/wink.gif[/img]

[AaronBrown]

There is a sound general principle in statistics that you should study what you care about. That's more important than theoretical considerations about the distribution. As several posters pointed out, poker outcomes do not follow a Normal distribution. Almost nothing follows a true Normal distribution. For some purposes that matters, for others the convenience of the Normal distribution outweighs the inaccuracy of the assumption.

If the main reason to estimate winrate is to decide if you can be a professional poker player, you probably care most about the lower tail: how much you can lose in (say) one month. There are a couple of reasons to think that might be greater than a Normal analysis based on mean and standard deviation would suggest. There is something called "extreme value theory" that worries about this, basically you look at your worst stretches and estimate from them rather than looking at all your results. For a similar reason, people often evaluate hedge funds based on maximum drawdown (how much you could have lost if you had bought and sold at the worst possible times) rather than mean and standard deviation of return.

Another reason to look at average win rate is to understand the relation between games played at various stakes and outcomes. If you have a winrate that is constant in BB/hour in all your games, and the distribution of outcomes is Normal, you can compute the full distribution of outcomes for any set of times at various games. That lets you decide whether you should play more or less, or move up or down in limit, or make other changes.

Again, the Normal distribution might not be the best model here, and the assumption of constant win rates in BB/hour is almost certainly false. You might be better off looking at some scatter plots of data and picking a specific pattern that works for your game.

[Gonso]

Aaron, you are just a good post machine.

[filsteal]

Aaron,

Great points. Extreme value theory as applied to poker seems like a particularly interesting idea.

Clearly if you could duplicate identical table conditions for 100k hands, and the hands were somehow independent of each other (maybe you play 100k one-hand sessions, all against opponents who don't remember you) the winrate over that sample will have to be approximately normal by the Central Limit Theorem, no matter what the distribution of outcomes for each hand is.

Obviously the above scenario is unrealistic, primarily because hand outcomes aren't independent of each other at all. (The cards dealt are independent from hand to hand, obviously, but that's not what I mean. I'm talking about how recent hands can affect the actions of you and your opponents on future hands.) I've wondered just how much this distorts the normality of the results, and in turn, any analysis that requires normality.

You seem to think that it distorts it quite a bit, and I think I might agree.

[Red_Diamond]

There actually is a main 2+2 thread on this article which is discussed here: 2+2 thread

I do have my own question there, as the variance of multiple-payout structures is something I haven't really carried out before. The main reason being I used to play strictly just cash games for the longest time.

But time to dust off something new, I think I'll re-break open some of Snyder's works now. Good thing I have not used the Poker Tournament Formula for kindling in the fireplace just yet, which other's here had recommended to me
[img]/images/graemlins/blush.gif[/img]

[Collin Moshman]

Hi Guys,

Thanks to everyone who posted for the feedback. Some very interesting discussion here. I too went to retrieve old stat texts, then realized all I had after the last move was Apostol. Fortunately he discusses, well, literally everything [img]/images/graemlins/smile.gif[/img]

I saw the thread in the magazine forum, and while I basically liked the article, my problem was in the author's presupposing that mean and standard deviation were applicable parameters. And indeed, while that particular thread questioned some of the stat definitions used, I don't believe anyone called the normal dist itself into question.

One comment I particularly liked from this thread:

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Again, the Normal distribution might not be the best model here, and the assumption of constant win rates in BB/hour is almost certainly false. You might be better off looking at some scatter plots of data and picking a specific pattern that works for your game.

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If I get around to doing this with my data I will post results, and if anyone does likewise, please share.

Best Regards,
Collin

[Red_Diamond]

[ QUOTE ]
As several posters pointed out, poker outcomes do not follow a Normal distribution. Almost nothing follows a true Normal distribution. For some purposes that matters, for others the convenience of the Normal distribution outweighs the inaccuracy of the assumption.


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I was doing some thinking on something regarding these issues today. Assume we have a BOT which understands normal distributions and calculates them out perfectly. In fact, it is so good, we simply tell it, use your knowledge of my variables and plan my next sequence of tournaments for next week, using the data you know for best risk/reward, etc.

We leave it up to our friendly Bot to make the best judgement here, and trust it knowingly computers don't make mistakes. That's great and all, but lets say..

We tell it that there is one condition, my first tournament must be the new Mr. DonkeyDoo. Which while we haven't played in, the Bot fills in our parameters and punches out our SD due to estimated placement frequencies.

Again, wonderful. Problem is we find out later that our whole scheduling is out of wack, because right off the start our buy-in for Mr. DonkeyDoo was much less than the SD (obviously very common), which the bot then wrongly concludes that we have a risk of losing MORE than our single buy-in, which obviously can't happen. So now it puts our lower tail scenarios worse than it should be by subtracting more off the BR than it should.

Granted, you could start adding in other checks in the coding here which wouldn't be rocket-science, but I'm wondering if it could really be so simple, or if this would be a never ending problem of scenarios where a BOT simply would keep going wrong.

[Troll_Inc]

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Hi guys,

It seems that in discussions of poker winrate calculations, it's usually taken for granted that the normal distribution applies. As a typical example, here is an article from this month's 2+2 magazine on SNG ROI calculations:


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I didn't see Chen and Ankenman's discussion on this subject listed here? They discuss this issue briefly and it's a good read for anyone in this thread.

I posted something along the lines of your question a little bit back in the Probability section and didn't get any sort of useful answer.

[Borys313]

I strongly believe that the fact how lucky we run has a major impact on the qulity of play. If you feel lucky, win alot you play better for many reasons.
So even if winrates would have normal distribution both ends will be more likely to happen due to the fluctuating strenght of play.

Also winning/losing streaks are alot longer then they should be.

Lets say on a given limit i win 25 days out of 30 played (I had good months like this). So I can estimate with 99% certenity that on any given day my chance to win is 79%-87%.

So then the chance that on the same limit I will lose 5 days in a row is 1/5*1/5*1/5*1/5*1/5 = 1 / 3000

Its really negligible but it happenes (far more often then it should). The extreme values show up alot cause we are only human and never stay cool enough.

PS.

The numbers are just my educated guesses, but if anyone cares to calculate it will be qutie close.