Standard Deviation of PT Stats (Light Math)

[yellowjack]

I'm on it sir
(stats major) hopefully not too uneducated from too much LHE

[jaxUp]

I will be posting helpful tidbits from the internet as I find them (I'll just keep editing here).

"The standard deviation is a statistic that tells you how tightly all the various examples are clustered around the mean in a set of data. When the examples are pretty tightly bunched together and the bell-shaped curve is steep, the standard deviation is small. When the examples are spread apart and the bell curve is relatively flat, that tells you you have a relatively large standard deviation."

"1. A measure of the dispersion of a set of data from its mean. The more spread apart the data is, the higher the deviation."

[yellowjack]

Ok, I'll try very hand to explain this in the simplest terms possible.

Getting the definitions of the way, standard deviation is a measure of our friend (and enemy), variance.

It means that on any other day, if you held the same cards but the community cards came out differently for all of your hands in the database:

You could easily lose 15.6327 BB/100 more from what you actually won because of horrible community cards coming up (bad luck).
Conversely, you could easily have won 15.6327 BBs more than what you have aactually won, due to great community cards coming up (good luck).

In fact, your true winrate for all your hands on average (average good and bad community cards) is

WR (+ and -) 15.6327 BBs/100
where WR is the winrate you happened to get for all your sessions.
i.e. PokerTracker says you won 3.2 BB/100 for all your sessions, then WR = 3.2

So your true winrate would be:
= 3.2 (+ and -) 15.6327 BBs/100
So the minimum true winrate is (3.2BB/100) - (15.6327BB/100) = -12.4327 BB/100
Your maximum true winrate is (3.2BB/100) + (15.6327BB/100) = 18.8327BB/100

The range of your true winrate is between -12.4327BB/100 and 18.8327BB/100.
You put as an interval like this (-12.4327, 18.8327). That's a huge range, because it doesn't even tell you if you're a winning or losing player yet!
Also, this huge range isn't even that accurate. We can only say that your true winrate is between (-12.4327, 18.8327 with a 68.4% level of confidence. We're 68.4% sure that your winrate is anywhere between there. So we're not even that sure. However we can say that your winrate is in (-28.0654, 34.4654) with a 95% level of confidence.

The theory behind the previous paragraph is something like this:

A standard deviation is used to show the variance (we've all heard that term, right?) of statistics of the same category, i.e. the statistic of hit avg. for all Yankee players.

In short, the sample mean (+ or -) 1 s.d. gives you a 68.4% probability that any randomly selected Yankee player has a hit avg. between the interval (mean-1sd, mean+1sd).

For more info that has minimal jargon, try this.

jaxup, reply as soon as you can so I can touch this up for other posters who are in the same boat as you.

[dozer]

Thanks for explaining it to me, I kinda understand it now. I would useThis to get a more acurate look at my true winrate, but I don't even have excel on my computer. I actually don't really care too much about my true winrate for now since I know that the sample size is small, I was just curious what the numbers in pokertracker meant under standard deviation.

[yellowjack]

Alright, np. I really do tend to get a little preachy, which is a shame because I enjoy sharing to people what they want to know.

[jaxUp]

sorry for the tardy reply. This seems to be pretty good to me. Haven't done stats for a bit, so not 100% sure. Maybe talk about how our certainty of WR changes as we play more hands. I would love to hear the math about that.

[yellowjack]

Ugh, I can no longer edit the original post.

As n, the number of hands you've played, increases, you've accounted for more and more possibilities of boards and opponent holdings to go with your 169 hands. This levels out your winrate to something more consistant because you have played through more of the possibilities.

[chris_a]

[ QUOTE ]
I love this post. (fish sucks).

[/ QUOTE ]

Thanks.
You don't like my icon?

[chris_a]

Effectively, this is how to think about it.

Really the true random variable is the number of times the event has occured out of N. We'll call this number X. But each event is chosen from a Bernoulli (binary) distribution with probability p. The resulting sum of indepedent binary trials with probability p is called a binomial random variable. X (the number of positive occurences, such as the number of times you won at the showdown) has standard deviation,

sig = Std. Dev. of X = sqrt(p*(1-p)*N)

But....

We are intereted in the standard deviation of the percentage X/N which is just a derived random variable with standard deviation sig/N

Std. Dev (X/N) = sqrt(p*(1-p)*N)/N = sqrt(p*(1-p)/N)

[jrz1972]

If you play 100 hands, about 95% of the time you will finish within a region given by your true winrate + or - two standard deviations.

In your case, that means that if you play 100 hands, 95% of the time you will end up somewhere between $27.4 down and $35 up.