ITH Poker Forum

[PauliF]

I hope I am not stepping on any ones toes here.
I have noticed some disquiet on here and other forums with regard to the standard deviation.

I can sense frustration in that many people have diligently studied the game and reach this point where technical jargon is banded about that makes them feel uncomfortable.

There is a big difference between probability (and odds) and statistics. I have only been playing for six months but I have read a dozen or so books and I am pretty sure that as far poker goes this is the hardest and most statsy thing you will encounter.

I am going to attempt to explain it here for the sake of those who do not have a statistics education. If you understand the odds and probability side of things then you are good to go. Don’t worry if you still don’t understand it after this. Its just a measure that can be useful to understand but it is highly unlikely to affect your behaviour.

I am going to “borrow” web resources and try to structure my explanation around poker.

I try to explain the following

1. What is a statistical distribution?
2. What does it tell me?
3. What does it tell me about my poker game?
4. What the funking hell is a standard deviation?
5. What does it tell me about my poker game?
6. What is a confidence interval?
7. What does it tell me about my poker game?
8. afterthoughts

1. What is a statistical distribution?

There are dozens if not hundreds of these. They describe things….

Lets start with an easy one.
Less toss a coin
Its either heads or its tails.
Right?
Toss it once.
Probability that it is a head is 50% or (1/2 or 1 to 1)
Same for a tail

Ok so lets toss it twenty times!!!
How many times out of twenty is it gonna be heads?
You would say on average 10?
You are correct.

But its not gonna be exactly 10 heads and 10 tails everytime you tossed a coin 20 times.
Its gonna be 11 and 9 sometimes
Or 12 and 8
Sometimes it will be 20 of them.
But the overall average will be 10 of each if you did loads and loads of groups of twenty tosses
Hope I haven’t lost you already.

As it turns out the probability of it being exactly 10 each is 17.6%.

This comes from this formula:



The N = 20 because we are doing it 20 times
This could be any number you wanted.
You might be interested in the probability of getting a certain number of tails or heads out of 100 or 1000 tosses. You also might need to get out more.

The little n the question we are asking.
ie if we want to know the probability of getting exactly 10 heads and 10 tails we plug in 10 to the formula whenever we see a “n”
If we wanted to know the probability of getting 5 heads and 15 tails we put a 5 in.
Note that this is the same as the probability of getting 5 tails and 15 heads.

The p and q are the probability of success or failure..
Don’t panic.
The p is the probability of the thing we are interested in happening on one occasion.
So here its interchangeable because the probability of getting a head or a tail is equal.
So p and q are both ½.
We might be interested in say getting a six on the roll of a die
So p = 1/6 = the probability of getting a 6
And q = 5/6 = the probability of not getting one.

The graph of the above formula would look something like this


The y axis is probability and the x axis is the number of successes.
So in our example above with the coins.
The numbers across the bottom would go from 0 to 20.
The middle one 10.
Would then read across to 17.6% on the y axis.
Note the symmetry ie as seen above the probability of getting 0 or 20 is the same.

Still with me?
Good for you.

Right then ………

This is a simple distribution
The second simplest as it goes.

Its what’s know as a discrete type because it’s a bar chart. Each integer number has a probability.

Lets look at a similar but continous distribution

Ladies and Gentlemen please be upstanding for the Normal distribution.




This is described by the following formula:


don’t panic its just an instruction on how to draw the graph.

All letters in there have preset numbers except for

this number is the mean (or average) and

this number is the variance.

As it turns out this formula can be used for all sorts of things.
The height of men say can be drawn on a graph like this with the x axis showing height and the y axis showing proportion of men at that height (proportion is the same as probability really).
The letters that are plugged in would be something like 5 foot 11 for the

This could be used for example:

The survival period for a fatal disease
For the age at which peeps loose their virginity etc etc etc

2. What does it tell me?

Well it tells us how likely things are?
How likely is a man to be between 6 foot and 6’3
How likely is it that the next 28 year old you come across is still a virgin.

That’s if you knew what the true numbers to plug into that formula where.
(you would also need to “standardise your distribution” this goes beyond the scope of this note).


3. What does it tell me about my poker game?

People are simply saying that if you play a hundred hands and your true earn rate is 2bb/100 you wont earn exactly 2 every hundred hands. It will be all sorts of numbers. But the average will be 2 per hundred.

The variance (or standard deviation) just describes how wide spread the numbers that come together to make that average are gonna be.

For a rock the will be mostly quite close to the earn rate
For a maniac they will be all over the place.



4. What the funking hell is a standard deviation?



The mean is easy to explain … it is the average number of what we are observing.
Its where the graph peaks.
In poker we are interested in the earn rate per 100 hands. So this number is your BB/100.

The variance describes how fat or thin that graph is.
The bigger it is the fatter it is.
The smaller it the thinner it is.

If you were looking at the height of all men the graph would be quite fat.
Lots of different heights out there from dwarfs to giants.
We got em all in there.

If you were looking at the height of basketball players the graph would be much thinner.
(and the peak to the right)

The 2 can be taken off (by square rooting it) and it is called the standard deviation.

5. What does it tell me about my poker game?

Not much
The lower the number the better … apparently.

http://www.pokertracker.com/forum/viewt ... +deviation

some one with a VPIP of 12% and a PFR of 2% (ie a rock) will probably have a lower earn rate than some one with a VPIP of 20% and a PFR of 8%.

But the rock will also have a lower variance (and standard deviation).
Thinner graph.

Compare yours with others
The lower the better!!

6.What is a confidence interval?


Remember these questions?

How likely is a man to be between 6 foot and 6’3
How likely is it that the next 28 year old you come across is still a virgin.

Well that how we use continuous distributions.
So you could use it to ask the question what is the probability that I will earn more than 5bb in the next 100 hands
Or between 0 and 10 bb
Or less than zero (ie a loosing streak)

And so on
A confidence interval is the other way round

So…it says 99% of the time my earn for the next 100 hands is gonna be what?
And the answer is between X and Y

PT works out earn rate and standard deviation and Bug’s post tells you how to turn that into a confidence interval.

You don’t need to remember all the crap behind it everytime you use these numbers…
Just think about the question that the number are answering.
That’s what I do anyway…


7.What does it tell me about my poker game?

Knowledge is power.


8.afterthoughts

I am not convinced that poker earn rate is normally distributed.
I can not find much text about poker that doesn’t start of implicitly assuming a normal distribution.

I think that playing styles and the nature of the game means that there is a third statistic to be considered.

Skewness

[Bugsbunny]

I am not convinced that poker earn rate is normally distributed.

There are some indications that short term poker earns involve distributions with fat tails on both ends of the spectrum. However over the long term your overall earn still trend towards a normal distribution.

Nice post by the way.

[Georgeair]

Yep - good post. Now where's my Advil? And how did I get two A's in statistics anyway? Must be the "soft" classes at Auburn......

[nsidestrate]

Outstanding post! I have also suspected that results aren't quite normal, but close enough to draw some conclusions. Mason Malmuth talked a bunch about SD in some essays and maybe one of the books (ToP?). He is the source for the original lower-better claim.

[beermantm]

Great thread!!! Thanks for the info!!!!

[jeffnc]

Nice post. Some things aren't black and white though. For example, the age people lose virginity. It's very rare for someone to lose their virginity at age 50 or 60 or 70. Also at very young ages. This gives a very nice bell shaped curve, as you've shown. The problem is, some people die without having lost it. How do you factor that in? Could make the graph very ugly.

[PauliF]

they are exlcuded from the sample

[jbharshaw]

You guys are going to love to beat on this one. I just passed this article to Totalbluff to be published next week, but I don't thing a early release here will hurt.



Most of our modern Geometry is based on Euclid’s Theorem, “Through a point outside a line, there is one and only one parallel line.” The problem is that it could never be proven. In fact an attempt to prove it by proving that the alternative was un-true not only failed but resulted in a whole new Geometry. Non Euclidean Geometry is the basis for most of our super-advanced mathematics like Quantum Mechanics and String Theory today. Maybe a better analogy would be: Theoretically a bumble bee can’t fly but empirical evidence seems to suggest otherwise. As most of you have probably guessed I’m about to suggest something that isn’t part of Poker theory and in many cases maybe the biggest sacrilege in Pokerdom.

Common wisdom, Poker Dogma and all the poker “authorities,” decry walking at a predetermined “win” point as ludicrous and foolish. If the game is good and you are the favorite, you should remain in the game. The poker player is paid at his hourly win rate and should continue to play as long as he is the favorite. By leaving at a predetermined “win” point, you limit your winnings not your losses. Play as long as the game is good.

Well I understand the reasoning and agree, but I do believe that the empirical evidence in this case may suggest another approach. Over the long run “luck” evens out and everyone will eventually get their share of cards. If they maximize their wins with good cards and minimize their losses with marginal cards, they will be “winners” over the long run. The problem is that the long run is divided into weeks, days, and sessions. And over that short period of time the cards often do not even out. In fact they tend to clump together in groups. Luck or short term variance may not matter in the long run, but it does matter in the short term and determines if you walk a winner today.

Let’s look at an example. (For those of you who work with Wilson’s Turbo Texas Hold ‘em, this is the Maverick profile.) Here is a graph of twenty 300 hand sessions and the expectation based on one million hands.

http://www.totalbluff.com/articles/images/mavrick.jpg

The red is the expectation based on one million hands. It continues to grow. The yellow is the change in total bankroll at the end of each session. The pink is the highest win point during the session and the purple is the lowest point during the session. The light blue is based on a stop win/loss point of one average variance. In this case $500 and an average win rate of $128 for the series of sessions. So if he won $628 stopped. If he lost $372 stopped. If at any time he had a swing of 500 one way or the other stopped. As you can see by the blue line a stop W/L point would have improved the player’s results over this rather bad streak of cards. I’ve conducted a dozen of these simulations so far and all indicate there might be a value to establishing a stop point.

Let’s make an assumption. If you could quit at 90% of your biggest stack during each session, wouldn’t your hourly “win rate” increase? The problem is how we calculate this target value. Like most players you should keep good records of all your sessions. You should be able to calculate your win rate and your variance. Usually we use the big bet as the measure of success, like 3 big bets per hour for a win rate. I’ve found that when making expectation calculations the average pot size is a better indication of the expectation than the Limits. Every poker player knows that their expectation in all $5-10 games is not equal, but in terms of the average pot size they can be compared more accurately.

So, could we improve our win rate if we could establish a viable stop win point and a good stop loss point? Definitely the problem is the calculation for these two points. In statistics 95% of all results occur within 2 standard deviations of the mean. Shouldn’t there be a way to calculate this value and establish meaningful stop points?

GL

jb

[AlamedaMike]

I try to explain the following

1. What is a statistical distribution?
2. What does it tell me?
3. What does it tell me about my poker game?
4. What the funking hell is a standard deviation?
5. What does it tell me about my poker game?
6. What is a confidence interval?
7. What does it tell me about my poker game?
8. afterthoughts

Very good write up - I have always been fascinated in Math and in my younger days quite good at it - I did get A's in stats in University.

But that was 35 years ago and the mind has gone soft.

My SD has aways been higher than I want that bugger just won't drop. It does mean that I can book a few huge wins however, it just those big losses that also occur that are a pain.

[AlamedaMike]

...

Common wisdom, Poker Dogma and all the poker “authorities,” decry walking at a predetermined “win” point as ludicrous and foolish. If the game is good and you are the favorite, you should remain in the game.

...
GL

jb

I saw this last year and did not respond. Every since I started playing poker I have been thinking about this very subject - I first read Malmuth's book and it worried me.

In fact just awhile ago at the table one Saturday morning when I was being handed my hat I let it slip that I should have left when I was way ahead. A player sitting across from me sitting on a stack of chips said

We would all leave if we knew that this was the most we would win today

Salient points are 1) if the game is good and 2) if you are still a favorite assuming you ever were in the first place.

When I have playing awhile I ask myself some questions (like running a system background check).

Am I winning or losing? Is the game still good? am I still a favorite to win? am I on tilt? Do I still want to play some more or has my daily appetite been satisfied? Has the game changed? Have a lot of chips left the table? Am I getting tired?

Not exactly a stop win/lose but close. I think it smooths out the luck factor some. If I leave the table I will never know the outcome of the next hand.