Are Winrates Normally Distributed?
 

BIG thanks to Justin A for coming to me about this. He was the one who first noticed it and most of this was his idea. I had some free time last month on my flight to Vegas so I did most of the tedious Excel stuff.

Also, Barron, I know this doesn't have anything to do with high stakes poker, but I feel that it is an interesting topic that many posters will be interested in. If you don't agree, please move it to MHHUSH.

I am not a statistician but based on my not-quite-intermediate understanding of statistics, this is what I've come up with.

Methodology: I took a database of about 150k hands at one limit. I took these hands and put them into Excel. I chopped them up into blocks of hands (in chronological order) and plotted them in a histogram.

Theory: BB/100 is the average BB you will win every 100 hands. If you've played a large enough sample of hands (technically 50 samples should be enough, so 5,000 hands), the sample should behave normally and form the shape of a bell curve.

Results: The following graphs are of BB/25, BB/50, BB/100, etc. Notice that they are skewed toward the low end. 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. I don't remember the exactly mean but figure it's somewhere between 2 and 2.5 BB/100. for BB/10 it would be between 20 and 25, etc.

Can someone with a better understanding of statistics explain this?

http://img.photobucket.com/albums/v6...ef09/per10.jpg

http://img.photobucket.com/albums/v6...ef09/per25.jpg

http://img.photobucket.com/albums/v6...ef09/per50.jpg

http://img.photobucket.com/albums/v6...f09/per100.jpg

http://img.photobucket.com/albums/v6...f09/per200.jpg

http://img.photobucket.com/albums/v6...f09/per500.jpg

http://img.photobucket.com/albums/v6...f09/per700.jpg

Re: Are Winrates Normally Distributed?
 

i can't speak on math, but i know that people tend to play a lot when they're down and tend to leave when they're up

Re: Are Winrates Normally Distributed?
 

It looks pretty close to normal here, but I think the way you did the sampling is not quite right.

You need to draw random samples from the total group of hands. Chopping them up into blocks is easier, but not appropriate. The way you have done it, we might find a different results using a different database.

Re: Are Winrates Normally Distributed?
 

My gut instinct says this is because the raw +$/hand won is greater than the -$/hand you lose (due to some pots being multiway). Generally speaking, your small chunks of hands will break down as follows:

1) Many where you don't win big pots and you are slightly -$

2) Slightly fewer where you win enough hands to have a small +$

3) A few where you win a big hand or two & are very +$.

4) A few where you lose a few big pots & are very -$ (less however than your big +$ chunks, since due to multiway pots, the amount you WIN if big pots is larger than the amount you LOSE in the same big pot)

This should leader to a bell curve is NOT evenly distributed, but peaks on the - side, with the difference being made up by the curve coming down less steeply on postive end.

I hope I described that well, I'd do one in MS paint, but I'm a very bad artist.

if i am correct however, if you took the same data from HU play, it SHOULD look like a normal bell curve, as the effect of multiway play is completely eliminated.

jvs

Re: Are Winrates Normally Distributed?
 

[ QUOTE ]
It looks pretty close to normal here, but I think the way you did the sampling is not quite right.

You need to draw random samples from the total group of hands. Chopping them up into blocks is easier, but not appropriate. The way you have done it, we might find a different results using a different database.

[/ QUOTE ]

i could easily randomize it

Re: Are Winrates Normally Distributed?
 

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

Re: Are Winrates Normally Distributed?
 

[ QUOTE ]
[ QUOTE ]
It looks pretty close to normal here, but I think the way you did the sampling is not quite right.

You need to draw random samples from the total group of hands. Chopping them up into blocks is easier, but not appropriate. The way you have done it, we might find a different results using a different database.

[/ QUOTE ]

i could easily randomize it

[/ QUOTE ]

You can easily randomize the order of the hands in your database, but that is not what I was suggesting.

You need to randomly select X number of hands and compute the win rate for that sample. Then you need to do it over and over and over again, each time selecting X number of hands from the total set of hands. Then you plot the win rates for each of those samples.

I don't know of any easy way to do that in Excel. It could be done in some statistical packages, but a data set of that size is too big for a desktop computer, it would need to be run off of a server or a mainframe computer.

There is type of statistics called Bootstrapping which does not rely on the assumption of normality. I assume there is some bootstrapping software that does this kind of thing, but I don't know enough about it.

Re: Are Winrates Normally Distributed?
 

[ QUOTE ]
i can't speak on math, but i know that people tend to play a lot when they're down and tend to leave when they're up

[/ QUOTE ]

Astro, that shouldn't matter since even if Josh leaves, his next session gets grouped in the stats (if I understand correctly). This would only matter if Josh tilts.

Re: Are Winrates Normally Distributed?
 

oh i see what youre getting at. they have to be completely independent. by choosing blocks of hands, it assumes dependence since each has an equal likelihood of being picked and thus isnt random?

Re: Are Winrates Normally Distributed?
 

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.