Basically, I'm saying that I'm too lazy to work out a formula that expresses the distribution of expected total winnings for a given winrate and standard deviation over a given number of hands, thus allowing calculation of the likelihood that you are actually beating a limit in a given number of hands.
I know it involves a Gaussian distribution. Probably.
Anyone got any thoughts?
Hi everyone, hi w34z3l,
Long-time lurker here but I thought I'd jump in and contribute as maths is something I know a thing or two about!
This is not too difficult actually. In fancy language our "bb/100" result over a certain number of hands is called a "sample statistic" and we're talking about the difference between the underlying distribution (our "true" bb/100 and standard deviation) and the distribution of the sample statistic (i.e. the mean and standard deviation of our actual results over a certain sample size). The relevant facts are:
1) the sample mean is the same as the true mean. This one is pretty obvious. If our true average winrate is +10bb/100, then on average we make +10bb/100 over any sample size.
2) the standard deviation of the sample mean improves (which means, becomes smaller) with the square root of the sample size. So if our standard deviation is 80bb/100, the standard deviation of our "bb/100" result over a 6400 hand sample (64 times as big) will be 10bb/100 (i.e. 8 times smaller).
3) as an approximation, 95% of the time an observation falls within 2 standard deviations either side of the mean.
As an example, suppose a person wins at +10bb/100 with a standard deviation of 80bb/100.
Over 100 hands, their observed winrate will be between -150bb/100 and +170bb/100 (i.e. 2 standard deviations above or below the mean) 95% of the time.
Over 6400 hands, their "bb/100" will still average +10bb/100 but with an SD of 10bb/100 (being 8 times smaller because this sample is 64 times bigger). So 95% of the time they will achieve between -10bb/100 to +10bb/100 over a 6400 hand sample.
With 25600 hands, the SD drops to 5bb/100 (the sample is 256 times bigger, square root of that is 16, so the standard deviation improves from 80 down to 80/16=5). The 95% range is from 0bb/100 to +20bb/100. So this is the smallest sample size for which this person would unlucky to have a worse than break-even stretch.
My winrates over NL2 and NL4 have been ~20BB/100 so far so how many hands should I play before deciding I'm not on an upswing and that I'm actually beating a limit?
On the face of it, 6400 hands would start to mean something. (Underlying SD is 80bb/100 which means the SD of the average bb/100 over a 6400-hand sample is 10bb/100. This means your actual result of +20bb/100 is two SD above zero.) Really important note: this does not mean that after 6400 hands you can be confident you are a +20bb/100 crushing player! It means that, if you achieve +20bb/100 over a 6400 hand sample with an SD of 80bb/100, you can just barely be confident that your true winrate is above zero at all!
No offence but it's probably unlikely you will maintain +20bb/100 over a bigger sample. Also, because of the square root thing, with lower winrates you will need many more hands in order to be confident that you are a winner.
+20bb/100 -> 6400 hands needed
+10bb/100 -> 25,600 hands needed
+5bb/100 -> 102,400 hands needed!
I've read about 100k break-even stretches
With a standard deviation of 80bb/100, a person's winrate over 100k hands would have a standard deviation of roughly 2.5 bb/100 (divide 80 by square root of 1000 to get approx 2.5). So yes, unless the person is crushing a lot more than +5bb/100, it's perfectly possible even for a winning player to break even or slightly lose over a 100k stretch.
Final comment, this is all a bit theoretical. In practice we'll never know our "true" winrate. It takes so long to get a big enough sample, that our skill and the skill of our player pool will have changed by then! So it is an impossible moving target. It's best to move up the limits based on a BRM plan and taking shots, not on waiting for statistical significance over a gazillion-hand sample.
Hope this helps!