To be clear, I am not a mathgician at all so please don’t stop reading! Hang in there as I will try to clarify the role of variance in poker in a very basic way. I got inspired to write this blogpost by questions from players that I came across. If you have similar questions like the ones below, please read the whole post because the purpose is to give you a better idea of how you can calculate the role of variance to your poker sessions as well as how to come up with an answer yourself to the following questions:
“Do I just run good or am I a winning player at this stake?”
“What’s a reasonable amount of hands to estimate a true bb/100 winrate?”
“I have a winrate of Xbb/100, am I ready to move up in stakes?”
It all comes down to be able to figure out what your actual winrate is and we will get an answer to that by using the following handy tool: http://pokerdope.com/poker-variance-calculator/
Why using EV bb/100
For the calculations we are going to use the EV bb/100 stat instead of the regular bb/100. EV stands for Expected Value and corrects for variance in a better way. Let’s assume you call an all-in on the flop in a heads-up pot and it turns out you have 80% equity to win a $200 pot. 4 Out of 5 times the result will be +$100, but 1 in 5 times it will be -$100. The EV bb/100 is a corrective factor for this and calculates it in this way: (0.8*100)+(0.2*-100)= +$60 will be the Expected Value in the long run. When you play ten thousands- or hundreds of thousands of hands this will give a more accurate approximation of your true winrate than the regular bb/100 does. Therefore, we will use EV bb/100.
The factors we need for the variance calculator to run the simulations are winrate in bb/100, where we will use EV bb/100, the standard deviation in bb/100 and the total number of hands.
This is a screenshot from Hold’em Manager 2 and shows what numbers are needed to use in the variance calculator. The ‘EV bb/100’ and ‘Std Dev bb per 100 hands’ stats can be imported so please look this up in your database if you aren’t using them yet.
The input we need here is the numbers of hands, in this case 25.000 rounded off, a standard deviation of 139 and an EV bb/100 of 2.89. I will use these numbers as starting point and the idea is to show a couple different scenarios so you will understand what’s happening and are able to do your own calculations after reading this article.
You see a lot of stats being simulated. The most important ones that I want to highlight are the 70%- and 95% confidence interval. For example, the 95% confidence interval tells us with 95% certainty that given the input, our true winrate will be between -14.69bb/100 and 20.47bb/100. With 70% certainty we find it will be between -5.90bb/100 and 11.68bb/100. To ‘prove’ we are actually beating a stake we want to have a higher winrate than 5bb/100 in general, that can be concluded with the highest possible certainty.
If you click the question mark after standard deviation in the simulator this example of standard deviations will pop up:
The Hold’em Manager screenshot is about PLO 6-max, so the standard deviation of 139 fits well between the regular 120-160bb/100. Other things that catch the eye are the standard deviations of PLO 6-max being higher than NLHE 6-max, and 6-max in general has higher standard deviations than full-ring. Higher standard deviations directly relate to a larger role of variance. (The square root of variance = the standard deviation)
So, what happens when we use the starting point input but lower the standard deviation from 139 to 120?
What you can see is that the 70% confidence interval is between -4.70 and 10.48bb/100 and the 95% confidence interval -12.29 and 18.07bb/100. Comparing this with the starting point input it can be concluded that the width of the confidence interval becomes smaller and this increases the likelihood of making accurate conclusions!
Two General Conditions
Two factors are leading to a smaller (and thus more reliable) confidence interval width:
1. As shown, the lower the standard deviation, the lesser the role of variance, and the smaller the width of the confidence interval becomes.
2. The same applies for increasing the sample size. The more hands you play, the more likely it is that the winrate converges to your true winrate (role of variance becomes smaller) and thereby the confidence interval becomes more zoomed in accurately.
Increasing The Sample Size
Keeping the winrate at 2.89 and the standard deviation but increasing the number of hands to 100.000 this is the outcome:
Comparing the results with the previous confidence intervals, we find that it became narrower again and thus more accurate. It can be concluded that with a 2.89 EV bb/100 winrate over 100.000 hands our true winrate is in between -4.70 and 10.48bb/100 with 95% certainty!
Now, when is one actually beating a stake?
Let’s assume our winrate in EV bb/100 is not 2.89 but 8. The standard deviation and the number of hands are unadjusted, 120 and 100.000. This is the output:
With 70% certainty our true winrate lies between 4.21 and 11.79bb/100. With 95% certainty it’s somewhere between 0.41 and 15.59bb/100.
Given a sample of 250.000 hands the 95% confidence interval is from 3.20 to 12.80bb/100 with an input of winrate 8bb/100 and standard deviation 120.
So, if you want to determine whether you’re actually beating a stake and ready to move up these are the numbers you should play around with. Generally speaking, I’d recommend to move up if you’re certain that you’re actually winning over 5bb/100 at a specific stake. In the examples above you saw it might take quite a sample before you’re able to come to this conclusion, but then you can confidently move up. This is purely based on winrate! Other factors like bankroll management and the pressure of playing higher stakes for example, can be a lot more important deciding if you want to move up, take shots first, or rather stick to the current stake and try to win 10bb/100. This is all very personal and whether to move up in stakes is not the purpose of this article. To be able to analyse what the impact of variance is on your winrate yourself, ís the purpose.
I hope you’re able to play around with your own stats now with the help of the simulator. A lot of players underestimate the role of variance and its impact. I think when you understand the magnitude of it in a better way, it can give you some peace in the short run. Swings can be very cruel and it can hurt you mentally in such a way that your true winrate can collapse. By focusing on the horizon of playing 50.000 hands or more in the best possible way, you can become more ‘immune’ to the whims of poker.