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Variance calculator.
Over a thousand hands the margin around your win rate is wider than the win rate itself. Not a little wider: at a normal spread it is about 62 big blinds per 100 either side, against a good win rate of five. This works out how wide the margin is for your own numbers, and how long it takes to close.
the calculator
80 to 110 for full ring, higher short handed
3,000 big blinds is 30 buy-ins
Your win rate could really be
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Chance of being at a loss
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Risk of ruin at this bankroll
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the same numbers, spelled out
The fields hold a solid winner over ten thousand hands, which most players would call a decent sample. Press the button and see what it actually proves.
how long the truth takes
A 5 big blind winner, sample by sample.
One player, a real win rate of exactly 5 big blinds per 100 hands and a spread of 100. The middle columns are what their measured win rate could honestly be after that many hands, nineteen times in twenty. Read the first row before you read anything else.
| Hands | Margin, plus or minus | Could really be | Still at a loss |
|---|---|---|---|
| 1,000 | 62.0 | -57.0 to 67.0 | 43.7% |
| 5,000 | 27.7 | -22.7 to 32.7 | 36.2% |
| 10,000 | 19.6 | -14.6 to 24.6 | 30.9% |
| 25,000 | 12.4 | -7.4 to 17.4 | 21.5% |
| 50,000 | 8.8 | -3.8 to 13.8 | 13.2% |
| 100,000 | 6.2 | -1.2 to 11.2 | 5.7% |
| 250,000 | 3.9 | 1.1 to 8.9 | 0.6% |
| 500,000 | 2.8 | 2.2 to 7.8 | 0.0% |
| 1,000,000 | 2.0 | 3.0 to 7.0 | 0.0% |
After 1,000 hands the margin is 62.0 big blinds per 100 in each direction, which means a genuine 5 big blind winner could measure anywhere from -57.0 to 67.0. The margin is 12 times the win rate. It does not shrink to the size of the win rate until about 153,658 hands, because the margin falls with the square root of the sample: four times the hands, half the margin. That is the whole reason a session, a day, or a week cannot tell you whether you are good.
the three formulas
The margin is 1.96 times your spread divided by the square root of your sample in hundreds of hands. The chance of being at a loss is the normal distribution evaluated at minus your win rate times the square root of that same sample, over the spread. The risk of ruin is e to the power of minus twice your win rate times your bankroll, divided by the spread squared. That last one is deliberately the same line as the bankroll calculator uses, because two pages of this site disagreeing about the same question would be worse than one page fewer.
All three assume your results are roughly normally distributed and your win rate is a fixed property of you. Over enough hands the first is close to true. The second never quite is: your win rate moves with the games you sit in, the hours you keep and how well you are playing that week. Treat the interval as a floor on your uncertainty rather than a ceiling.
What follows from all this is not fatalism, it is where to look instead. If results cannot tell you whether you are playing well over any sample you will realistically reach, the only thing left is the quality of the decisions themselves. Whether a call was priced correctly is knowable in the moment: the expected value of the decision does not care how the card came, and the price you were being laid is a fact you can check afterwards. That is a sample of one per hand instead of one per hundred thousand.
And the inputs matter more than the arithmetic. A win rate typed in from a few thousand hands is itself a guess with this same margin around it, which is why the honest version of this calculator is a range rather than a number. If you have never measured your spread, 100 big blinds per 100 is a reasonable stand in for a full ring cash game, and the swings you have actually lived through are a better guide than any default.
next
Swings this wide need funding.
If a real winner can sit at a loss after a hundred thousand hands, the money behind you is not a comfort, it is a requirement. The bankroll that goes with your win rate follows from the same two numbers you just typed in.
questions about variance
How many hands do I need to know my win rate?
More than almost anyone plays. With a standard deviation of 100 big blinds per 100 hands, the 95 percent margin after 1,000 hands is 62.0 big blinds per 100, which is wider than any real win rate. The margin only shrinks to the size of a 5 big blind win rate after about 153,658 hands, and even then it is a margin rather than a measurement.
Why is the confidence interval so wide?
Because it shrinks with the square root of the sample, not the sample. Four times as many hands halves the margin, so going from 10,000 to 40,000 hands only buys you a factor of two. That square root is the single most important fact about poker results, and it is why a good month proves nothing.
Can a winning player lose over 100,000 hands?
Yes, and the chance is not small. A 5 big blind per 100 winner with a standard spread is still at a loss after 100,000 hands about 5.7 percent of the time. After 1,000 hands it is 43.7 percent, which is close enough to a coin flip that a thousand hand sample tells you nothing at all.
What is a normal standard deviation in poker?
Around 80 to 110 big blinds per 100 hands in a full ring cash game, and higher in short handed and in tournaments, where most entries return nothing. A loose aggressive style raises it, a tight one lowers it, and the style with the lower spread is not automatically better: the win rate is what pays, the spread only decides how long the truth takes to appear.
Does a lower variance style make me more money?
Not by itself. Lower variance means the same win rate arrives more smoothly, which lets you play a bigger game on the same bankroll and keeps you calmer while you do it. Those are real advantages. But if reducing the swings costs you win rate, you have traded the thing that pays for the thing that feels better.
How does this relate to risk of ruin?
It is the same distribution asked a different question. The confidence interval asks where your results are likely to be; risk of ruin asks how likely the path is to touch zero on the way. This page uses the same formula for it as the bankroll calculator, so the two never disagree.
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