Poker Variance Calculator

What this variance calculator does

Enter your edge, your variance and how much you plan to play, and it tells you what the sample can look like: the profit to expect, how far above or below it 95% of outcomes land, the chance of finishing the sample behind, and the downswings to expect along the way. The chart draws thirty simulated run-outs at your numbers so the spread is something you can see rather than a figure to take on trust.

The maths behind it

Poker results over a sample behave like a sum of many independent results, so two numbers describe the whole distribution:

• Expected profit = edge × volume. For cash games that is win rate (bb/100) × hands ÷ 100; for tournaments, ROI × number of tournaments.
• Standard deviation of the total = per-unit standard deviation × √volume. For cash, the bb/100 figure times the square root of the number of hundreds of hands; for tournaments, the buy-in figure times the square root of the number of events.

Expected profit grows in proportion to volume while the spread grows only with its square root, which is the whole reason variance fades with volume. Double the sample and the expected profit doubles, but the standard deviation rises by only 41%. The chance of finishing behind is the probability that a normal variable falls more than EV ÷ SD standard deviations below its mean, which the calculator reads from the normal distribution. The downswing figures come from the simulated runs: for each, the largest drop from a high point to a later low point.

Cash games: worked examples

A $1/$2 player with a $200 buy-in, winning 5 bb/100 with a standard deviation of 80 bb/100, over 50,000 hands:

• Expected profit: 5 × 500 = 2,500 bb, or 25 buy-ins, $5,000.
• Standard deviation: 80 × √500 = 1,789 bb, or 18 buy-ins.
• Chance of being down after the sample: about 8%.
• 95% of outcomes: between roughly −10 and +60 buy-ins.

Over 100,000 hands the same player expects 50 buy-ins with a standard deviation of 25, and the chance of a losing sample falls to about 2%. A player winning 2 bb/100 with a standard deviation of 90, over the same 100,000 hands, expects 20 buy-ins with a standard deviation of 28, and is behind about 24% of the time: a quarter of all 100,000-hand samples for a genuine winner at that rate are losing ones.

Tournaments: worked examples

A $109 player with a 20% ROI and a standard deviation of 7 buy-ins per tournament, over 500 tournaments:

• Expected profit: 0.2 × 500 = 100 buy-ins, $10,900.
• Standard deviation: 7 × √500 = 157 buy-ins.
• Chance of being down after 500 tournaments: about 26%.

Over 2,000 tournaments the expectation is 400 buy-ins with a standard deviation of 313, and the chance of a losing stretch falls to about 10%. A 30% ROI player in smaller fields, with a standard deviation of 5, is behind after 500 events about 9% of the time. This is the arithmetic behind the advice that tournament players need years, not months, to know whether they are winning.

Which standard deviation to use

Read it from your tracking software if you can. Without a figure, 70 bb/100 is typical for full-ring no-limit hold'em, 80 to 100 for 6-max, with heads-up and pot-limit Omaha higher still. For tournaments the standard deviation per event rises with field size and with how top-heavy the payouts are: around 5 buy-ins for small fields and flatter structures, 7 for mid-sized fields, 10 or more for large-field events with a thousand or more entries. Progressive knockout events run a little lower than their field size suggests, because bounties pay out along the way.

Reading the downswing figures

The typical worst downswing is the median, across the simulated runs, of the largest peak-to-trough drop in each run: the downswing a player at your numbers should expect to go through once in a sample this size. The bad-luck figure is the 95th percentile: one run in twenty sees a drop at least that large. Both grow with the sample, because a longer sample has more chances to hit a bad stretch, and both are the figures to hold in mind when deciding whether a run is ordinary or a sign that your edge is not what you think. The bankroll calculator turns the same inputs into the bankroll that keeps the risk of ruin where you want it.

What the model leaves out

• Your edge is an estimate. The calculator takes the win rate you enter as true. In practice it is the most uncertain input, and a 2 bb/100 error at the top of a 5 bb/100 estimate changes every figure on this page.
• The normal approximation. Tournament results are skewed: most events lose a buy-in, a few win many. The normal curve is a good fit for the total over hundreds of tournaments and a poor one for a handful.
• Nothing changes. The model assumes the same games and the same edge for the whole sample. Moving up, moving down, tilt and game selection all break that.

Questions

How is poker variance calculated? From two numbers per unit of play: your edge (win rate in bb/100, or ROI per tournament) and your standard deviation (bb/100, or buy-ins per tournament). Over a sample, expected profit is edge times volume and the standard deviation is the per-unit figure times the square root of the volume. The chance of being down is the probability that a normal variable falls more than EV divided by SD below its mean.

How many hands does it take to know if you are a winning player? More than most players expect. At 5 bb/100 with a standard deviation of 80, a 50,000-hand sample finishes behind about 8% of the time and a 100,000-hand sample about 2%. At 2 bb/100 with a standard deviation of 90, a quarter of 100,000-hand samples are losing ones. Tournament players need thousands of events.

What is a normal downswing in poker? It depends on edge, variance and sample. For a 5 bb/100 cash player with a standard deviation of 80, the typical worst downswing in a 50,000-hand sample is in the region of 15 to 20 buy-ins, and one sample in twenty sees 30 or more. A 20% ROI tournament player with a standard deviation of 7 should expect a worst downswing over 500 events well above 100 buy-ins.

What standard deviation should I enter for tournaments? Buy-ins per tournament, from your tracker if possible. Typical values are about 5 for small fields, 7 for mid-sized fields and 10 or more for large-field events with top-heavy payouts.

Why does the chance of losing fall as the sample grows? Because expected profit grows with volume while the spread grows only with its square root. Doubling the sample doubles the expectation and raises the standard deviation by 41%, so the expectation pulls away from zero.

Does a bigger bankroll reduce variance? No. Variance is a property of the game and your style; the bankroll only decides whether a normal downswing is survivable. The bankroll calculator gives the figure that keeps your risk of ruin below a chosen level.