Part 1 · Lesson 2 of 25 · Free
Poker variance: why short-term results fluctuate
Two players with exactly the same skill play 20,000 hands each, and one can finish thousands of bb ahead of the other. Neither played better. Variance measures how far results spread around the expected value; its square root, the standard deviation, is usually given per 100 hands.
The hand
You and the Regular are equally good: each of you wins 5 bb per 100 hands, with a standard deviation of 90 bb per 100. You each play 20,000 hands.
Predict first
Both of you win 5 bb per 100 hands on average. After 20,000 hands, how far apart are your results likely to be?
- Within about 100 bb
- A few hundred bb apart
- Possibly thousands of bb apart
Show the answer
Answer: Possibly thousands of bb apart
Each of you expects +1,000 bb, but each result has a 95% range from about −1,495 to +3,495 bb. Two equally good players can end thousands of bb apart.
Working it out
The expected result grows in a straight line: 5 × 200 = +1,000 bb over 20,000 hands, which are 200 blocks of 100. The spread grows with the square root: 90 × √200 ≈ 1,273 bb. A 95% range is about 1.96 standard deviations either side: 1,000 ± 2,495, from −1,495 to +3,495 bb. About one winning player in 17 is still at least 1,000 bb down after these 20,000 hands.
Use the 95% range to set expectations, to tell a normal downswing from a leak, and to read your own results honestly. The uncertainty shrinks only with the square root of the sample, so an honest read needs far more hands than intuition suggests.
Formula
After N hands: mean = win rate × N/100; spread = 1.96 × SD × √(N/100); measured win rate ± 1.96 × SD ÷ √(N/100)
Check your understanding
You play twice as many hands. How wide does the 95% range of your measured win rate become?
- Half as wide
- About 0.71 times as wide
- About 1.41 times as wide
Show the answer
Answer: About 0.71 times as wide
The width scales with 1 ÷ √hands: doubling the hands multiplies it by 1 ÷ √2 ≈ 0.71. It is the range of your total result that grows, by √2 ≈ 1.41.
Win rate 4 bb/100, SD 80 bb/100, over 40,000 hands. The measured win rate is 4 plus or minus how many bb/100 (95%)?
Show the answer
Answer: 7.84
1.96 × 80 ÷ √400 = 156.8 ÷ 20 = 7.84 bb/100: the true rate could be anywhere from about −3.8 to +11.8.