The core concept
Your "true" win rate is what you would win per 100 hands over an infinite sample. Your realized win rate is what you actually won so far. They are not the same number, because every hand has a random component on top of the skill edge.
Standard deviation captures the size of that random component. For NLHE cash, SD is roughly 80-110 BB/100. That means a typical 100-hand session has results scattered with SD ≈ 100 BB around the expected win rate. If your true rate is 5 BB/100, a 100-hand session has expected result 5 BB, but actual results land anywhere from −195 to +205 BB (95% range). Multi-buy-in swings are normal in a single session.
How variance scales with sample size
Here is the math that breaks intuition. Variance is linear in sample size. Standard deviation is the square root of variance — so SD scales with the square root of N. Translation: as you play more hands, your expected result grows linearly, but your spread grows only with the square root.
Concrete example. 5 BB/100 NLHE winner, SD 100:
| Hands | Expected BB | 95% range | P(breakeven or worse) |
|---|---|---|---|
| 1,000 | 50 | −570 to +670 | ~44% |
| 10,000 | 500 | −1,460 to +2,460 | ~31% |
| 50,000 | 2,500 | −1,884 to +6,884 | ~13% |
| 100,000 | 5,000 | −1,200 to +11,200 | ~6% |
| 1,000,000 | 50,000 | +30,400 to +69,600 | ~0% |
At 10,000 hands, a 5 BB/100 winner has a roughly one-in-three chance of being break-even or worse. At 50,000 hands, one in eight. At 100,000 hands, one in seventeen. Variance does not stop being painful at large samples — but the truth slowly emerges.
What this means for your downswing
Most "broken" winning players are running normal variance and panicking. The check: calculate the 95% confidence interval for your sample size and assumed win rate. If your downswing is inside that interval, you are normal. If it is outside, something has changed — your play, the games, your mental state, or your assumed win rate is wrong.
Use the variance calculator to run the numbers for your own sample. The output: expected result, 95% range, and the probability you are running break-even or worse. Often the answer is "this is normal." Sometimes the answer is "this is actually outside what variance can explain — look at your play." Either is useful.
Tournament variance is much larger
Tournament variance is bigger than cash variance because most of your equity sits in the top three finishing positions, and you reach those positions in roughly 1-in-10 events. A positive-ROI MTT player can easily go 200+ events without a four-figure score. The bankroll requirement for tournaments — 100+ buy-ins live, 200+ online — is direct downstream of this.
The five practical takeaways
- Your realized win rate is signal+noise. Trust the long-run; do not react to short stretches.
- Sample size determines confidence. 10,000 hands is not enough. 50,000 is meaningful. 100,000+ is reliable.
- Bankroll management lets you survive variance. A winner with too little bankroll will go broke during normal variance, with certainty.
- Different formats have different variance profiles. PLO is 50%+ higher than NLHE. Tournaments are dramatically higher than cash.
- Calculate, do not feel. The math will tell you whether you should worry or whether your downswing is normal. Tilt usually loses more money than the downswing itself.
