Understanding poker variance
Why 'running bad' usually isn't bad luck, the math behind every long downswing.
Variance is the single most misunderstood concept in poker. Most players who think they're "running bad" are actually running normally. Most players who think they're great are short-term lucky. Both are missing the same calculation, what does a normal sample look like for someone with their win rate?
The simple math
Imagine a true 5bb/100 winner. With a typical 100bb standard deviation, here's what 95% of 100,000-hand stretches look like for that player: anywhere from -15bb/100 to +25bb/100 observed. That's an observed win rate range of 40 big blinds. A 5bb/100 winner can run as a small loser over 100,000 hands and still be a great player.
Open the variance calculator and plug in 5/100/100,000. The 95% confidence interval is roughly ±32,000bb. That's three buy-ins of swing in either direction over 100k hands. Normal. Predicted. Expected. Nothing wrong.
The three lessons
- Sample size matters more than result. 10,000 hands tells you almost nothing about your true win rate. 100,000 starts to narrow it. 500,000 gets you within ±1bb/100 if your true rate is around 5.
- Don't move down on a downswing. If the math says you're a 5bb/100 winner, 30 buy-ins of swing is a normal Tuesday. Moving down stakes after a normal downswing locks in the loss.
- Don't move up on a heater. Same math, reversed. Running 20bb/100 over 20k hands does not mean you're a 20bb/100 player.
For the formal treatment, the risk of ruin calculator uses the closed-form exponential formula. The bankroll calculator tells you how many buy-ins your variance demands. The variance calculator draws the picture.