This approximation assumes identically distributed independent wagers.
Variance and standard deviation
Variance is the probability-weighted squared distance from expected value. Standard deviation is its square root and returns to the original result units.
Larger standard deviation means wider typical swings, not necessarily a worse return.
Scaling with play
Expected loss grows roughly in direct proportion to repeated action. Standard deviation grows with the square root of independent repetitions.
Over sufficiently large samples, expectation can dominate relative noise even while absolute swings remain large.
Comparing games
Two 96% slots can have radically different standard deviations. Even-money roulette and straight-up numbers share an edge on the same wheel but distribute outcomes differently.
Choose the metric that answers the question: price, swing, or probability of a threshold.
Limitations
Changing bets, correlated side wagers, finite-deck effects, and stopping rules complicate the simple square-root model.
Simulation can estimate complex distributions, but it must disclose assumptions and uncertainty.
Working checklist
- State wager unit
- Use the correct outcome distribution
- Check independence
- Separate edge from volatility
- Do not call one standard deviation a guarantee
This lesson explains general gambling mathematics and uses rounded examples. A game-specific figure applies only to the rules, pay table, strategy, and denominator stated with it. Last reviewed September 20, 2026.