Standard Deviation Calculator

Understanding Standard Deviation in Gambling

While expected value tells you what will happen on average over thousands of bets, standard deviation tells you how much variance to expect along the way. Understanding both is crucial for proper bankroll management.

The Formula

For a simple bet with probability p of winning W and probability (1-p) of losing L:

Why Standard Deviation Matters

Imagine two bets, both with -$0.05 expected value per dollar bet. Bet A has a 49% chance to win $1 and 51% chance to lose $1. Bet B has a 0.05% chance to win $99 and 99.95% chance to lose $1. Both have the same EV, but vastly different standard deviations:

• Bet A (low variance): Standard deviation ≈ $1. Your results will cluster close to the expected loss. Predictable, grinding losses.
• Bet B (high variance): Standard deviation ≈ $9.95. Wild swings between big wins and small losses. You might be ahead for a long time before the negative EV catches up.

The Normal Distribution and Risk Zones

When you make many independent bets, the Central Limit Theorem tells us your total outcome follows a bell curve (normal distribution). This lets us make probabilistic predictions:

• 68.3% of outcomes fall within ±1 standard deviation (Safe Zone)
• 95.4% of outcomes fall within ±2 standard deviations (adds Caution Zone)
• 99.7% of outcomes fall within ±3 standard deviations (adds Danger Zone)

Bankroll Sizing Strategies

For Negative EV (Casino Games): Since you will lose money long-term, bankroll sizing is about entertainment duration. A larger bankroll (20+ standard deviations) lets you play longer and survive more variance. But remember: no bankroll size can make a losing game profitable.

For Positive EV (Skilled Betting): The Kelly Criterion provides mathematically optimal bankroll sizing. Full Kelly maximizes long-term growth but has high variance. Most professionals use fractional Kelly (1/4 to 1/2) for smoother results with lower risk of ruin.

Example: Fair Coin Flip vs. Roulette

Consider betting $10 per bet with a $100 bankroll (10 bets possible):

Notice how similar the standard deviations are, but the expected values differ. The house edge in roulette (-2.7%) shifts all outcomes slightly negative. Over time, this small shift guarantees the casino profits.