Variance Calculator

Understanding Variance and Risk

While expected value tells you what will happen on average in the long run, variance tells you how unpredictable individual outcomes will be. Variance is crucial for understanding risk.

The Formula

Variance measures the average squared deviation from the expected value:

Example: Fair Coin Flip

Consider a coin flip where you win $1 on heads and lose $1 on tails. We already know the EV is $0. Let's calculate variance:

• Outcome 1: Win $1 with 50% probability, deviation from EV = $1 - $0 = $1
• Outcome 2: Lose $1 with 50% probability, deviation from EV = -$1 - $0 = -$1

A standard deviation of $1 means outcomes typically deviate by $1 from the expected value of $0. This makes intuitive sense: you either win $1 or lose $1.

Example: Lottery Ticket

Now consider a lottery where you have a 0.0001% chance to win $99,999 and a 99.9999% chance to lose $1. The EV is -$0.90 (negative, as expected). But what's the variance?

Even though the expected value is only -$0.90, the standard deviation is $100! This extreme variance is why lotteries feel so different from other negative-EV games. You'll almost always lose $1, but on the rare occasion you win, it's a massive payout.

Variance vs. Expected Value

Compare these two bets, both with -$0.027 EV (same as roulette):

• Bet A: 51.35% chance to lose $1, 48.65% chance to win $1. Very low variance—outcomes are close to breaking even.
• Bet B: 99.9973% chance to lose $1, 0.0027% chance to win $999. Very high variance—you'll almost certainly lose, but could win big.

Both have the same expected value, but Bet B has far higher variance. This is why some people prefer slots (high variance) over blackjack (lower variance), even though both have negative EV. The gambling experience is fundamentally different.

The Normal Distribution

When you make many independent bets, the Central Limit Theorem states that your total outcome follows a normal (bell curve) distribution, regardless of the individual bet's distribution. This allows us to make probabilistic statements:

• 68.3% of the time, your outcome will be within ±1 standard deviation of the expected value
• 95.4% of the time, within ±2 standard deviations
• 99.7% of the time, within ±3 standard deviations

For example, if you make 100 coin flips at $1 each, your EV is $0 and your standard deviation is $10. This means:

• 68% of the time, you'll end up between -$10 and +$10
• 95% of the time, between -$20 and +$20
• 99.7% of the time, between -$30 and +$30

Variance Scaling Rules

• Bet size: Variance scales with the square of bet size. If you double your bet, variance quadruples.
• Number of bets: Total variance scales linearly. If you make 100 bets instead of 1, total variance is 100× larger.
• Standard deviation: Grows with the square root of the number of bets. 100 bets = 10× the standard deviation of 1 bet.

Why Casinos Love Variance

Casinos rely on the law of large numbers: over millions of bets, their actual results converge to the expected value. But individual players experience high variance:

• A player might get lucky and win big (positive variance), encouraging them and others to keep playing.
• Most players lose money (negative EV), but attribute losses to "bad luck" rather than mathematical certainty.
• The casino makes so many bets that variance averages out—they're virtually guaranteed to profit.