Variance in Gambling: Standard Deviation and Swings

Variance in gambling measures how widely results scatter around the average, and the standard deviation (SD) is its square root, expressed in dollars. An even-money bet on an American roulette wheel has an SD of about 1 unit per bet, so after 100 bets of $10 the typical swing is around ±$100, while the expected result is only −$52.63. The swing is almost twice the expected loss, which is why short sessions look so unpredictable.

Variance answers a different question from expected value and the house edge. Those describe the average. Variance describes the spread. A game with the same edge can feel calm or wild depending on how large and how rare its wins are, and the same game feels calmer over a long run than a short one when measured as a share of the money wagered.

This guide gives the variance and SD formulas with every symbol defined, three worked examples, and a table showing how the swings grow with the number of bets. It also lists the mistakes that cause the most confusion. All figures are hypothetical and calculated on the page.

What Variance Means in Gambling

Every bet has a set of possible results. The expected value is their weighted average, and variance measures the spread around that average. A coin-flip bet that wins or loses $1 has an average of $0 but results of exactly +$1 or −$1, so every single result is a full dollar away from the average.

Variance is measured in squared dollars, which is awkward to read. The standard deviation is the square root of the variance, which brings it back to dollars. A bet with an SD of $1 produces results that typically sit about $1 from the average.

In casino terms, high-variance bets pay rarely but pay a lot, and low-variance bets pay often but pay little. A single-number bet on roulette is high variance, while red or black is low variance. Slot machines are commonly described as high or low volatility in the same sense, and the numbers for a specific machine come from its own published math, not from a rule of thumb.

Two points keep variance in perspective. First, variance is a measure of risk, not a forecast. It describes how scattered outcomes tend to be, not what the next one will be. Second, variance does not change the house edge. A high-variance bet and a low-variance bet can carry exactly the same edge, as the roulette examples below show.

The Variance and Standard Deviation Formulas

For a single bet, variance is the average of the squared results minus the square of the average result.

where:

X is the net result of one bet in dollars (for example +1 or −1).

E[X] is the expected value of X, which is Σ (probability × result).

E[X²] is the expected value of the squared result, Σ (probability × result²).

√ means square root.

Over many bets, expected values add up and spreads grow more slowly. For n independent, identical bets, the total expected result and total SD are as follows.

where:

n is the number of independent bets.

EV and SD are the expected value and standard deviation of one bet.

√n is the square root of the number of bets.

The key consequence is that the expected loss grows in proportion to n, while the typical swing grows only with the square root of n. This is the law of large numbers at work: as n rises, the average result per bet gets closer to the expected value.

A common way to describe the range of a session uses the normal approximation: about 68% of outcomes fall within one SD of the expected result and about 95% within two SDs. This is an approximation. It is less reliable for few bets and for bets with rare, large wins, where the results are lopsided rather than bell-shaped.

Worked Examples, Step by Step

Example 1: one even-money bet. Suppose you bet $1 on red on an American wheel. It wins +1 with probability 18 ÷ 38 and loses −1 with probability 20 ÷ 38.

Step 1: E[X] = (18 ÷ 38) × 1 + (20 ÷ 38) × (−1) = −2 ÷ 38 ≈ −0.0526.

Step 2: E[X²] = (18 ÷ 38) × 1 + (20 ÷ 38) × 1 = 1.

Step 3: variance = 1 − (−0.0526)² ≈ 1 − 0.0028 = 0.9972.

Step 4: SD = √0.9972 ≈ 0.9986, which is about $1 per bet.

Example 2: 100 bets of $10. Using the same bet scaled to $10 stakes, total EV = 100 × (−0.0526) × 10 ≈ −$52.63, and total SD = 0.9986 × 10 × √100 ≈ $99.86.

Step 1: one SD below and above: −52.63 − 99.86 = −$152.49 and −52.63 + 99.86 = +$47.23.

Step 2: by the normal approximation, about 68% of such sessions end between −$152.49 and +$47.23, and the chance of finishing ahead is roughly 30%.

Example 3: a single number, European wheel. A $1 single-number bet wins +35 with probability 1 ÷ 37 and loses −1 with probability 36 ÷ 37. E[X] = −1 ÷ 37 ≈ −0.0270. E[X²] = (1 ÷ 37) × 35² + (36 ÷ 37) × 1 ≈ 34.081. Variance ≈ 34.081 − 0.0007 ≈ 34.080, and SD ≈ 5.84. The same edge produces a much wider spread: a $10 single-number bettor has a one-bet SD of about $58.38 against an average loss of $0.27.

Swings Across Session Lengths

The table follows $10 bets on red at an American wheel. It uses the formulas above (total EV = n × EV and total SD = SD × √n) and the approximate range of two SDs either side of the expected result. These ranges are approximations.

Read the table from the top. With 10 bets, the range includes large wins and large losses, and winning overall is common. By 100 bets the expected loss is still smaller than the typical swing. By 1,000 bets the range barely reaches break-even, and by 10,000 bets it sits entirely below zero. The edge dominates over time because the loss grows with n and the spread with √n.

The spread is also shaped by the type of bet. Per $1 wagered, the SD of an even-money roulette bet is about 1, while a single-number bet is about 5.84. The edge is lower for the single number on a European wheel (2.70%) than for red on an American wheel (5.26%), yet its results are far more scattered.

Common Mistakes and Misconceptions

The first mistake is adding standard deviations instead of scaling by √n. Ten bets do not have ten times the SD of one bet. They have √10 ≈ 3.16 times. Only the expected result is multiplied by n.

The second is believing high variance means higher expected return. A wide spread says nothing about the average. A single-number bet is far more volatile than red, yet on a European wheel its edge equals the edge on red.

A third is treating a good or bad streak as proof of anything. With an SD this large, a short run of wins is an ordinary outcome, not evidence of skill or of a working system, and a losing run does not make a win due.

Finally, the normal approximation is easy to over-trust. For a few bets or for bets with rare large wins, the real distribution is lopsided, so the 68% and 95% ranges are only a rough guide.

Quick Reference and How to Check Your Answer

Check a result in three steps. Confirm variance is not negative and that E[X²] is at least (E[X])², which must hold for every bet. Confirm that a bet with only +1 or −1 outcomes has an SD close to 1. Finally, test the scaling: quadrupling the number of bets should only double the total SD, so 400 bets should have twice the SD of 100.

Related Concepts and Next Steps

Variance builds directly on the expected value formula, since both use the same probabilities and results. The average loss per dollar comes from the roulette house edge, and variance explains why a real session can land far from it.

The next questions are practical ones about risk: how large a bankroll is needed to survive the swings, and whether a staking system changes the picture. Later articles show that systems reshape the swings but never the average.