KEY TAKEAWAYS
To calculate the standard deviation of a bet, list every outcome with its probability and net result, find the expected value, find the average of the squared results, subtract the square of the expected value to get the variance, and take the square root. A $1 bet on one number on a European roulette wheel has an expected value of −$0.027 and a standard deviation of about $5.84, because it usually loses $1 but occasionally wins $35.
The standard deviation measures how far a single result typically sits from the expected value. It is a measure of swing, not of edge: two bets can have the same house edge and very different standard deviations. Over many bets, the total standard deviation grows with the square root of the number of bets.
This guide gives the five steps with each symbol defined, works two complete examples with different inputs, shows how to scale the result to a session, and ends with a reference table and a way to check your answer. Every step can be done by hand with a basic calculator.
What the Standard Deviation Tells You
A bet has a list of possible results. Each result has a probability, and the expected value is their weighted average. The standard deviation is the typical distance from that average, measured in the same units as the bet, such as dollars per bet.
A bet that pays 35 to 1 and one that pays even money can have nearly the same house edge. The even-money bet stays close to its average from spin to spin, because every result is +$1 or −$1. The 35-to-1 bet is far more spread out, because one result is +$35 and the other is −$1. A larger standard deviation means wider swings around the same average cost, which is the idea behind variance in gambling.
The standard deviation is not the largest possible loss, and it is not a promise about any one session. It is a summary of spread for a stated bet and a stated set of probabilities.
It also depends on the unit. A $10 bet has ten times the standard deviation of a $1 bet on the same event. Always state the stake, the game and the rule set with the result, such as a single-zero wheel.
The Standard Deviation Formula, Step by Step
Write each outcome’s net result as x and its probability as p. The calculation has five steps.
where:
x = the net result of one outcome (profit is positive, a lost stake is negative).
p = the probability of that outcome, with all probabilities adding to 1.
Σ = add up the values over every outcome.
EV = the expected value of one bet, as in the expected value formula guide.
E[X²] = the average of the squared results.
√ = the square root.
The idea behind step 4 is simple. Squaring each result makes every distance positive, and variance is the average squared distance from the average. Taking the square root returns the answer to dollars. The shortcut E[X²] − EV² gives the same value as averaging the squared distances directly.
For several independent bets of the same kind, total EV = n × EV and total SD = SD × √n, where n is the number of bets. The assumptions are independent bets, fixed probabilities and a fixed stake.
Worked Examples: Two Different Bets
Example 1. A $1 straight-up bet on a European single-zero wheel, paying 35 to 1. The outcomes are a win with p = 1 ÷ 37 and x = +35, and a loss with p = 36 ÷ 37 and x = −1.
Step 2. EV = (1 ÷ 37) × 35 + (36 ÷ 37) × (−1) = (35 − 36) ÷ 37 = −1 ÷ 37 ≈ −0.0270.
Step 3. E[X²] = (1 ÷ 37) × 35² + (36 ÷ 37) × 1 = (1,225 + 36) ÷ 37 = 1,261 ÷ 37 ≈ 34.0811.
Step 4. Variance = 34.0811 − (0.0270)² ≈ 34.0811 − 0.0007 = 34.0804.
Step 5. SD = √34.0804 ≈ 5.84. The typical swing is about $5.84 per $1 bet, over 200 times the 2.7-cent average cost.
Example 2. A hypothetical $110 bet at -110 that wins $100, with an assumed true win probability of 50%.
Step 2. EV = 0.5 × 100 + 0.5 × (−110) = −$5.00.
Step 3. E[X²] = 0.5 × 10,000 + 0.5 × 12,100 = 11,050.
Step 4. Variance = 11,050 − (−5)² = 11,050 − 25 = 11,025.
Step 5. SD = √11,025 = $105.00. Both results are exact, and the swing of $105 is 21 times the $5 average loss.
Scaling to a Session and Comparing Bets
The table gives the standard deviation per $1 bet for common wagers, calculated with the same five steps. House edge is the expected loss per $1.
| Bet (per $1) | House edge | SD per bet |
|---|---|---|
| European wheel, red (even money) | 2.70% | $1.00 (0.9996) |
| American wheel, red (even money) | 5.26% | $1.00 (0.9986) |
| European wheel, single number (35 to 1) | 2.70% | $5.84 |
| American wheel, single number (35 to 1) | 5.26% | $5.76 |
Now a session. Suppose 100 bets of $10 on red at an American wheel. The SD per bet is $10 × 0.9986 ≈ $9.99, so the session SD is 9.986 × √100 = $99.86. The expected total is 100 × $10 × (−5.26%) ≈ −$52.63.
As a rough normal approximation, about 95% of sessions land within two standard deviations of the average: −52.63 ± 2 × 99.86, or about −$252 to +$147. An exact binomial calculation puts the coverage at about 95.5%. That approximation is less reliable for few bets or heavily skewed payouts, such as the single-number bet.
A positive result in this session is not unusual: the chance of finishing ahead after 100 bets is about 26.5%.
Common Mistakes When Calculating SD
The first mistake is forgetting to square the results. Averaging the results themselves just returns the expected value, with no information about spread.
The second is subtracting the wrong quantity. Variance is E[X²] minus the square of EV, not minus EV itself. In Example 2 the correct subtraction is 25, not 5.
The third is scaling the SD by n instead of √n. For 100 independent bets the SD is ten times the single-bet SD, not one hundred times. The variance scales with n, and the SD scales with its square root, as the law of large numbers guide explains.
The fourth is mixing units. A result in units of the stake must be converted to dollars before it is compared with a dollar bankroll.
The fifth is reading the SD as a limit. Results beyond two standard deviations still happen, and the SD says nothing about how a stake plan changes the picture. For ruin probabilities, see risk of ruin.
Quick Reference and Check Your Answer
| Step | Action | Example 1 result |
|---|---|---|
| 1 | List outcomes: p and x | 1/37 at +35; 36/37 at −1 |
| 2 | EV = Σ p × x | −0.0270 |
| 3 | E[X²] = Σ p × x² | 34.0811 |
| 4 | Variance = E[X²] − EV² | 34.0804 |
| 5 | SD = √variance | 5.84 |
Check the answer in three ways: confirm the probabilities add to 1, confirm the EV matches the known house edge for the bet (2.70% here), and confirm that the SD is sensible for the payout. An even-money bet should land near 1 unit, and a 35-to-1 bet near 5.8 units.
Related Concepts and Next Steps
The first two steps of the method are the EV calculation from the expected value formula guide, applied to a casino bet, and the expected value in sports betting guide applies it to a betting market. The meaning of spread is developed in variance in gambling.
The standard deviation is the input to a session range and to a ruin estimate. It describes how widely results vary, never how to remove the average cost, and gambling is best treated as paid entertainment with a known average price.
Frequently Asked Questions
