What is variance in betting?
Variance is the statistical measure of how far individual results deviate from the expected value. In betting, it quantifies the degree to which actual outcomes fluctuate around the long-term average. A player with positive expected value will profit in the long term, but in the short term, his results will oscillate — sometimes substantially — above and below the expected value. This oscillation is variance.
Understanding variance is essential because it explains why a mathematically sound strategy can produce losing streaks, and why a poor strategy can produce winning streaks. Neither outcome, in isolation, proves or disproves the quality of the strategy. Only over a large number of bets does the actual average converge to the expected value.
Standard deviation
The standard deviation is the square root of the variance and is expressed in the same units as the original measurement, making it more intuitive to interpret. For a single bet with decimal odd d and actual probability of winning w, the standard deviation of the profit per unit wagered is:
p: prize (d − 1)
w: actual probability of winning
l: actual probability of losing (1 − w)
σ = √(w · p² + l · 1²) − (w · p − l)²
Which simplifies to:
σ = √(w · l · (p + 1)²)
σ = (p + 1) · √(w · l)
σ = d · √(w · l)
For example, a bet at decimal odd 2.50 with an actual probability of 45%:
σ = 2.50 · √(.45 · .55) = 2.50 · √(.2475) = 2.50 · .4975 ≈ 1.244
This means the profit from a single bet will typically deviate from the expected value by approximately ¤1.244 per ¤1 wagered. Compared to the expected value of this bet:
e = .45 · 1.50 − .55 = .675 − .55 = .125 = 12.5%
The standard deviation (124.4%) is nearly ten times larger than the expected value (12.5%). This ratio illustrates why short-term results are dominated by variance rather than by the expected value.
Variance over multiple bets
When a player places n independent bets of equal size, the standard deviation of the average result per bet decreases by a factor of √n:
σₙ = σ / √n
This is the mechanism by which the law of large numbers operates. As n increases, the average result per bet converges toward the expected value. Using the example above:
After 1 bet: σ₁ ≈ 1.244
After 100 bets: σ₁₀₀ ≈ 1.244 / √100 = .1244
After 10,000 bets: σ₁₀₀₀₀ ≈ 1.244 / √10000 = .01244
After 10,000 bets, the standard deviation of the average result per bet is just .01244, meaning the average will almost certainly be very close to the expected value of .125. But after only 100 bets, the standard deviation is still .1244 — comparable in magnitude to the expected value itself — which means the player might easily be in the negative despite having placed bets with positive expected value.
Confidence intervals
A confidence interval provides a range within which the actual result is expected to fall with a specified probability. For a normal distribution, approximately 68% of outcomes fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three.
For a player who has placed 100 bets of ¤10 each, with an expected value of 12.5% per bet and a standard deviation per bet of 1.244:
Total expected profit: 100 · ¤10 · .125 = ¤125
Standard deviation of total profit: √100 · ¤10 · 1.244 = ¤124.4
The 95% confidence interval for the total profit after 100 bets is approximately:
¤125 ± 2 · ¤124.4 = ¤125 ± ¤248.8
This means that after 100 bets, the player's total profit is expected to be between −¤123.80 and +¤373.80, with 95% confidence. The fact that the lower bound is negative — despite every bet having positive expected value — demonstrates the practical reality of variance: 100 bets is not enough to guarantee a positive result.
After 1000 bets of ¤10 each:
Total expected profit: ¤1250
Standard deviation of total profit: √1000 · ¤10 · 1.244 ≈ ¤393.4
95% interval: ¤1250 ± ¤786.8 → from ¤463.20 to ¤2036.80
Now the lower bound is positive. After 1000 bets with this expected value, the player can be 95% confident of being in profit.
Variance and odds level
The level of variance is directly related to the decimal odd. Higher odds produce higher variance. A bet at a decimal odd of 10.00 has a much larger standard deviation per bet than a bet at 1.50, even if both have the same expected value as a percentage of the wager.
This means that a player who primarily bets on options with high odds (low probability events) will experience much larger swings in his bankroll than a player who bets on options with low odds (high probability events), even if both have the same long-term expected value per bet. The first player will need a much larger number of bets before his results converge to the expected value.
Practical implications
Variance is not an obstacle to be eliminated — it is an inherent property of any probabilistic activity. The player cannot control variance; he can only manage his exposure to it through proper bankroll management and staking strategies.
A player who understands variance will not abandon a sound strategy after a losing streak, nor will he develop overconfidence after a winning streak. He will evaluate his strategy based on whether his bets consistently have positive expected value, not on whether any particular sequence of bets happened to produce a profit or a loss.