The question
"I had a good system last month and now I'm £300 down. How much should my results swing?"
With betting variance explained properly, a losing month stops being a shock. Every strategy has a normal amount of swing built into it, and you can work it out before you place a bet. This lesson shows you how.
The idea in one sentence
Standard deviation is the typical distance between the result you get and the result you should expect, and it tells you in advance how big a normal good or bad run looks.
The picture
Imagine twenty bettors with exactly the same edge, placing exactly the same bets at the same prices. After 1,000 bets they won't all have the same profit. Plot their bank balances bet by bet and you get twenty lines that fan out from zero like a funnel.
The centre of the funnel is your expected profit: the average you would get if you could replay the season thousands of times. The width of the funnel is set by the standard deviation. Most lines stay within one standard deviation of the centre, nearly all stay within two.
Two things decide how wide the funnel is:
- The odds. At long odds you win rarely and big, so each bet is lumpy and the funnel is wide. At short odds you win often and small, so it's narrow.
- The number of bets. The funnel widens as you go, but more slowly than your expected profit grows. That's why a real edge eventually climbs out of the noise.
The rule from Lesson 1.1 applies here too. About 68% of results land within 1 standard deviation of the expected profit, about 95% within 2, and 99.7% within 3.
Worked Betfair example
You back selections in Match Odds and Over/Under markets at average odds of 3.0, £10 a bet. Your true win rate is 35%, a 5% edge on the price, and Betfair takes 2% commission on net winnings. (Illustrative figures.)
- What a winner pays. A £10 bet at 3.0 wins £20. Commission is 2% of that, £0.40, so you keep £19.60. A loser costs £10.
- Expected profit per bet. 0.35 × £19.60 − 0.65 × £10 = £6.86 − £6.50 = +£0.36.
- Variance per bet. Variance is the average squared distance from the expected result. Here it works out as (£19.60 + £10)² × 0.35 × 0.65 = 876.16 × 0.2275 ≈ 199.3 (in "pounds squared", which is why nobody quotes it directly).
- Standard deviation per bet. The square root of the variance: √199.3 ≈ £14.12. One bet swings by about £14 around an expected +36p. The noise is forty times the signal.
- After 100 bets. Expected profit is 100 × £0.36 = £36. The standard deviation is £14.12 × √100 = £141. The normal range (±2 standard deviations) runs from about −£246 to +£318.
- After 1,000 bets. Expected profit is £360. The standard deviation is £14.12 × √1,000 ≈ £446. The normal range is about −£533 to +£1,253.
Verdict: being £300 down after 100 bets is well inside the normal range for this strategy. Even after 1,000 bets, about 1 in 5 bettors with this genuine edge are still showing a loss (20.4% by the exact binomial calculation).
How the odds change the swing
The same £10 stakes, each strategy with a 5% edge on the price, 2% commission:
| Average odds | Win rate | Expected profit per bet | SD per bet | SD after 1,000 bets | Chance of a loss after 1,000 bets |
|---|---|---|---|---|---|
| 1.5 | 70.0% | £0.43 | £6.83 | £216 | about 2% |
| 3.0 | 35.0% | £0.36 | £14.12 | £446 | about 21% |
| 10.0 | 10.5% | £0.31 | £30.10 | £952 | about 37% |
The chance-of-a-loss column uses the bell-curve approximation. Same edge, very different ride. A Correct Score backer at 10.0 has more than four times the swing of a short-priced backer at 1.5.
The formula
Variance of one bet
- σ² is the variance of one bet's profit, per £1 staked.
- p is your true win rate.
- W is what £1 wins after commission: (odds − 1) × 0.98 on Betfair.
In plain English: a bet has two outcomes that sit W + 1 apart. The variance is that gap squared, multiplied by how uncertain the outcome is.
Standard deviation of one bet
- σ (sigma) is the standard deviation, in £ per £1 staked.
- W + 1 is the net odds after commission (3.0 becomes 2.96).
In plain English: this is the same formula as in Lesson 1.1, with commission taken off the odds. Longer odds mean a bigger gap between winning and losing, so a bigger swing.
Swing over many bets
- n is the number of bets.
- S is your stake per bet.
- μ (mu) is the expected profit per £1 staked.
In plain English: your expected profit grows in a straight line with the number of bets, but the swing only grows with the square root. Double the bets and the swing grows by about 1.4 times, while the expected profit doubles.
Why variance adds and standard deviation doesn't
- X₁ … Xₙ are the profits of each bet.
In plain English: variances of independent bets simply add up, which is why statisticians like them. Take the square root at the end to get back to £ you can picture.
Try it
Set odds 3.0, edge 5%, 1,000 bets and £10 stakes, and check the readout shows an expected profit of £360 and a standard deviation of about £446. Then drop the odds to 1.5 and watch the funnel tighten, and push them to 10.0 and watch how many of the twenty bettors finish below zero.
Common mistakes
- Judging a strategy on one month. At odds of 3.0, 100 bets can land anywhere from −£246 to +£318 on £10 stakes. A month tells you almost nothing about the edge.
- Ignoring the odds when comparing strategies. A 20-bet losing run is routine for a Correct Score backer and alarming for someone backing at 1.5. Compare swings, not raw results.
- Thinking variance evens out quickly. It evens out relative to your turnover, not in pounds. After 1,000 bets the swing in pounds is bigger than after 100; it's only smaller as a share of what you staked.
- Changing stakes after a bad run. Doubling stakes to "win it back" doubles the swing too. Staking belongs in Module 6, decided before the run starts.
- Forgetting that the maths assumes independent bets. Five bets on the same match or the same weekend move together. Linked bets swing more than this formula says, which is the subject of Lesson 1.4.
Why swings sink so many good models: why good models still lose money.