In betting, standard deviation (SD) measures how far the result of a run of bets typically lands from its expected value. For an illustrative 100 bets of $10 at $2.00 on true 50% chances, the expected profit is $0 and one standard deviation is $100, so finishing $100 up or down is ordinary luck.
Standard deviation is the square root of variance: variance adds up neatly across bets, while standard deviation comes out in dollars.
Standard deviation for a run of bets
For one bet of stake s at decimal odds o, with a true win chance p:
- expected profit = s x (p x o - 1)
- standard deviation = s x o x square root of (p x (1 - p))
Over n such bets the expected profit grows n times, but the standard deviation only by the square root of n. In winners, it is the square root of (n x p x (1 - p)).
Worked for 100 bets at $2.00
Example: Illustrative: 100 bets of $10, each at $2.00, at three true chances. Ranges are rounded to the dollar.
| True win chance | Edge | Expected profit | One standard deviation | Within one SD | Within two SD |
|---|---|---|---|---|---|
| 47.5% | -5% | -$50 | $99.87 | -$150 to +$50 | -$250 to +$150 |
| 50% | 0% | $0 | $100 | -$100 to +$100 | -$200 to +$200 |
| 52.5% | +5% | +$50 | $99.87 | -$50 to +$150 | -$150 to +$250 |
At 50%: standard deviation = $10 x 2.00 x square root of (0.5 x 0.5) x square root of 100 = $100. In winners it is the square root of (100 x 0.5 x 0.5) = 5 either side of 50, each winner being worth $20. At 52.5% it is $10 x 2.00 x square root of (0.525 x 0.475) x 10 = $99.87.
Counted exactly (the binomial distribution), 72.9% of runs at 50% land between 45 and 55 winners and 96.5% between 40 and 60; the bell-curve rule of thumb says about 68% and 95%. A 5% edge moves the expected profit by $50, half a standard deviation, and barely changes the spread, so luck decides most 100-bet records. Variance in betting covers longer prices.
Using it to judge your own results
Divide the gap between your result and the expected result by the standard deviation, a figure often called a z-score.
Example: Illustrative: the same 100 bets at $2.00 finish with 58 winners, a profit of 58 x $20 - $1,000 = $160.
- Against no edge: (160 - 0) / 100 = 1.6 standard deviations. With no edge, the exact chance of 58 or more winners is 6.7%, about 1 in 15.
- Against a 5% edge: (160 - 50) / 99.87 = 1.1 standard deviations, an unremarkable result.
The record fits a real edge and no edge at all, which is why a betting strategy judges a method on more than its profit. Within one standard deviation is ordinary, beyond two is worth a closer look, and only something near three is hard to put down to luck. Sample size in betting turns this into the bets a record needs.
Risk: Betting involves risk. A result two standard deviations above zero can still be luck, more so if it is the best of several methods tried. See responsible gambling for limits and support.
Where the formula misleads
- Stakes that vary: add each bet's variance, stake x stake x odds x odds x p x (1 - p), then take the square root.
- Linked bets: a win and a place bet on one runner, or multis sharing a leg, move together, so the real spread is wider.
- Long prices: 100 bets at $15.00 bring only a handful of winners, and the bell-curve shares turn rough.
- The worst case: runs three standard deviations out happen, and a drawdown, the fall from a peak, can be deep inside an ordinary final result.