Two games with the same house edge can feel completely different. Standard deviation is why — and here is how it is computed per hand.
I. Two numbers describe every bet, not one
The house edge is the mean of a bet's outcome distribution — the centre of gravity your bankroll drifts toward. But a distribution has a width as well as a centre, and the width is what you actually feel session to session. That width is the variance, and its square root, the standard deviation (SD), is measured in the same units as the bet itself. Edge sets the destination; standard deviation sets the size of the detours.
II. The definition
Var = E[X²] − (E[X])² ; SD = √Var
X is the net result of a one-unit bet. Because the mean (E[X]) is tiny for table games, the variance is dominated by E[X²] — the size of the payouts, weighted by how often they land.III. Roulette, derived exactly
Compare two bets on the same single-zero wheel, both carrying the identical 2.70% edge. First, a straight-up bet on one number, paying 35 to 1:
E[X²] = 35² × (1/37) + 1² × (36/37) = 1261/37 ≈ 34.08
Var ≈ 34.08 − (0.027)² ≈ 34.08 → SD ≈ 5.84 units
Now an even-money bet — red, say — on the same wheel:
E[X²] = 1² × (18/37) + 1² × (19/37) = 1.0
Var ≈ 1.0 − (0.027)² ≈ 0.999 → SD ≈ 1.0 unit
Identical 2.70% edge; standard deviation differs almost sixfold. The straight-up bet is the same game, priced identically, delivered as a completely different experience.IV. Standard deviation by bet
| Bet | House edge | Std. deviation (units) |
|---|---|---|
| Baccarat — Banker | ≈ 1.06% | ≈ 0.93 |
| Blackjack — one hand, basic strategy | ≈ 0.5% | ≈ 1.15 |
| Roulette — even money (single zero) | ≈ 2.70% | ≈ 1.0 |
| Craps — pass line | ≈ 1.41% | ≈ 1.0 |
| Roulette — straight-up (single zero) | ≈ 2.70% | ≈ 5.84 |
Premise. Roulette figures are derived exactly above. Blackjack's ≈ 1.15 reflects the extra swing from doubles, splits and the 3:2 natural on a standard six-deck game; baccarat's ≈ 0.93 comes from the Banker payoff distribution. These are the standard reference values for their stated rule sets, not estimates.
V. The square-root law — scaling to a session
The reason a short session feels like luck and a long one feels like arithmetic is that the two numbers scale differently with the number of bets, N:
expected result = −N × edge (grows in proportion to N)
standard deviation of the total = SD × √N (grows only as √N)
Because the edge grows with N and the swing only with √N, the edge inevitably overtakes variance as play lengthens. Early on, the √N spread dwarfs the tiny expected loss — which is precisely why short sessions can end well ahead, and why the house needs volume, not any single hand.Worked example: 100 pass-line bets of one unit. Expected loss ≈ 100 × 0.0141 ≈ 1.4 units; standard deviation ≈ 1.0 × √100 = 10 units. The typical swing is roughly seven times the expected loss — so over 100 hands the result is dominated by variance, not edge. Play 10,000 hands and the expected loss (≈ 141 units) finally exceeds the swing (≈ 100 units). The house is playing the long N; the player is usually playing the short one.
- Every bet needs two numbers: edge (the mean) and standard deviation (the spread).
- Same-edge bets can differ enormously in SD — roulette straight-up (≈ 5.84) vs even-money (≈ 1.0).
- Expected loss scales with N; the swing scales only with √N. That is the whole story of short vs long play.
- Short sessions are governed by variance; long sessions by the edge.
- Standard deviation is one input to risk of ruin, not the same thing as it.