Concepts → Fundamentals → Variance and standard deviation

Variance and standard deviation, in dice — how to measure spread.

Once you know the mean of a distribution, the next question is: how far do typical outcomes wander from it? Variance answers that in squared units; standard deviation answers it in the same units as the roll, so you can compare it directly to the mean.

Definition

The variance of a random variable X is the expected squared distance from the mean:

Var(X) = E[(X − μ)²], where μ = E[X].

Algebraically equivalent and usually easier to compute:

Var(X) = E[X²] − (E[X])²

Standard deviation is just the square root of variance:

SD(X) = √Var(X)

Variance is in squared units (squared damage, squared HP), which makes the raw number hard to read. Standard deviation puts it back into native units, which is why most rules of thumb (the "68-95-99.7 rule" you may have seen for normal distributions) are stated in standard deviations.

Worked example: 1d6

For 1d6, the mean is μ = 7/2. Variance:

E[X²] = (1² + 2² + 3² + 4² + 5² + 6²) / 6 = 91/6

Var(1d6) = 91/6 − (7/2)² = 91/6 − 49/4 = 182/12 − 147/12 = 35/12

So Var(1d6) = 35/12 ≈ 2.917, and SD(1d6) = √(35/12) ≈ 1.708. Tap the mean on the panel below to read off the rational; the engine carries the same 35/12 internally for any expression that involves a d6.

1d6

min 1 max 6

1 16.67%
2 16.67%
3 16.67%
4 16.67%
5 16.67%
6 16.67%

The shortcut for fair dice

For a fair die with M faces:

Var(1dM) = (M² − 1) / 12

Drop in M = 6: (36 − 1)/12 = 35/12. Drop in M = 20: (400 − 1)/12 = 399/12 ≈ 33.25. Drop in M = 4: (16 − 1)/12 = 15/12 = 5/4 = 1.25.

Variance grows quadratically in the face count: a d20 has roughly 27× the variance of a d4, even though the means differ by a factor of 4.2. That ratio is why high-face-count dice produce wildly different fight outcomes from low-face-count ones at the same mean — the spread is in different leagues.

Variance under sums

For independent random variables, variance adds:

Var(X + Y) = Var(X) + Var(Y) (when X, Y independent)

Two fair 1d6s are independent, so:

Var(2d6) = Var(1d6) + Var(1d6) = 35/12 + 35/12 = 70/12 = 35/6

Standard deviation: SD(2d6) = √(35/6) ≈ 2.415. Compared to SD(1d6) ≈ 1.708 — note that the variance doubled but the standard deviation grew only by a factor of √2 ≈ 1.414.

2d6

min 2 max 12

2 2.78%
3 5.56%
4 8.33%
5 11.11%
6 13.89%
7 16.67%
8 13.89%
9 11.11%
10 8.33%
11 5.56%
12 2.78%

Constants don't add variance — they just shift the mean. So Var(2d6 + 5) = Var(2d6) = 35/6, and the +5 only moves the centre. That is a foundational fact: when you stack a flat damage modifier onto a roll, you change the mean linearly but the spread not at all.

Same mean, different spread

2d6+5 and 3d4+4 have nearly identical means (12 vs 11.5) but very different variances. Build a quick mental table:

Var(2d6+5) = 2·(35/12) = 35/6 ≈ 5.83 (SD ≈ 2.41)
Var(3d4+4) = 3·(15/12) = 45/12 = 15/4 = 3.75 (SD ≈ 1.94)

Same ballpark mean. 2d6+5 has roughly 1.55× the variance. That's the entire mechanical reason for the surprising result in variance and kill probability: against a target whose HP sits below the mean, the higher-variance roll wins; above the mean, the lower-variance roll wins.

Try it yourself

↦ /strike/1d6 — Var = 35/12 ↦ /strike/2d6 — Var = 35/6 (twice 1d6) ↦ /strike/3d4+4 — Var = 15/4, mean 23/2 ↦ /strike/2d6+5 — Var = 35/6, mean 12 ↦ /strike/1d20 — Var = 399/12 ≈ 33.25