Variance of a Sum of Dice

The average tells you where a roll sits; the variance tells you how far it wanders, and for a game designer it is often the more important of the two. A single die of $d$ faces has variance $(d^{2} - 1)/12$ — which is the variance of a uniform distribution, with the 12 coming from the integral that defines it — and because the dice are independent, their variances add. Hence $n(d^{2}-1)/12$, with no cross terms.

The number you actually want is the square root of that, because the standard deviation is in pips and can be compared directly with the average. Three six-sided dice: average 10.5, standard deviation 2.96. One twenty-sided die: average 10.5, standard deviation 5.77. Identical averages, one nearly twice as swingy as the other. That single comparison is the reason a designer who wants results to feel reliable rolls a handful of small dice and one who wants drama rolls a single large one, and it is why the two feel nothing alike at the table despite agreeing on paper.

Notice how the two quantities grow at different rates. The average is proportional to the number of dice; the standard deviation grows as its square root. Double the dice and the total doubles while the spread grows only by a factor of about 1.41, so the roll becomes relatively tighter — the coefficient of variation falls. Push that far enough and the sum of many dice becomes very nearly a bell curve clustered close to its mean, which is the central limit theorem arriving in a form you can hold in your hand.

Two limitations worth stating. This is the variance of the TOTAL, not of an average roll, and it applies only to independent dice added together — the moment a rule says "reroll ones" or "keep the highest three", the variance is a different calculation entirely and this one does not cover it. And variance is a summary, not a boundary: a distribution with a standard deviation of three pips still produces results six pips out, regularly and by design.

The same warning belongs here as on every dice page. A spread describes the shape of the long run and predicts nothing about the next roll. A sequence of results at one end of the distribution is exactly what a distribution with that spread produces from time to time, and it does not make the other end more likely afterwards.