Dice Average Roll Calculator: Mean, Range, Odds and Full Distribution

How it works

one die: mean = (s + 1) ÷ 2, variance = (s² − 1) ÷ 12 · n dice: mean = n × (s + 1) ÷ 2 + modifier, SD = √[n × (s² − 1) ÷ 12] · advantage: P(max ≤ t) = F(t)² · disadvantage: P(min > t) = (1 − F(t))²

A fair die with s faces averages the midpoint of its faces, (s + 1) ÷ 2, so a d6 averages 3.5 and a d20 averages 10.5. Rolling several dice adds those averages, and a flat modifier shifts the whole total. The spread comes from the variance of one die, (s² − 1) ÷ 12, which also adds across dice; its square root is the standard deviation. Sums of several dice bunch toward the middle in a bell shape, so more, smaller dice give steadier totals than one big die with the same average. For the full picture the calculator builds the exact probability of every possible total by convolving the dice one at a time, then reads off the chance of hitting your target, the most likely total, the median and the middle half of results. Advantage and disadvantage square the cumulative distribution (roll twice, keep the higher or lower total). Drop-and-keep rolls such as 4d6 drop lowest are counted exactly by sorting the dice faces from high to low, so the averages are exact rather than simulated. An optional second group of dice lets you model rolls like 2d6 + 1d8 + 3. All dice are assumed fair and independent; rerolls and exploding dice are not modelled.

Worked example

For 3d6+2 with a target of 15: average = 3 × (6 + 1) ÷ 2 + 2 = 10.5 + 2 = 12.5. The minimum is 3 + 2 = 5 and the maximum 18 + 2 = 20. The variance is 3 × (6² − 1) ÷ 12 = 8.75, so the standard deviation is √8.75 ≈ 2.96. Reaching 15 or more needs 13 or more on 3d6, which 56 of the 216 outcomes do, a 25.93% chance. The most likely totals are 12 and 13, the median is 12, and the middle half of rolls falls between 10 and 15.

Sources: Omni Calculator: dice average · Omni Calculator: dice probability · AnyDice: dice probability calculator

Frequently asked questions