Dice Average Calculator — Expected Value & Distribution
Calculate the expected average, standard deviation, and full probability distribution for any dice roll — pick a standard die (d4–d100) or enter custom sides, and choose how many dice to roll at once.
Die type
The long-run average result across many rolls
- 1
Single die expected value
(s + 1) ÷ 2 = (6 + 1) ÷ 2 = 3.5 - 2
Expected total (mean)
n × (s + 1) ÷ 2 = 2 × 3.5 = 7 - 3
Per-die variance
(s² − 1) ÷ 12 = (36 − 1) ÷ 12 = 2.9167 - 4
Standard deviation
√(n × per-die variance) = √(2 × 2.9167) = 2.42Roughly two-thirds of all rolls land within ±1 SD of the mean.
How does this calculator work?
For n fair dice each with s sides (1 to s): expected total = n(s+1)/2, standard deviation = √[n(s²−1)/12], minimum = n, maximum = ns. Enter dice count and type above for the exact figures plus an approximate normal distribution curve.
Formula
How this is calculated
When you roll a single fair die with s sides numbered 1 to s, each face is equally likely with probability 1/s. The expected value (long-run average) of one die is the simple mean of 1 through s: (1 + 2 + … + s) / s = (s + 1) / 2. So a standard d6 has an expected value of 3.5, and a d20 has an expected value of 10.5. When you roll n dice and add them, expectations add: total mean = n × (s + 1) / 2.
The spread of results is measured by variance. For one die the variance is (s² − 1) / 12 — a standard result from the uniform discrete distribution. For n independent dice the variances also add, giving total variance = n × (s² − 1) / 12 and standard deviation σ = √(n × (s² − 1) / 12). Roughly two-thirds of all rolls will fall within one standard deviation of the mean, and about 95% within two standard deviations.
The shape of the distribution changes with n. For a single die the distribution is flat (uniform). For two dice the distribution becomes triangular (think of 2d6: 7 is most likely). For n ≥ 2 the Central Limit Theorem drives the distribution toward a bell curve, which is why the visualisation shows a normal approximation — accurate for practical dice pool sizes and useful for estimating probabilities without enumeration.
Frequently asked questions
A standard six-sided die (d6) has an expected value of (6 + 1) / 2 = 3.5. You cannot actually roll 3.5, but over many rolls the average converges to this value.
Multiply the expected value of one die by the number of dice. For 3d6: average = 3 × 3.5 = 10.5. The standard deviation scales as the square root of the number of dice, so the distribution becomes relatively narrower (as a fraction of the mean) with more dice.
Yes — for n ≥ 2 the sum of independent, identically distributed dice rolls converges toward a normal distribution. By the time you are rolling 4 or more dice of the same type, the bell-curve approximation is quite accurate for most probability estimates. A single die (n=1) remains uniformly distributed.
Also known as
TG we-Calculate Editorial Team. (2026). Dice Average Calculator — Expected Value & Distribution [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/dice-average-calculator
TG we-Calculate Editorial Team. "Dice Average Calculator — Expected Value & Distribution." TG we-Calculate. 2026. https://we-calculate.com/calculator/dice-average-calculator.
TG we-Calculate Editorial Team, "Dice Average Calculator — Expected Value & Distribution," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/dice-average-calculator
@misc{wecalculate_dice_average_calculator, title = {Dice Average Calculator — Expected Value & Distribution}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/dice-average-calculator}}, year = {2026}, note = {TG we-Calculate} }
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