Dice Calculator — Expected Value & Statistics
Enter the number of dice, sides per die, and any flat modifier to see the mean, min, max, variance, and standard deviation of the roll total.
Average result of 2d6
- 1
Single die expected value
(s + 1) ÷ 2 = (6 + 1) ÷ 2 = 3.5 - 2
Dice sum expected value
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) = 2.42
此計算機如何運作？
For n dice with s sides and modifier m: mean = n(s+1)/2 + m, variance = n(s²−1)/12, std dev = √variance. Min = n+m, max = n·s+m. The bell-curve approximation works well for 3+ dice. For 2d6+0 the mean is 7 and the standard deviation is about 2.42.
公式
How this is calculated
For a fair s-sided die, every face from 1 to s is equally likely, so the expected value of one die is (s+1)/2. Rolling n independent identical dice multiplies both the mean and the variance by n, giving mean = n(s+1)/2 and variance = n(s²−1)/12. A flat modifier shifts the mean by a constant without changing the spread. Standard deviation is the square root of variance.
The minimum possible total is n (all dice show 1) plus any modifier, and the maximum is n·s (all dice show the highest face) plus any modifier. By the central limit theorem, the distribution of a dice-sum approaches a normal bell curve as n grows — the approximation becomes quite close for n ≥ 3 even for small s, and is excellent for n ≥ 10.
For dice expressions where n·s ≤ 500 total outcomes, the calculator uses the exact convolution distribution to find the probability of the modal (most-likely) sum. All dice are assumed to be fair and independent; fractional modifier values are not meaningful for integer dice rolls and should be entered as integers.
常見問題
Mean = n(s+1)/2 = 2×(6+1)/2 = 7. Seven is the most likely result with 2d6, appearing with probability 6/36 ≈ 16.7%.
A constant modifier shifts the mean, minimum, and maximum by exactly that amount, but leaves the variance and standard deviation unchanged. Adding +3 to 2d6 gives mean 10, min 5, max 15, but the same standard deviation ≈ 2.42.
About 68% of rolls land within ±1 standard deviation of the mean, and about 95% within ±2. A larger standard deviation means more variability and less predictable results. High-variance rolls (many-sided dice) are wilder than low-variance ones.
也稱為
TG we-Calculate Editorial Team. (2026). Dice Calculator — Expected Value & Statistics [Online calculator]. TG we-Calculate. https://we-calculate.com/zh-tw/calculator/dice-calculator
TG we-Calculate Editorial Team. "Dice Calculator — Expected Value & Statistics." TG we-Calculate. 2026. https://we-calculate.com/zh-tw/calculator/dice-calculator.
TG we-Calculate Editorial Team, "Dice Calculator — Expected Value & Statistics," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/zh-tw/calculator/dice-calculator
@misc{wecalculate_dice_calculator, title = {Dice Calculator — Expected Value & Statistics}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/zh-tw/calculator/dice-calculator}}, year = {2026}, note = {TG we-Calculate} }
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