Dice Roll Expected Value Calculator
Calculate the expected value, variance, and standard deviation when rolling one or more dice with any number of sides.
Results will appear here.
Formula
For n dice each with s sides and a modifier m:
- Expected Value of one die: E[X] = (s + 1) / 2
- Expected Value of total roll: E[T] = n × (s + 1) / 2 + m
- Variance of one die: Var[X] = (s² − 1) / 12
- Variance of total roll: Var[T] = n × (s² − 1) / 12
- Standard Deviation: σ = √Var[T]
- Minimum Roll: n × 1 + m
- Maximum Roll: n × s + m
Example: 2d6 → E[T] = 2 × (6+1)/2 = 7, Var[T] = 2 × (36−1)/12 ≈ 5.833, σ ≈ 2.415
Assumptions & References
- Each die is fair (all faces equally likely with probability 1/s).
- All dice rolls are independent of each other.
- Sides are numbered consecutively from 1 to s.
- The modifier is a fixed integer added to the sum of all dice (common in tabletop RPGs).
- Variance of a sum of independent random variables equals the sum of their individual variances.
- Formula derivation: E[X] = Σ(i=1 to s) i/s = (s+1)/2; Var[X] = E[X²] − (E[X])² = (s+1)(2s+1)/6 − ((s+1)/2)² = (s²−1)/12.
- Reference: Probability and Statistics for Engineering and the Sciences, Jay Devore.