Dice Average Calculator
Calculate the expected average, variance, and standard deviation for rolling any number of dice. Works with all standard polyhedral dice from d4 to d20.
When you roll dice, the expected average tells you the long-run mean outcome. Knowing the variance and standard deviation tells you how spread out the results will be.
Expected Value of One Die:
E(X) = (s + 1) / 2
where s is the number of sides. A d6 averages 3.5, a d20 averages 10.5.
For Multiple Dice:
- Expected total = n x (s + 1) / 2
- Variance of one die = (s^2 - 1) / 12
- Total variance = n x (s^2 - 1) / 12
- Standard deviation = square root of total variance
Quick Reference:
| Die | Average | Variance |
|---|---|---|
| d4 | 2.5 | 1.25 |
| d6 | 3.5 | 2.917 |
| d8 | 4.5 | 5.25 |
| d10 | 5.5 | 8.25 |
| d12 | 6.5 | 11.917 |
| d20 | 10.5 | 33.25 |
Example: Rolling 3d6 (three six-sided dice) gives an expected average of 10.5 with a standard deviation of about 2.96. That means most rolls land between roughly 8 and 13.