Mixed polyhedral dice and a single fixed wooden counter on a gaming mat. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Add means and variances for independent dice
A fair s-sided die numbered 1 through s has mean (s+1)/2 and variance (s²−1)/12. For independent dice, expected values add and variances add. Standard deviations do not add directly: add variances first, then take the square root. A fixed modifier shifts the mean and every possible total but adds no variance.
Work through 2d6 + 1d8 + 3
Two six-sided dice contribute mean 7 and variance 35/6. One eight-sided die contributes mean 4.5 and variance 63/12 = 5.25. Adding the fixed modifier gives total mean 14.5. Total variance is 133/12, about 11.0833333, and standard deviation is approximately 3.3292. The minimum possible total is 6 and the maximum is 23.
| Term | Mean | Variance |
|---|---|---|
| 2d6 | 7 | 35/6 |
| 1d8 | 4.5 | 5.25 |
| +3 | 3 | 0 |
Changing a modifier differs from adding a die
Replacing +3 with +5 raises the mean by two without changing the spread. Replacing +3 with 1d6 changes both mean and variance. Two expressions can have the same average but different minimums, maximums and probabilities of crossing a target. Average alone is not enough to compare a threshold-based game mechanic.
Keep special mechanics outside the ordinary expression
The Dice Average Calculator supports sums of standard fair dice and integer modifiers. Keep-highest, exploding dice, rerolls, advantage and custom face values need different models. For a threshold probability on identical ordinary dice, use the Dice Probability Calculator. Read why an average need not be a possible or most likely roll before interpreting a decimal expectation as a suggested single outcome.
Check the model behind your result
Use the Dice Average Calculator for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26