How this calculator works
The mean of the consecutive faces 1 through s is (s + 1)/2. Expectations add across dice. Under the independence assumption, variances add as well. A fixed modifier shifts the mean and both range endpoints but leaves variance unchanged. Mixed mode pairs each count with the side count in the same position. The mean can be fractional even when every attainable total is a whole number.
Formula
For count c_i of s_i-sided dice and modifier b: E(total) = b + sum[c_i(s_i + 1)/2]. Var(total) = sum[c_i(s_i² - 1)/12]. Standard deviation = sqrt(variance).
Worked example
Two d6 plus one have mean 2 × 3.5 + 1 = 8, variance 2 × 35/12 = 35/6 and range 3 to 13. One d4 plus two d8 has mean 2.5 + 9 = 11.5.
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Frequently asked questions
Does average mean the most likely total?
No. It is the expected value over repeated rolls. Several totals can tie for the highest probability.
Do modifiers change standard deviation?
A fixed modifier does not. A random modifier would need its own probability model.
Can I drop the lowest die?
This calculator adds all entered dice. Keeping only selected dice changes the distribution and needs a different model.