Find expected total, variance, standard deviation and range for mixed fair-dice expressions with fixed integer modifiers.
How to use this tool
- Choose the event or model and review the assumptions below.
- Enter the labeled inputs.
- Select Calculate and read the labeled outputs.
- Change an input and calculate again; Reset restores the example inputs.
Calculation method
For n fair s-sided dice plus modifier b: mean = n(s+1)/2+b; variance = n(s²−1)/12. Mixed independent dice means and variances add.
Assumptions and limits
Supports standard dice numbered 1 through s and fixed integer modifiers. No keep-highest, exploding, advantage, custom face values or reroll rules. Up to 1,000 dice with 2–1,000 sides.
Worked example
For 2d6+1d8+3, expected total is 14.5. Independent variances add to 133/12, approximately 11.0833, giving standard deviation about 3.3292. Possible totals range from 6 to 23. The +3 modifier shifts the center and range but contributes no variance. Keep-highest, rerolls and exploding dice are not part of this ordinary sum model.
Worked reading and practical questions
- Mixed Dice and Fixed Modifiers: Mean Changes Differently from Spread
- An Average Roll Need Not Be a Possible or Most Likely Roll
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Frequently asked questions
What does this calculator assume?
Supports standard dice numbered 1 through s and fixed integer modifiers. No keep-highest, exploding, advantage, custom face values or reroll rules. Up to 1,000 dice with 2–1,000 sides.
How do I enter and update the inputs?
Use the labeled fields, then select Calculate. Percent fields use 0 to 100. After editing, calculate again; Reset restores the example inputs.