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Expected Damage Roll Calculator

Find the average damage of any dice roll

How it is calculated

For a roll written NdS+M you enter the number of dice N, the number of sides S and the flat modifier M. The tool gives three numbers. Average: N × (S + 1) / 2 + M, because one fair die with sides 1 to S averages (S + 1) / 2. Minimum: N × 1 + M. Maximum: N × S + M.

Dice and sides are cut to whole numbers, the number of dice cannot be negative (0 dice leaves just the modifier) and a die needs at least 1 side. The modifier may be negative. The tool does not limit results to zero, so a large negative modifier can give a negative minimum, and it does not model critical hits, rerolls, advantage or resistance; the average is the plain long-run mean of a single roll.

Worked example

Roll 3d8+4. Each d8 averages (8 + 1) / 2 = 4.5, so three dice average 3 × 4.5 = 13.5. Add the modifier: 13.5 + 4 = 17.50. The minimum is 3 × 1 + 4 = 7 and the maximum is 3 × 8 + 4 = 28, so a single roll can land anywhere from 7 to 28, but 17 or 18 is most common.

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Common mistakes

  • Using the maximum as the expected damage; a roll of 3d8+4 averages 17.5, not 28, and hits the maximum only about once in 512 rolls.
  • Adding the modifier once per die; M is added a single time to the whole roll, not to each die.
  • Assuming a d10 averages 5; its faces run from 1 to 10 in this tool, so the average is 5.5.
  • Expecting the average to be a possible result; 17.5 can never come up on a single roll of whole numbers, it is the mean over many rolls.
  • Applying the average to a hit that can miss; the tool gives damage per roll that lands, so multiply by the hit chance yourself to get damage per attack.

About this calculator

Dice notation NdS+M means rolling N dice with S sides each and adding a flat modifier M. The average (expected) result is N × (S+1)/2 + M — because each single die averages (S+1)/2 over many rolls — while the minimum is N×1+M (all dice show 1) and the maximum is N×S+M (all dice show S).

Tabletop RPG players and dungeon masters use this to estimate whether a monster's or spell's damage (like 8d6 for a fireball) is balanced for a given character level, and game designers use the same math to tune weapons and abilities before publishing a ruleset.

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Comments (1)

  • Editorial team

    Averages let you compare options that look different on paper: 2d6 averages 7.00 and 1d12 averages 6.50, but the minimum and maximum lines show 2 to 12 against 1 to 12, so the two-dice roll clusters nearer its average. The result ignores hit chance and critical hits, so multiply by your chance to hit for expected damage per attack. Tell us which game system's rules, such as advantage, you would like supported.

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