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Dice Notation Calculator

Enter the number of dice, sides, and a modifier to analyze an NdM+K roll.

How it is calculated

For the roll NdM+K, n is how many dice you throw, m the number of faces on each and k a flat modifier added once to the total (it can be negative, and defaults to 0). The lowest total happens when every die shows 1, min = n + k; the highest when every die shows its top face, max = n × m + k. The average is n × (m + 1) ÷ 2 + k, because one fair die averages (m + 1) ÷ 2. Needs at least one die and at least two faces, otherwise there is no result. The average is shown to one decimal. The tool does not floor results (a negative K can give a minimum below 1 or 0), it does not handle exploding, dropped or re-rolled dice, and it gives no distribution, only the three summary values.

Worked example

Take 4d8+3: four eight-sided dice plus 3. Step 1: the average of one d8 is (8 + 1) ÷ 2 = 4.5, so four dice average 4 × 4.5 = 18, and the modifier lifts that to 18 + 3 = 21.0. Step 2: the minimum is four 1s plus 3, which is 4 + 3 = 7. Step 3: the maximum is four 8s plus 3, which is 4 × 8 + 3 = 35. So the roll spans 7 to 35 with a typical total in the low twenties.

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Common mistakes

  • Reading the average as the most likely single result. It is the mean; with several dice the totals cluster around it, but you cannot get a fractional total such as 12.5 from a real roll.
  • Adding the modifier to every die. In NdM+K, K is added once to the whole roll, not to each die; (3d6)+2 is not 3(d6+2).
  • Treating the range as equally likely. With more than one die, middle totals are far more likely than the minimum or maximum, which each need all dice to land the same way.
  • Entering a one-sided die or zero dice. The tool needs at least 1 die and 2 faces, so a flat bonus belongs in K.
  • Forgetting game rules that floor the result. A negative modifier can push the tool's minimum under 1 or 0, while many games treat damage as at least 0 or 1.

About this calculator

Dice notation like “3d6+2” is shorthand used across tabletop games for “roll N dice with M sides each, then add a flat modifier K” — written generally as NdM+K. This calculator computes the statistical range of such a roll: the minimum possible result (all dice show 1, plus K), the maximum (all dice show M, plus K), and the average or expected value, which for a single fair die of M sides is (M+1)/2, scaled by N dice and offset by K.

Game designers, dungeon masters, and players of tabletop RPGs like Dungeons & Dragons and board games use dice notation constantly to describe damage rolls, ability checks, and random loot tables, and knowing a roll's average and spread helps with game balancing — a weapon dealing “2d6” damage has the same average as one dealing a flat “7” but with much more variance, which matters for both game design and in-the-moment player decisions.

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Comments (1)

  • Editorial team

    Averages make different options directly comparable. A single d20 averages 10.5, spanning 1 to 20, while 2d6 averages 7.0 and spans only 2 to 12, so the second is far steadier. For a flat bonus choice such as 1d12+7, the tool gives 13.5 on average with a range of 8 to 19. It shows only the summary, not how likely each total is. What roll are you trying to balance or compare?

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