Custom Dice Roller Calculator

Enter dice in NdM+K form — number of dice, sides per die, and a flat modifier — to get the minimum, maximum, mean, standard deviation, and number of possible outcomes for the total.

Quick Facts

Method
Exact discrete-uniform dice statistics: mean = N(M+1)/2 + K, variance = N(M²−1)/12
Each die is a fair, independent, uniform 1..M outcome.

Your Results

Calculated
Mean total
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Expected value of the roll
Range (min – max)
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Lowest and highest possible totals
Standard deviation
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Typical spread around the mean
Possible outcomes
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Distinct ordered rolls (MN)

Ready

Enter your dice notation and press Calculate.

Understanding custom dice rolls

Tabletop games, probability homework, and game design all use the shorthand NdM+K: roll N dice, each with M sides numbered 1 through M, sum them, then add a flat modifier K. This calculator takes any N, M, and K and returns the exact statistics of the total — its lowest and highest possible values, its average, its spread, and how many distinct ways the dice can land. "3d6+2" means three six-sided dice plus 2; "1d20" means a single twenty-sided die; "4d4-1" means four four-sided dice minus 1.

The formulas

A single fair die of M sides is a discrete uniform variable on {1, 2, …, M}. Because the N dice are independent, means and variances add. Let S be the total of NdM+K:

  • Minimum: Smin = N + K (every die shows 1).
  • Maximum: Smax = N×M + K (every die shows M).
  • Mean (expected total): E[S] = N×(M+1)/2 + K. One die averages (M+1)/2, so N of them average N×(M+1)/2, and the modifier just shifts the center by K.
  • Variance: Var[S] = N×(M²−1)/12. A single die has variance (M²−1)/12; adding a constant K does not change variance.
  • Standard deviation: σ = √(N×(M²−1)/12).
  • Distinct ordered outcomes: MN (each die has M faces, chosen independently). Note that many of these map to the same total, which is why middle totals are far more likely than extremes.

Common reference points

  • 1d6: min 1, max 6, mean 3.5, σ ≈ 1.708. The classic single cube.
  • 2d6: min 2, max 12, mean 7, σ ≈ 2.415. The most common roll in board games; 7 is the single most likely total.
  • 3d6: min 3, max 18, mean 10.5, σ ≈ 2.958. The traditional ability-score roll in many role-playing games.
  • 1d20: min 1, max 20, mean 10.5, σ ≈ 5.766. Flat and swingy — every value is equally likely.
  • 4d6: min 4, max 24, mean 14, σ ≈ 3.416.

Why the distribution matters

The single die (1dM) is flat: every face is equally likely. But adding dice together produces a bell-shaped distribution by the central limit theorem — the more dice, the tighter and more symmetric the hump around the mean. That is why 3d6 (rolls cluster near 10–11) feels very different from 1d18 (any value 1–18 equally likely) even though both have the same average of 10.5. The mean tells you the center; the standard deviation tells you how far a typical roll strays from it.

Frequently Asked Questions

What does NdM+K notation mean?
N is the number of dice, M is the number of sides on each die, and K is a flat modifier added to the total. For example, 3d6+2 means roll three six-sided dice, add them together, then add 2. The possible totals run from 3×1+2 = 5 up to 3×6+2 = 20.
What is the average roll of NdM+K?
The mean total is N×(M+1)/2 + K. A single d6 averages 3.5, so 2d6 averages 7 and 4d6 averages 14. The modifier K shifts the average up (positive K) or down (negative K) by exactly K, without changing the spread.
Why is 7 the most likely result on 2d6?
There are 36 equally likely ordered outcomes on two dice. Six of them sum to 7 (1-6, 2-5, 3-4, 4-3, 5-2, 6-1), more than any other total, so 7 has probability 6/36 = 1/6. Totals of 2 and 12 each have just one combination, so they are the rarest at 1/36 apiece.
Does adding dice make the result more or less predictable?
More predictable relative to its range. As you add dice, the distribution of the total becomes bell-shaped and concentrates near the mean. The standard deviation grows only with the square root of N, while the range grows linearly with N, so the spread shrinks as a fraction of the total. That is why 10d6 lands near its average of 35 far more reliably than 1d60 lands near 30.5.

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