Dice Average Calculator
Find the average result of any dice roll in NdS notation. Enter the number of dice, pick the die type (d4 through d100), add a flat modifier if your roll has one, and get the expected roll, the possible range, and how much a typical roll swings.
Example: with Number of dice (N) 13 · Die type (S sides) d8 · Modifier (+/-, optional) 0 → Average roll: 58.5 (13d8).
- Possible range13 to 104
- Typical swing (standard deviation)± 8.26 — about 2 in 3 rolls land between 50.24 and 66.76
Computed by the calculator below using its default values. Change any input to see your own numbers.
Each fair die averages (S + 1) / 2, so a d8 averages 4.5 and 13d8 averages 13 × 4.5 = 58.5. The modifier shifts everything by a constant.
Why the average of a die is (S + 1) / 2
A fair die lands on each face equally often, so its long-run average is the mean of the numbers 1 through S. That mean is (S + 1) / 2: a d6 averages 3.5, a d8 averages 4.5, a d20 averages 10.5. Averages add, so N dice average N × (S + 1) / 2, and a flat modifier just shifts the total. For the classic 13d8 roll, that is 13 × 4.5 = 58.5.
The standard deviation tells you how far real rolls stray from that average. One die has variance (S² − 1) / 12, and independent dice add variance, so 13d8 swings about ±8.3 around 58.5. More dice make the total more predictable relative to its size — 13d8 is far steadier than 1d100, even though both can reach triple digits.
How it’s calculated
Average = N × (S + 1) / 2 + modifier. Minimum = N + modifier; maximum = N × S + modifier. Standard deviation = √(N × (S² − 1) / 12), from the variance of a discrete uniform die (S² − 1) / 12 summed across independent dice. For 13d8: average 58.5, range 13 to 104, SD √(13 × 5.25) ≈ 8.26.
Assumes fair, independent dice with faces 1 through S — it does not model exploding dice, rerolls, advantage, or drop-lowest mechanics.
Average roll of common dice
| Die | Average per die | Average of 2 | Average of 10 |
|---|---|---|---|
| d4 | 2.5 | 5 | 25 |
| d6 | 3.5 | 7 | 35 |
| d8 | 4.5 | 9 | 45 |
| d10 | 5.5 | 11 | 55 |
| d12 | 6.5 | 13 | 65 |
| d20 | 10.5 | 21 | 105 |
| d100 | 50.5 | 101 | 505 |
Computed with (S + 1) / 2 per die; averages add across dice.
Common mistakes
- Using S / 2 instead of (S + 1) / 2 — a d6 averages 3.5, not 3, because the faces start at 1, not 0.
- Multiplying the modifier by the number of dice: 2d6+3 averages 10, not 13. The +3 is added once.
- Expecting the average on every roll — 13d8 averages 58.5, but about a third of rolls land more than 8 points away.
- Adding standard deviations instead of variances when combining different dice pools.
Frequently asked questions
What is the average dice roll formula?
Average = N × (S + 1) / 2 + modifier, where N is the number of dice and S the sides per die. Each fair die averages (S + 1) / 2 because the faces 1 through S are equally likely.
What does 13d8 average?
13d8 averages 13 × 4.5 = 58.5, with a possible range of 13 to 104 and a standard deviation of about 8.3. Roughly two-thirds of rolls land between 50 and 67.
Why is a d6 average 3.5 and not 3?
The faces run 1 to 6, so the mean is (1 + 6) / 2 = 3.5. Halving the top face ignores that the die cannot roll a 0 — the most common mistake with dice math.
Does rolling more dice make results more predictable?
Yes, relative to the total. The average grows with N but the standard deviation only grows with √N, so a 20-dice pool clusters much more tightly around its average than a single big die.