The average roll of a die is the mean of its faces. A six-sided die averages (1 + 6)/2 = 3.5 and a twenty-sided die averages 10.5. You will never roll 3.5, of course – the expected value describes the long-run average over many rolls, not a single throw.
Averages add up, so three six-sided dice average 3 × 3.5 = 10.5 and a 2d6 + 3 damage roll averages 10. The spread of the total grows more slowly: the variance of one die is (s² − 1)/12, variances add for independent dice, and the standard deviation is the square root, so 3d6 has a standard deviation of about 2.96. As you add dice the distribution of the total becomes the familiar bell shape centred on the average.
Game designers use these numbers to balance weapons and spells, players use them to compare options such as 1d12 against 2d6, and teachers use them to introduce expected value. Choose a die type, the number of dice and an optional modifier; the calculator gives all the summary figures and draws the exact distribution with the most likely totals highlighted.
How to use the Dice Average Calculator
- Dice type
- Number of sides (custom die)
- Number of dice
- Modifier added to the total — For example +3 for 2d6 + 3.
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The expected value and variance follow from the uniform distribution on 1 … s and the rules E(X + Y) = E(X) + E(Y) and Var(X + Y) = Var(X) + Var(Y) for independent dice. The chart convolves the single-die distributions to get the exact probability of every total.
Worked example
Roll 3d6. One die averages (6 + 1)/2 = 3.5, so the total averages 3 × 3.5 = 10.5. One die has variance (36 − 1)/12 ≈ 2.917, so the total has variance 8.75 and standard deviation √8.75 ≈ 2.958. Totals run from 3 to 18; 10 and 11 can each be rolled 27 ways out of 216 (12.5%), and 67.6% of rolls land between 7.5 and 13.5.
Average and spread of common dice
| Die | Faces | Average roll | Standard deviation | Typical use |
|---|---|---|---|---|
| d4 | 1–4 | 2.5 | 1.118 | Small weapons, healing potions |
| d6 | 1–6 | 3.5 | 1.708 | Board games, fireballs, ability scores |
| d8 | 1–8 | 4.5 | 2.291 | Longswords, cleric hit dice |
| d10 | 1–10 (or 0–9) | 5.5 | 2.872 | Percentile rolls, heavy weapons |
| d12 | 1–12 | 6.5 | 3.452 | Greataxes, barbarian hit dice |
| d20 | 1–20 | 10.5 | 5.766 | Attack rolls, checks and saves |
| d100 | 1–100 | 50.5 | 28.866 | Percentage tables, random events |
Tips
- Write probabilities as decimals between 0 and 1, or percentages between 0% and 100% – the field label says which.
- Check whether the events are independent before multiplying probabilities.
- Small probabilities add up: a 1% chance repeated 100 times happens at least once 63% of the time.
- Everything is calculated in your browser, so you can try as many scenarios as you like.
Common mistakes to avoid
- Adding probabilities of events that can happen together (use P(A) + P(B) − P(A and B)).
- Confusing odds (3 to 1) with probability (75% or 25%).
- Believing a streak makes the next independent outcome more or less likely (gambler’s fallacy).
Formula
E(one die) = (s + 1)/2 · E(total) = n·(s + 1)/2 + modifier · Var(total) = n·(s² − 1)/12 · SD = √Var
Worked example
Inputs — Dice type: Cube – d6 (6 faces); Number of dice: 3; Modifier added to the total: 0.
Results — Average total = 10.5; Average of one die = 3.5; Standard deviation of the total = 2.958; Variance of the total = 8.75; Lowest possible total = 3; Highest possible total = 18.
Key terms
- Expected value
- The long-run average outcome of a random experiment.
- Variance
- The average squared distance from the mean; for one fair s-sided die it is (s² − 1)/12.
- Standard deviation
- The square root of the variance, in the same units as the total.
- Modifier
- A fixed number added to a dice roll, such as the +3 in 2d6 + 3.
Frequently asked questions
What is the average roll of a d6?
3.5, the mean of 1, 2, 3, 4, 5 and 6.
What is the average of 2d6?
7 – and 7 is also the most likely total, with probability 1/6.
What is the average of 3d6?
10.5, with a standard deviation of about 2.958; 10 and 11 are equally the most likely totals (12.5% each).
What are the averages of the other dice?
d4 2.5, d8 4.5, d10 5.5, d12 6.5, d20 10.5 and d100 50.5 – always (sides + 1)/2.
Is 1d12 or 2d6 better?
Both average about the same (6.5 vs 7), but 2d6 is much more consistent: its standard deviation is 2.42 against 3.45 for 1d12.
What does the standard deviation tell me?
How far a typical total lands from the average. For 3d6 about two thirds of rolls fall within 10.5 ± 2.96.
Does a modifier change the spread?
No. Adding +3 shifts every total by 3 but leaves the standard deviation unchanged.
What formula does the Dice Average Calculator use?
E(one die) = (s + 1)/2 · E(total) = n·(s + 1)/2 + modifier · Var(total) = n·(s² − 1)/12 · SD = √Var.