Quick answer
The average of dice is n × (sides + 1) ÷ 2 + modifier. Two six-sided dice average 7, with a minimum of 2 and a maximum of 12.
Enter the number of dice, sides per die and an optional modifier. The calculator returns the average, minimum and maximum.
Related tools: Probability Calculator: Probability of A and B, and A or B, for independent or mutually exclusive events. Permutation and Combination Calculator: Number of permutations and combinations of r items from n. Expected Value Calculator: Expected value and variance of a discrete distribution.
How to use the Dice Average Calculator
- Enter number of dice (a whole number, between 1 and 1000).
- Enter sides per die (a whole number, between 2 and 1000).
- Optionally, enter modifier (a whole number).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Number of dice | A whole number | ≥ 1, ≤ 1000 |
| Sides per die | A whole number | ≥ 2, ≤ 1000 |
| Modifier (optional) | A whole number | — |
Formula and method
Formula
average = n × (sides + 1) ÷ 2 + modifier
IEEE-754 double precision; shown to at most 6 decimal places.
Expected value of the sum of fair dice.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Number of dice = 2; Sides per die = 6.
| Result | Value |
|---|---|
| Average roll | 7 |
| Minimum | 2 |
| Maximum | 12 |
Example 2
Inputs: Number of dice = 3; Sides per die = 8; Modifier = 2.
| Result | Value |
|---|---|
| Average roll | 15.5 |
| Minimum | 5 |
| Maximum | 26 |
The first line is the headline result (average roll). Below it the dice Average Calculator lists minimum and maximum, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Per die
One d6 averages 3.5, a d20 averages 10.5 and a d8 4.5. Two dice give 7, and 3d6 average 10.5. A +3 modifier adds exactly 3 to the average.
Fair dice
The formula assumes fair dice. The average is not the most likely roll for every case, but for 2d6 the most likely sum is 7 (probability 1/6). Tabletop games use averages to compare weapons: 2d6 (7) beats 1d10 (5.5).
How the result changes with number of dice
Holding every other input at the first worked example, this table changes only number of dice. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Number of dice | Average roll | Minimum | Maximum |
|---|---|---|---|
| 1 | 3.5 | 1 | 6 |
| 2 | 7 | 2 | 12 |
| 4 | 14 | 4 | 24 |
| 8 | 28 | 8 | 48 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Number of dice: entering "0" shows "Number of dice must be at least 1." beside that field and no result.
Choosing between the Dice Average Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Probability Calculator](/statistics/probability-calculator/) | p, p and events are | P(A and B) |
| [Permutation and Combination Calculator](/statistics/permutation-and-combination-calculator/) | total items and chosen items | Combinations (nCr) |
| [Expected Value Calculator](/statistics/expected-value/) | values and probabilities | Expected value E[X] |
| [Bayes' Theorem Calculator](/statistics/bayes-theorem-calculator/) | prior P, p and p | P(A | B) |
| [Implied Probability Calculator](/statistics/implied-probability/) | odds format and odds | Implied probability |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- Fair dice
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for number of dice to confirm this.
Frequently asked questions
What is the average of 1d20 + 5?
1d20 averages 10.5, so 1d20 + 5 averages 15.5.
What is the average of 4d6?
Each d6 averages 3.5, so 4d6 averages 14.
What is the minimum of 3d8?
The minimum of 3d8 is 3, when every die shows 1 (the maximum is 24).
What does the Dice Average Calculator calculate?
Average, minimum and maximum of rolling several dice. It returns Average roll, Minimum and Maximum.
What formula does the Dice Average Calculator use?
average = n × (sides + 1) ÷ 2 + modifier. Expected value of the sum of fair dice.
What do I need to enter in the dice Average Calculator?
You enter Number of dice and Sides per die. Optional: Modifier.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Fair dice.
What is an example of a dice Average result?
With number of dice 2 and sides per die 6, the average roll is 7.
Related calculators
- Probability Calculator — Probability of A and B, and A or B, for independent or mutually exclusive events.
- Permutation and Combination Calculator — Number of permutations and combinations of r items from n.
- Expected Value Calculator — Expected value and variance of a discrete distribution.
- Bayes' Theorem Calculator — Posterior probability from a prior and test accuracy.
- Implied Probability Calculator — Probability implied by betting odds.
- Average Calculator — Arithmetic mean, sum, count, minimum and maximum of a list of numbers.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.