# Dice Average Calculator

Free dice average calculator. Find the expected average, variance, and standard deviation for any number of polyhedral dice (d4, d6, d8, d10, d12, d20).

## What this calculates

Calculate the expected average, variance, and standard deviation for rolling any number of dice. Works with all standard polyhedral dice from d4 to d20.

## Inputs

- **Number of Dice** — min 1, max 100 — How many dice to roll (1-100).
- **Sides per Die** — options: 4 (d4), 6 (d6), 8 (d8), 10 (d10), 12 (d12), 20 (d20) — Number of sides on each die.

## Outputs

- **Expected Average** — The expected value (mean) of the total dice roll.
- **Average per Die** — The expected value for a single die.
- **Variance** — The variance of the total dice roll.
- **Standard Deviation** — The standard deviation of the total dice roll.
- **Minimum Possible** — The lowest possible total (all ones).
- **Maximum Possible** — The highest possible total (all max).
- **Formula** — formatted as text — Step-by-step calculation breakdown.

## Details

When you roll dice, the expected average tells you the long-run mean outcome. Knowing the variance and standard deviation tells you how spread out the results will be.

**Expected Value of One Die:**

E(X) = (s + 1) / 2

where s is the number of sides. A d6 averages 3.5, a d20 averages 10.5.

**For Multiple Dice:**

- Expected total = n x (s + 1) / 2
- Variance of one die = (s^2 - 1) / 12
- Total variance = n x (s^2 - 1) / 12
- Standard deviation = square root of total variance

**Quick Reference:**

| Die | Average | Variance |
|-----|---------|----------|
| d4  | 2.5     | 1.25     |
| d6  | 3.5     | 2.917    |
| d8  | 4.5     | 5.25     |
| d10 | 5.5     | 8.25     |
| d12 | 6.5     | 11.917   |
| d20 | 10.5    | 33.25    |

**Example:** Rolling 3d6 (three six-sided dice) gives an expected average of 10.5 with a standard deviation of about 2.96. That means most rolls land between roughly 8 and 13.

## Frequently Asked Questions

**Q: Why is the average of a d6 equal to 3.5 and not 3?**

A: The average is the sum of all faces divided by the number of faces: (1+2+3+4+5+6) / 6 = 21 / 6 = 3.5. Even though you can never roll a 3.5, it represents the long-run average. Roll a d6 thousands of times and your average will converge on 3.5.

**Q: How does adding more dice affect the spread of results?**

A: Each additional die adds more variance, but the standard deviation grows slower than the mean. With 1d6 the SD is about 1.71 and the mean is 3.5. With 10d6 the SD is about 5.40 and the mean is 35. The relative spread (coefficient of variation) shrinks as you add dice, making extreme results less likely compared to the average.

**Q: Can I use this for mixed dice (e.g., 1d6 + 1d8)?**

A: This calculator handles identical dice. For mixed dice, calculate each type separately and add the results. The expected value of 1d6 + 1d8 is 3.5 + 4.5 = 8.0. The variance is 2.917 + 5.25 = 8.167 (variances add for independent rolls).

**Q: What is the relationship between dice averages and game balance?**

A: Game designers use expected values and variance to balance encounters. A weapon doing 2d6 damage averages 7 with low variance (tight clustering around 7), while 1d12 also averages 6.5 but with higher variance (more extreme highs and lows). The choice between them affects gameplay feel and reliability.

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Source: https://vastcalc.com/calculators/statistics/dice-average
Category: Statistics
Last updated: 2026-04-08