# Dice Average Calculator URL: https://speedycalculator.com/calculators/statistics/dice-average/ Last updated: 2026-10-30 > The dice average calculator gives the expected value of a single die, the average total of any number of dice plus a modifier, the standard deviation and variance, the possible range, the most likely totals and the full probability distribution of the sum. ## Inputs - Dice type - Number of sides (custom die) - Number of dice - Modifier added to the total — For example +3 for 2d6 + 3. ## Results - Average total - Average of one die - Standard deviation of the total - Variance of the total - Lowest possible total - Highest possible total - Most likely total(s) - Chance the total lands within one standard deviation of the average - Distribution of the total ## Formula ``` E(one die) = (s + 1)/2 · E(total) = n·(s + 1)/2 + modifier · Var(total) = n·(s² − 1)/12 · SD = √Var ``` ## Explanation 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 DieFacesAverage rollStandard deviationTypical used41–42.51.118Small weapons, healing potionsd61–63.51.708Board games, fireballs, ability scoresd81–84.52.291Longswords, cleric hit diced101–10 (or 0–9)5.52.872Percentile rolls, heavy weaponsd121–126.53.452Greataxes, barbarian hit diced201–2010.55.766Attack rolls, checks and savesd1001–10050.528.866Percentage tables, random events ## 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. ## FAQ **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. ## Sources - [Wolfram MathWorld – Dice](https://mathworld.wolfram.com/Dice.html) - [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/) - [OpenStax – Introductory Statistics 2e](https://openstax.org/details/books/introductory-statistics-2e)