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Expected Dice Average Calculator — result sheet
Calculate the expected value, variance, standard deviation, and possible range for a sum of fair identical dice.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For d independent fair s-sided dice, E[sum]=d(s+1)/2, Var(sum)=d(s²−1)/12, SD=√Var, minimum=d, and maximum=ds.
This page describes the distribution of the sum before any particular roll is made. A single fair die has a mean halfway between one and its number of sides; independent dice add their expected values and variances. The range is shown separately so a theoretical average is not mistaken for a guaranteed or most-common single outcome.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Sides per die — minimum 2; maximum 1000
- Number of dice — minimum 1; maximum 1000
Worked example
| Input | Value |
|---|---|
| Sides per die | 6 |
| Number of dice | 2 |
Two fair six-sided dice have expected sum 7, variance 35/6 ≈ 5.8333, standard deviation ≈ 2.4152, and range 2–12.
Assumptions and limits
- Every die has the same integer number of sides and each face is equally likely.
- Dice are independent, so variances add across dice.
- The result describes the sum of face values, not a custom scoring system or reroll rule.
- Expected value is a long-run average and need not be an attainable integer outcome.
- No advantage, disadvantage, exploding die, modifier, or critical rule is included.
- Physical bias, wear, rolling method, and table effects are not modeled.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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