Free · Probability and statistical models

Expected Total and Spread of Fair Dice

Find the mean and standard deviation of a sum of independent fair dice.

  • Formula & worked example
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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

Two fair six-face dice have expected total 7 and variance 35/6.

E(S) = n(s+1)/2; SD(S) = √(n(s²−1)/12)

Worked example

Enter these known values and leave the other values blank.

Dice type
Cube (6 faces)
Total number of dice
2
Die expected value
3.5
Dice roll average
7
Dice roll standard deviation
2.415

Assumptions and limitations

  • Each die is independent and its faces are the integers from 1 through the selected face count.
  • The mean is a long-run expectation; a single total need not equal it. Dice counts must be whole positive numbers.

Common questions

Which assumptions govern this model?

Each die is independent and its faces are the integers from 1 through the selected face count.

References

Replace captured external image with an original accessible description preserving its complete probability/count relationships and all variant conditions.

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