# Is there an easy way to calculate standard deviation for dice rolls?

Is there an easy way to calculate standard deviation for dice rolls?

So I’ve got all these polyhedral dice for role playing games. I can use classical statistics to calculate quite easily the assumed uniform distribution for a single die roll, and what the mean is.

Now if I roll the same die several times, and add the results, the probability for any particular number starts to form a bell shaped curve.

For example, if I roll a six-sided die three times, there would be a sort of bell shaped curve, with 10.5 being the mean, with 3 and 18 being the extremes.

It seems to me that I should be able to calculate the standard deviation for these sort of bell shaped curves without too much trouble, but our statistics teacher thinks the only way to do it is to roll the dice about a billion times, and go through these complicated equations for each individual roll. Isn’t there an easier way?

What a mess! I might as well make a table for each one and estimate it off of that!

## Answer

Yes, there is an easier way to calculate the standard deviation for dice rolls without having to perform numerous rolls or create complicated tables. The standard deviation can be calculated mathematically.

For a single die roll, the mean (µ) is the average of all possible outcomes. For an n-sided die, the mean is:

µ = (1 + 2 + 3 + … + n) / n

For a six-sided die, the mean is:

µ = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5

To calculate the variance, you need to find the average of the squared differences from the mean. For a single n-sided die, the variance (?²) is:

?² = [(1 – µ)² + (2 – µ)² + … + (n – µ)²] / n

For a six-sided die, the variance is:

?² = [(1 – 3.5)² + (2 – 3.5)² + (3 – 3.5)² + (4 – 3.5)² + (5 – 3.5)² + (6 – 3.5)²] / 6 ? 2.92

The standard deviation (?) is simply the square root of the variance:

? = ?(?²) ? 1.71 for a six-sided die.

Now, if you roll the same die k times and sum the results, the mean will be k times the mean of a single die:

µ_sum = k * µ

The variance of the sum will be k times the variance of a single die:

?²_sum = k * ?²

Finally, the standard deviation of the sum is the square root of the variance of the sum:

?_sum = ?(?²_sum)

In your example, you roll a six-sided die three times (k = 3). The standard deviation of the sum will be:

?_sum = ?(3 * 2.92) ? 2.96

So, the standard deviation for rolling a six-sided die three times and summing the results is approximately 2.96. This method can be applied to any polyhedral die and any number of rolls.