Standard deviation

The standard deviation is equal to the square root of the variance.

What the standard deviation is

The standard deviation, like the variance, tells us how spread out the values are around the mean. It is equal to the square root of the variance. If you don't know what the variance is, read this article. Here is a short summary:

The variance, which we write as Var, gives us the mean of the squares of the distances from the mean. Suppose we have a set of values X = [x₁, …, x_N], where $\bar{x}$ is the mean. We work out the variance like this:

$$\text{Var}(X)=\frac{1}{N}((x_1-\bar{x}{)}^2+(x_2-\bar{x}{)}^2+\dots +(x_N-\bar{x}{)}^2)$$

We can also write the formula using sigma notation:

$$\text{Var}(X)=\frac{1}{N}\sum _{i=1}^N(x_i-\bar{x}{)}^2$$

We write the standard deviation with the lower-case Greek letter sigma, σ, and since it is equal to the square root of the variance, we work it out like this:

$$\sigma =\sqrt{\text{Var}(X)}$$

Instead of the variance we can substitute the formula for the variance directly, which gives the formula:

$$\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^N(x_i-\bar{x}{)}^2}$$

Sometimes we also write the variance itself as σ², because the variance is equal to the square of the standard deviation.

How to work out the standard deviation in Excel

In Excel you can use the function STDEV.P, or one of its variants.