Standard deviation

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

What is standard deviation

Standard deviation, like variance, determines how much the values are spread out or deviated from the mean of the values. The standard deviation is equal to the square root of the variance, if you don't know what the variance is, read this article. A brief summary follows:

The variance, denoted by $\mbox{Var}$, gives us the average of the squared distances from the mean. If we have a set of values X = [x₁, …, x_N], where $\overline{x}$ is the average value, then we calculate the variance as follows:

$$ \mbox{Var}(X) = \frac1N \left((x_1-\overline{x})^2 + (x_2-\overline{x})^2 + … + (x_N-\overline{x})^2 \right) $$

We can also write the formula using sum:

$$ \mbox{Var}(X) = \frac1N\sum_{i=1}^N (x_i-\overline{x})^2 $$

We denote the standard deviation by the lowercase letter sigma $\sigma$ and since the deviation is equal to the square root of the variance, we calculate it as follows:

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

In place of the variance, we can substitute the formula for calculating the variance directly to obtain the formula:

$$ \sigma = \sqrt{\frac1N\sum_{i=1}^N (x_i-\overline{x})^2} $$

Sometimes we also refer to the variance itself as $\sigma^2$, because the variance is equal to the square of the standard deviation.

How to calculate the standard deviation in Excel

In both Czech and English Excel, the function smodch, or some variant of it, e.g. 'smodch.p', is used for this.