Standard Deviation

The standard deviation is a measure of the spread of the data around the mean. The standard deviation of a population, $x$ is symbolised as $\sigma_x$ whereas the standard deviation of a sample is represented by $s_x$ .

In the graph on the left, the standard deviation of both sets of data is the same. This is observed by the same spread. The graph on the right shows two sets of data with very different standard deviations. The taller graph represents a smaller standard deviation than the shorter graph.

The formula for the standard deviation, $s$ , is

$s=\sqrt{\frac{\sum{(x-\bar{x})^2}}{n-1}}$
where $n$ is the number of data values (sample size) and $\bar{x}$ is the sample mean.

This means that the standard deviation is an average of the squared deviations of each data value from the mean.

If a data set is small, a rule for estimating the standard deviation for small data sets is

standard deviation $\approx \frac{range}{4}$ or $s \approx \frac{x}{4}$

If the distribution is highly skewed or with outliers, the IQR is a preferred measurement of spread.

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