## 5. Standard deviation

Variance is a measure of variability or spread in a set of observations that uses the square of the deviation (difference from the mean value). Whilst variance is a measure of overall variability, it is difficult to relate it directly to the observations, as is possible for the range, because the calculations are based on squares of deviations.

The '**standard deviation**' is given simply by the square root of the variance, and has the same units the observations:

s = √s^{2} = √{Σ(x – x_{mean})^{2} / (n – 1)}

or using power notation:

s = {Σ(x – x_{mean})^{2} / (n – 1)}^{0.5}

The square root of a number has two values - a positive and a negative ('minus times minus equals plus'). So the typical way to write the standard deviation is in association with the mean as in:

Mean (± 1 SD) = 11.34 ± 1.06 mm

This indicates that the mean value for a series of observations of lengths (of something) is 11.34 mm, and that the standard deviation is 1.06 mm either side of the mean (ie 10.28 – 12.40 mm).

As with variance, it is only appropriate to use standard deviation with scale data, and there is an underlying assumption that data are distributed normally about the mean (the interval bounded by the standard deviation is symmetrical about the mean).