standard deviation

Every set of measurements has a middle and a spread. The standard deviation is the spread. The NIST handbook of statistical methods defines it as the square root of the variance, and the variance as the average of the squared distances between each value and the mean. Taking that square root matters for a practical reason the handbook states plainly: it restores the units of the spread to the units of the original data, where the variance leaves them squared. So a standard deviation of four points on a memory test is measured in points, and can be read against the scores themselves.

That property is what lets researchers report a result in standard deviations rather than in the units of any one test. When a group's score is said to have moved by a third of a standard deviation, the movement is being expressed relative to how widely the scores in that group are spread. It is a way of putting findings from different tests, and different studies, on a single scale. The cost is that the reader loses the original units: a change of a third of a standard deviation says nothing about how many questions anyone answered correctly.

The measure has limits. The handbook notes that the standard deviation is sensitive to values far out in the tails of a distribution, because squaring the distances weights the distant ones heavily, so a few unusual results can widen it noticeably. It also assumes the spread is worth summarising with one number, which holds less well the further a distribution departs from a symmetrical bell shape.