Deviation (statistics)

Not to be confused with Deviance (statistics).

In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation (positive or negative), reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference.

When deviations refer to the difference between a variable and its value implied by some function of other variables, or between a variable and its value implied by the estimated version of that function, they are also known as errors or residuals respectively, and are applicable for data at the interval and ratio levels of measurement. When the reference point is simply a mean, deviations from the population mean are errors while deviations from the sample mean are residuals.

The sum of the deviations across the entire set of all observations from the overall sample mean is always zero, and the average deviation is zero.

Measures of deviation

Dispersion

Statistics of the distribution of deviations are used as measures of statistical dispersion.

Standard deviation is the frequently used measure of dispersion: it uses squared deviations, and has desirable properties, but is not robust.

Average absolute deviation, sometimes called the "average deviation" is calculated using the absolute value of deviation – it is the sum of absolute values of the deviations divided by the number of observations.

Median absolute deviation is a robust statistic which uses the median, not the mean, of absolute deviations.

Maximum absolute deviation is a highly non-robust measure, which uses the maximum absolute deviation.

Standardizing and Studentizing

For more on Studentizing, see Studentization, Studentized residual, and Studentized range.

Deviations have units of the measurement scale (for instance, meters if measuring lengths); one can nondimensionalize them by dividing by a measure of scale (statistical dispersion), most often either the population standard deviation, in standardizing, or the sample standard deviation, in studentizing.

One can also scale by location, not dispersion: the formula for a percent deviation is the observed value minus accepted value divided by the accepted value multiplied by 100%.