# Deviation (statistics)

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Deviation (statistics)

In mathematics and statistics, deviation is a measure of difference for interval and ratio variables between the observed value and the mean. The sign of deviation (positive or negative), reports the direction of that difference (it is larger when the sign is positive, and smaller if it is negative). The magnitude of the value indicates the size of the difference.

Deviations are known as errors or residuals: 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 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.

### Dimensional analysis

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 accepted value minus observed value divided by the observed value multiplied by 100.