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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.

Types of deviation
Main article: Errors and residuals

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.
Unsigned or absolute deviation
"Absolute deviation" redirects here. It is not to be confused with Average absolute deviation.

In statistics, the absolute deviation of an element of a data set is the absolute difference between that element and a given point. Typically the deviation is reckoned from the central value, being construed as some type of average, most often the median or sometimes the mean of the data set.

$$D_i = |x_i-m(X)|$$

where

$$D_i$$ is the absolute deviation,
$$x_i$$ is the data element
and m(X) is the chosen measure of central tendency of the data set—sometimes the mean $$(\overline{x})$$, but most often the median.

Measures of deviation
Central tendency
Main article: Mean signed deviation

For an unbiased estimate, the sum of the signed deviations across the entire set of all observations from the overall sample mean is always zero, and the expected, mean, or average deviation is also zero; conversely, a nonzero average deviation quantifies the bias exhibited by the estimate.

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.

Mean absolute deviation, 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.

Normalization
Further information: Normalization (statistics)

Deviations have units of the measurement scale (for instance, meters if measuring lengths). One can nondimensionalize in two ways.

One way is 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 (e.g., Studentized residual).

One can scale instead 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%.