Signed deviations

A deviation that is a difference between an observed value and the true value of a quantity of interest (such as the population mean) is an error.

A deviation that is the difference between the observed value and an estimate of the true value (e.g. the sample mean) is a residual. These concepts are applicable for data at the interval and ratio levels of measurement.

Unsigned or 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:

$${\displaystyle D_{i}=|x_{i}-m(X)|,}$$

where

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

Mean signed deviation

For an unbiased estimator, the average of the signed deviations across the entire set of all observations from the unobserved population parameter value averages zero over an arbitrarily large number of samples. However, by construction the average of signed deviations of values from the sample mean value is always zero, though the average signed deviation from another measure of central tendency, such as the sample median, need not be zero.

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.^[1]
• Average 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.^[2]
• Maximum absolute deviation is a highly non-robust measure, which uses the maximum absolute deviation.