In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.
Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, 1, 2, 4, 7)). So the median absolute deviation for this data is 1.
The median absolute deviation is a measure of statistical dispersion. Moreover, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the deviations of a small number of outliers are irrelevant.
Relation to standard deviation
For normally distributed data is taken to be
i.e., the reciprocal of the quantile function (also known as the inverse of the cumulative distribution function) for the standard normal distribution . The argument 3/4 is such that covers 50% (between 1/4 and 3/4) of the standard normal cumulative distribution function, i.e.
Therefore, we must have that
we have that , from which we obtain the scale factor .
Another way of establishing the relationship is noting that MAD equals the half-normal distribution median:
This form is used in, e.g., the probable error.
In the case of complex values (X+iY), the relation of MAD to the standard deviation is unchanged for normally distributed data.
MAD using geometric median
Analogously to how the median generalizes to the geometric median (gm) in multivariate data, MAD can be generalized to MADGM (median of distances to gm) in n dimensions. This is done by replacing the absolute differences in one dimension by euclidian distances of the data points to the geometric median in n dimensions. This gives the identical result as the univariate MAD in 1 dimension and generalizes to any number of dimensions. MADGM needs the geometric median to be found, which is done by an iterative process.
The population MAD
The population MAD is defined analogously to the sample MAD, but is based on the complete distribution rather than on a sample. For a symmetric distribution with zero mean, the population MAD is the 75th percentile of the distribution.
- Deviation (statistics)
- Interquartile range
- Probable error
- Robust measures of scale
- Relative mean absolute difference
- Average absolute deviation
- Least absolute deviations
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