Median absolute deviation
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
where is a constant scale factor, which depends on the distribution.
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.
Geometric median absolute deviation
Similarly to how the median generalizes to the geometric median in multivariate data, a geometric MAD can be constructed that generalizes the MAD. Given a 2 dimensional paired set of data (X1,Y1), (X2,Y2),..., (Xn,Yn) and a suitably calculated geometric median , the geometric median absolute deviation is given by:
This gives the identical result as the univariate MAD in 1 dimension and extends easily to higher dimensions. In the case of complex values (X+iY), the relation of MAD to the standard deviation is unchanged for normally distributed data.
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.
The earliest known mention of the concept of the MAD occurred in 1816, in a paper by Carl Friedrich Gauss on the determination of the accuracy of numerical observations.
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