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.The MAD may be used similarly to how one would use the deviation for the average.In order to use the MAD as a consistent estimator for the estimation of the standard deviation σ {displaystyle sigma }  , one takesSimilarly 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 ( X ~ , Y ~ ) {displaystyle ({ ilde {X}},{ ilde {Y}})}  , the geometric median absolute deviation is given by: MAD = ( median ⁡ ( | X i − X ~ | ) 2 + median ⁡ ( | Y i − Y ~ | ) 2 ) 1 / 2 {displaystyle operatorname {MAD} ={Bigl (}operatorname {median} (|X_{i}-{ ilde {X}}|)^{2}+operatorname {median} (|Y_{i}-{ ilde {Y}}|)^{2}{Bigr )}^{1/2}}  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.