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.
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 MAD, the deviations of a small number of outliers are irrelevant.
Because the MAD is a more robust estimator of scale than the sample variance or standard deviation, it works better with distributions without a mean or variance, such as the Cauchy distribution.
is a constant scale factor, which depends on the distribution.
covers 50% (between 1/4 and 3/4) of the standard normal cumulative distribution function, i.e.
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.
MADGM needs the geometric median to be found, which is done by an iterative process.
Unlike the variance, which may be infinite or undefined, the population MAD is always a finite number.
For example, the standard Cauchy distribution has undefined variance, but its MAD is 1.
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.