Editing Average absolute deviation (section)

=== Mean absolute deviation around the median ===

The [[median]] is the point about which the mean deviation is minimized. The MAD median offers a direct measure of the scale of a random variable around its median
<math display="block">D_\text{med} = E |X-\text{median}| </math>

This is the [[maximum likelihood]] estimator of the scale parameter <math>b</math> of the [[Laplace distribution]]. 

Since the median minimizes the average absolute distance, we have <math>D_\text{med} \le D_\text{mean}</math>.
The mean absolute deviation from the median is less than or equal to the mean absolute deviation from the mean. In fact, the mean absolute deviation from the median is always less than or equal to the mean absolute deviation from any other fixed number.

By using the general dispersion function, Habib (2011) defined MAD about median as
<math display="block">D_\text{med} = E |X-\text{median}| = 2\operatorname{Cov}(X,I_O) </math>
where the indicator function is
<math display="block">\mathbf{I}_O := \begin{cases}
1 &\text{if } x > \text{median}, \\
0 &\text{otherwise}.
\end{cases}
</math>

This representation allows for obtaining MAD median correlation coefficients.{{citation needed|date=November 2019}}