Average Absolute Deviation from the Mean (Population):
\begin{equation*}
\frac{\sum_{k=1}^n \left | x_k-\mu \right |}{n}
\end{equation*}
which, although nicely stated, is difficult to deal with algebraically since the absolute values do not simplify well algebraically. Indeed, it is easy to see that, for example when n=3, the mean lies somewhere between \(y_1\) and \(y_3\) (using the ordered data) but could be on either side of \(y_2\) and so
\begin{equation*}
|y_1-\mu| + |y_2 - \mu| + |y_3 - \mu| = -(y_1-\mu) + ?(y_2 - \mu) + |y_3 - \mu|
\end{equation*}
where the ? is either a + or a - but it could be either in general. To avoid this algebraic roadblock, we can look for another way to nearly accomplish the same goal by squaring and then square rooting.