Consider the expression for one half of the mean absolute difference divided by the mean $\mu$, i.e., $\frac{1}{2\mu{n^2}}\Sigma^n_{i=1} \Sigma^n_{j=1} |y_i-y_j|$ (the Gini coefficient). How does it change if we add one more entry $y_{n+1}$ to the array $y_1$, $y_2$, ..., $y_n$ that we are considering? Can we say when it is going to increase, decrease or stay constant, depending on $y_{n+1}$? Or, in other words, can you solve the following for $y_{n+1}$?
$\frac{1}{2\mu{n^2}}$ $\Sigma^n_{i=1} \Sigma^n_{j=1} |y_i-y_j| = \frac{1}{2\mu^\prime{(n+1)^2}}$ $\Sigma^{n+1}_{i=1} \Sigma^{n+1}_{j=1} |y_i-y_j| $,
where $\mu=\frac{1}{n}\Sigma^n_{i=1}y_i$ and $\mu^\prime=\frac{1}{n+1}\Sigma^{n+1}_{i=1}y_i$.
Thanks.