Consider a sample x1, ... , xn with n even. Let L and U denote the average of the smallest n/2 and the largest n/2 observations, respectively. Show that the mean absolute deviation from the median for this sample satisfies
∑ |xi - | / n = (U - L) / 2
Then show that if n is odd and the two averages are calculated after excluding the median from each half, replacing n on the left with n - 1 gives the correct result.