Mean of the deviations from the mean

Question

I ve been struggling to understand the below problem.If I could get help with this problem, it would be greatly appreciated. Consider the data y, y+a, y+2a,….,y+na and the deviations of these observations from their mean. The mean of these deviations is,

A] {n(n+1)a}/(2n+1)
B] {n(n-1)a}/(2n+1)
C] (n+1)a/2n
D] None of these

So mean of the deviations of each observation from their mean? Im a little lost.

Answer 1

Here is a direct approach. The mean is

$$ \frac{1}{n+1} \sum_{k=0}^n y + ka = y + \frac{a}{n+1} \sum_{k=0}^n k = y + \frac{an(n+1)}{2(n+1)} = y + \frac{an}{2} $$

The mean of deviations is $$ \frac{1}{n+1} \sum_{k=0}^n |y + ka - y - an/2| = \frac{a}{n+1} \sum_{k=0}^n |k - n/2| $$ can you finish this?