# Calculate the mean deviation about the mean of the set of first n natural numbers where numbers where n is an even number.

$\begin{array}{1 1}(A)\;\large\frac{n}{4}\\(B)\;(n+1)\\(C)\;0\\(D)\;\large\frac{n-1}{12}\end{array}$

Mean of first n natural numbers where n is an even number is $= \large\frac{1}{2} \bigg[ \frac{n}{2} +\frac{n+1}{2} \bigg]$
$\qquad= \large\frac{2n+1}{4}$
mean deviation $=\large\frac{\sum |x_i -x_n|}{n}$
$\qquad= \large\frac{1}{4n} $$[1+3+5+9+.......+nth\; odd number] now sum of odd numbers =n^2 \qquad=\large\frac{1}{4n}$$n^2$
$\qquad= \large\frac{n}{4}$
Hence A is the correct answer.
how can the mean of first n even natural numbers be (2n+1)/4.
eg. 1, 2, 3, 4, 5, 6 are first 6(even) natural numbers.
mean= (1+2+3+4+5+6)/6 =  21/6 = 3.5  -----------  (1)

now (2*6 + 1)/4 = 13/4 = 3.25   ----------------------- (2)

clearly 1&2 are not equal