Q. The mean deviation from mean of the observation $a,\ a + d,\ a + 2d,\ \ldots a + 2nd$ is
A) $\frac{n(n + 1)d^{2}}{3}$
B) $\frac{n(n + 1)}{2}d^{2}$
C) $a + \frac{n(n + 1)d^{2}}{2}$
D) None of these
Explanation
(d)
$\overline{x} = \frac{1}{2n + 1}\left\lbrack a + (a + d) + \ldots + (a + 2nd) \right\rbrack$
$= \frac{1}{2n + 1}\left\lbrack (2n + 1)a + d(1 + 2 + \ldots + 2n) \right\rbrack$
$= a + d\frac{2n}{2}.\frac{(2n + 1)}{2n + 1} = a + nd$
$\therefore MD\ from\ mean = \frac{1}{2n + 1}\sum_{}^{}\left| x_{1} - \overline{x} \right|$
$= \frac{1}{2n + 1}2|d|(1 + 2 + ... + n)$
$= \frac{n(n + 1)|d|}{(2n + 1)}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.