First slide

Introduction to statistics

Question

The mean deviation from the mean of the series $a, a + d, a + 2 d, \dots, a + 2 n d$ is

Moderate

A

$n (n + 1) d$
B

$\frac{n (n + 1) d}{2 n + 1}$
C

$\frac{n (n + 1) d}{2 n}$
D

$\frac{n (n - 1) d}{2 n + 1}$

Solution

The mean of the series $a, a + d, a + 2 d, \dots, a + 2$ nd is
given by
$\begin{aligned} \bar{X} = \frac{1}{2 n + 1} {a + a + d + a + 2 d + \dots + a + 2 n d} \\ \Rightarrow \bar{X} = \frac{1}{2 n + 1} \{\frac{2 n + 1}{2} (a + a + 2 n d)\} = a + n d \end{aligned}$
The mean deviation from mean is given by
$\begin{array}{l} M.D. = \frac{1}{2 n + 1} \sum_{r = 0}^{2 n} | (a + r d) - (a + n d) | \\ \Rightarrow M.D. = \frac{1}{2 n + 1} \sum_{r = 0}^{2 n} | r - n | d \\ \Rightarrow M.D. = \frac{1}{2 n + 1} {2 d (1 + 2 + \dots + n)} = \frac{n (n + 1)}{2 n + 1} d \end{array}$

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