First slide

Introduction to statistics

Question

The mean deviation from the mean of the AP a, a + d , a+2d , ……..a+ nd 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
$a, a + d, a + 2 d, . . . . . . . . \dots, a + 2 nd$
$\begin{matrix} \bar{X} = \frac{1}{2 n + 1} [a + (a + d) + (a + 2 d) + \dots + (a + 2 nd)] \\ = \frac{1}{2 n + 1} \{\frac{2 n + 1}{2} (a + a + 2 nd)\} = a + nd \end{matrix}$
Therefore, mean deviation from mean
$\begin{matrix} = \frac{1}{2 n + 1} \sum_{r = 0}^{2 n} | (a + rd) - (a + nd) | = \frac{1}{2 n + 1} \sum_{r = 0}^{2 n} | r - n | d \\ = \frac{1}{2 n + 1} \cdot 2 d (1 + 2 + \dots + n) = \frac{n (n + 1)}{2 n + 1} d \end{matrix}$

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