First slide

Introduction to statistics

Question

The mean deviation of the series $a, a + d, a + 2 d, . . . ., a + (2 n - 1) d, a + 2 n d$ about the mean is

Moderate

A

$\frac{n + 1}{2 n + 1}$
B

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

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

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

Solution

The mean of the given series is $a + n d .$
$∴$ Mean deviation about mean
$\begin{array}{l} = \frac{1}{(2 n + 1)} \sum_{r = 0}^{2 n} | (a + r d) - (a + n d) | \\ = \frac{d}{2 n + 1} \sum_{r = 0}^{2 n} | r - n | \\ = \frac{2 d}{2 n + 1} \sum_{r = 1}^{n} r = \frac{2 d}{2 n + 1} \times \frac{n (n + 1)}{2} = \frac{n (n + 1) d}{2 n + 1} \end{array}$

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