First slide

Introduction to statistics

Question

The mean of n items is $\bar{x}$ . If the first term is increased by 1, second by 2 and so on, then the new mean is

Moderate

A

$\bar{x} + n$
B

$\bar{x} + \frac{n}{2}$
C

$\bar{x} + \frac{n + 1}{2}$
D

None of these

Solution

Let $x_{1}, x_{2}, \dots x_{n}$ be the n items.
Given, $\bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n}$
$\begin{matrix} ∴ New mean = \frac{(x_{1} + 1) + (x_{2} + 2) + \dots + (x_{n} + n)}{n} \\ = \frac{(x_{1} + x_{2} + \dots + x_{n}) + (1 + 2 + \dots + n)}{n} \\ = \bar{x} + \frac{n (n + 1)}{2 n} [∵ from Eq. (i)] \\ = \bar{x} + \frac{(n + 1)}{2} \end{matrix}$

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