First slide

Introduction to statistics

Question

The mean of the series $x_{1}, x_{2}, \dots, x_{n} is \bar{x}$ . If x₂ is replaced by $λ$ , then the new mean is

Easy

A

$\bar{x} - x_{2} + λ$
B

$\frac{\bar{x} - x_{2} - λ}{n}$
C

$\frac{(n - 1) \bar{x} + λ}{n}$
D

$\frac{n \bar{x} - x_{2} + λ}{n}$

Solution

$\begin{array}{l} We have, \bar{x} = \frac{x_{1} + x_{2} + \dots + x_{n - 1} + x_{n}}{n} \\ \Rightarrow n \bar{x} = x_{1} + x_{2} \dots + x_{n - 1} + x_{n} \end{array}$
Let $\vec{y}$ be the new mean when x₂ is replaced by $λ$ Then,
$\begin{array}{l} \Rightarrow \bar{y} = \frac{(x_{1} + x_{2} + \dots + x_{n}) - x_{2} + λ}{n} \\ ∴ \bar{y} = \frac{n \bar{x} - x_{2} + λ}{n} \end{array}$

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