First slide

Introduction to statistics

Question

If the mean of the set of numbers $x_{1}, x_{2}, x_{3}, \dots, x_{n} is \bar{x}$ , then the mean of the numbers $x_{i} + 2 i, 1 \leq i \leq n$ is

Moderate

A

$\bar{x} + 2 n$
B

$\bar{x} + n + 1$
C

$\bar{x} + 2$
D

$\bar{x} + n$

Solution

We know that $\bar{x} = \frac{\sum_{i = 1}^{n} x_{i}}{n}$
$\begin{array}{l} \Rightarrow \sum_{i = 1}^{n} x_{i} = n \bar{x} \\ ∴ \frac{\sum_{i = 1}^{n} (x_{i} + 2 i)}{n} = \frac{\sum_{i = 1}^{n} x_{i} + 2 \sum_{i = 1}^{n} i}{n} = \frac{n \bar{x} + 2 (1 + 2 + \dots n)}{n} \\ = \frac{n \bar{x} + 2 \frac{n (n + 1)}{2}}{n} = \bar{x} + (n + 1) \end{array}$

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