First slide

Introduction to statistics

Question

If the first item 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 n values of variable X. Then
$\bar{X} = \frac{1}{n} Σ x_{i}$
Let $y_{1} = x_{1} + 1, y_{2} = x_{2} + 2, y_{3} = x_{3} + 3, \dots, y_{n} = x_{n} + n$ Then the mean of the new series is given by
$\begin{array}{l} \bar{X^{'}} = \frac{1}{n} \sum y_{i} \\ \Rightarrow \bar{X^{'}} = \frac{1}{n} \sum_{i} (x_{i} + i) \\ \Rightarrow \bar{X^{'}} = \frac{1}{n} \sum_{i} x_{i} + \frac{1}{n} (1 + 2 + 3 + \dots + n) \\ \Rightarrow \bar{X^{'}} = \bar{X} + \frac{1}{n} \cdot \frac{n (n + 1)}{2} = \bar{X} + \frac{n + 1}{2} \end{array}$

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