First slide

Introduction to statistics

Question

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

Moderate

A

$X + n$
B

$X + \frac{n}{2}$
C

$X + \frac{n + 1}{2}$
D

None of these

Solution

$\begin{aligned} Let x_{1}, x_{2}, \dots, x_{n} be n items. Then \bar{X} = \frac{1}{n} Σ x_{i} . Let \\ y_{1} = x_{1} + 1, y_{2} = x_{2} + 2, y_{3} = x_{3} + 3, \dots, y_{n} = x_{n} + n \end{aligned}$
Then the mean of the new series is
$\begin{aligned} \frac{1}{n} \sum (x_{i} + i) = \frac{1}{n} \sum x_{i} + \frac{1}{n} (1 + 2 + 3 + \dots + n) \\ = \bar{X} + \frac{1}{n} \frac{n (n + 1)}{2} = \bar{X} + \frac{n + 1}{2} \end{aligned}$

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