First slide

Introduction to statistics

Question

If each observation of a raw data whose variance is $σ^{2}$ is multiplied by h, then the variance of the new set is

Moderate

A

$σ^{2}$
B

$h^{2} σ^{2}$
C

$h σ^{2}$
D

$h + σ^{2}$

Solution

$\begin{aligned} Let x_{1}, x_{2}, \dots, x_{n} be the raw data. Then, \\ σ^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2} \end{aligned}$
$If each value is multiplied by h, then the values become h x_{1},$
$h x_{2}, \dots, h x_{n} . Then AM of the values is$
$\frac{h x_{1} + h x_{2} + \dots + h x_{n}}{n} = h \bar{x}$
Therefore, the variance of the new set of values is
$\frac{1}{n} \sum_{i = 1}^{n} {(h x_{i} - h \bar{x})}^{2} = h^{2} (\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}) = h^{2} σ^{2}$

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