First slide

Introduction to statistics

Question

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

Moderate

A

$σ^{2}$
B

$h^{2} σ^{2}$
C

${hσ}^{2}$
D

$h + σ^{2}$

Solution

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

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