Gujarati
Hindi
13.Statistics
medium

The variance of $20$ observation is $5$ . If each observation is multiplied by $2$ , then the new variance of the resulting observations, is 

A

$5$

B

$10$

C

$20$

D

$40$

Solution

$\frac{1}{{20}}\sum\limits_{i = 1}^{20} {{{\left( {{x_i} – \bar x} \right)}^2} = 5} $

$\sum\limits_{i = 1}^{20} {{{\left( {{x_i} – \bar x} \right)}^2} = 100} $

new observations are $2 \mathrm{x}_{1}, 2 \mathrm{x}_{2}, \ldots \ldots, 2 \mathrm{x}_{20}$

Their mean $=\overline{\mathrm{x}}_{1}=\frac{2\left(\mathrm{x}_{1}+\mathrm{x}_{2}+\ldots+\mathrm{x}_{20}\right)}{20}=2 \overline{\mathrm{x}}$

Now, variance$ = \frac{1}{{20}}\sum\limits_{i = 1}^{20} {{{\left( {2{x_i} – 2\bar x} \right)}^2}} $

$ = \frac{1}{{20}} \times 4\sum\limits_{i = 1}^{20} {{{\left( {{x_i} – \bar x} \right)}^2}} $

$ = \frac{1}{{20}} \times 4 \times 100 = 20$

Standard 11
Mathematics

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