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The variance of $20$ observation is $5$ . If each observation is multiplied by $2$ , then the new variance of the resulting observations, is
$5$
$10$
$20$
$40$
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$
Similar Questions
The variance $\sigma^2$ of the data is $ . . . . . .$
$x_i$ | $0$ | $1$ | $5$ | $6$ | $10$ | $12$ | $17$ |
$f_i$ | $3$ | $2$ | $3$ | $2$ | $6$ | $3$ | $3$ |