Question

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

Answer 1

Let observations are $x_1,x_2,x_3\dots \dots x_{20}$
$\frac{1}{20}\sum _{i=1}^{20}{(x_i-\overline{x})}^2=5$
$\Rightarrow \sum _{i=1}^{20}{(x_i-\overline{x})}^2=100\dots \left(1\right)$
New, observations are, $2x_1,2x_2,2x_3,.........,2x_{20}$
Their mean, ${\overline{x}}_{\text{new}}=\frac{2(x_1+x_2+.........x_{20})}{20}=2\overline{x}$
Now, variance $=\frac{1}{20}\sum _{i=1}^{20}{(2x_i-2\overline{x})}^2$
$=\frac{1}{20}\times 4\sum _{i=1}^{20}{(x_i-\overline{x})}^2$
from equation $\left(1\right)$
$=\frac{1}{20}\times 4\times 100=20$


Alternate Solution
If variance of $n$ obsevations is ${\sigma }^2$
Now each observation is multiplied by $k,k\in R$,
then variance of new observations $:{\sigma }_{\text{new}}^2=k^2{\sigma }^2$
Here, $k=2,{\sigma }^2=5$
Now, ${\sigma }_{\text{new}}^2=2^2\times 5=20$