Question

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

Answer 1

Let $x_1,x_2,x_3.....x_{20}$ be the 20 observations

Variance (X) = 5

Variance (X) $=\frac{1}{20}\times \sum (x_i-\overline{X}{)}^2=5$

(Here, $\overline{X}$ is the mean of the given observations)

Let $u_1,u_2,u_3.....u_{20}$ be the new observations, such that

$u_1=2x_i(fori=1,2,3...20)$

Mean = $\overline{U}=\frac{{\sum }_{i=1}^{20}{u}_i}{n}=\frac{{\sum }_{i=1}^{20}2x_i}{20}$ [Substituting $u_i$ from eq . (i) and taking n as 20]

$=2\times \frac{{\sum }_{i=1}^{20}2x_i}{20}=2\overline{X}$

$u_i-\overline{U}=2x_i=2\overline{X}$ ( for i = 1, 2, .... 20 )

= $2\left(x_i\right)\overline{X})$

$u_i-\overline{U}=\left(2\right(x_i=\overline{X}){)}^2$ (squaring both sides)

= $4(x_i-\overline{X}{)}^2$

$\therefore {\sum }_{i=1}^{20}(u_i-\overline{U}{)}^2={\sum }_{i=1}^{20}4(x_i-\overline{X}{)}^2$

${\sum }_{i=1}^{20}\frac{(u_i-\overline{U}{)}^2}{20}=={\sum }_{i=1}^{20}\frac{4(x_i-\overline{X}{)}^2}{20}$

$={\sum }_{i=1}^{20}\frac{4(x_i-\overline{X}{)}^2}{20}$

Variance (U) = Variance (X)

= $4\times 5=20$

Thus variance of the new observations is 20.