Question

The variance of 20 observations 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 given observations.

$\mathrm{Variance}\left(X\right)=5$

$\mathrm{Variance}\left(X\right)=\frac{1}{20}\times \sum {\left(x_i-\overline{X}\right)}^2=5(\mathrm{Here},\overline{X}\mathrm{is}\mathrm{the}\mathrm{mean}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{observations}.)$


Let u_1,u_2,,u₃, ..., u₂₀ be the new observations, such that
$u_{\mathrm{i}}=2x_{\mathrm{i}}(\mathrm{for}i=1,2,3,...,20)...\left(1\right)$

$$\begin{gathered} \mathrm{Mean}=\overline{\mathrm{U}}=\frac{{\displaystyle \sum _{i=1}^{20}}{u}_i}{n} \\ =\frac{{\displaystyle \sum _{i=1}^{20}}2x_i}{20}\left[\mathrm{substituting}{u}_{\mathrm{i}}\mathrm{from}\mathrm{eq}\hspace{0.17em}\left(1\right)\mathrm{and}\mathrm{taking}n\mathrm{as}20\right] \\ =2\times \frac{{\displaystyle \underset{i=1}{\overset{20}{\sum x_i}}}}{20} \\ =2\overline{X} \end{gathered}$$


$$\begin{gathered} u_i-\overline{U}=2x_i-2\overline{X}(\mathrm{for}i=1,2,...,20) \\ =2\left(x_i-\overline{X}\right) \\ {\left(u_i-\overline{U}\right)}^2={\left(2\left(x_i-\overline{X}\right)\right)}^2\left(\mathrm{squaring}\mathrm{both}\mathrm{the}\mathrm{sides}\right) \\ =4{\left(x_i-\overline{X}\right)}^2 \\ \therefore \sum _{i=1}^{20}{\left(u_i-\overline{U}\right)}^2=\underset{i=1}{\overset{20}{\sum 4}}{\left(x_i-\overline{X}\right)}^2 \\ \frac{{\displaystyle \sum _{i=1}^{20}}{\left(u_i-\overline{U}\right)}^2}{20}=\frac{\underset{i=1}{\overset{20}{\sum 4}}{\left(x_i-\overline{X}\right)}^2}{20} \\ =4\frac{{\displaystyle \sum _{i=1}^{20}}{\left(x_i-\overline{X}\right)}^2}{20} \\ \mathrm{Variance}\left(U\right)=4\times \mathrm{Variance}\left(X\right) \\ =4\times 5 \\ =20 \end{gathered}$$


Thus, variance of the new observations is 20.