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

$\text{ Let the observations be }{x}_1,x_2,\dots ,x_{20}\text{ and }\overline{x}\text{ be their mean. }$

Given that, variance = 5 and n =20. 

We know that

$\begin{array}{r}\text{ Variance }\left({\sigma }^2\right)=\frac{1}{n}\sum _{i=1}^{20} {\left(x_i-\overline{x}\right)}^2\text{ , }\\ \text{ Or }5=\frac{1}{20}\sum _{i=1}^{20} {\left(x_i-\overline{x}\right)}^2\end{array}$

$\text{ Or }\sum _{i=1}^{20} {\left(x_i-\overline{x}\right)}^2=100-------\left(1\right)$

$\text{ If each observation is multiplied by }2\text{ , and the new resulting observations are }{y}_i\text{ , then }$

$y_{\mathrm{i}}=2x_{\mathrm{i}}\text{ i.e., }{x}_{\mathrm{i}}=\frac{1}{2}{y}_{\mathrm{i}}$

$\text{ Therefore, }\overline{y}=\frac{1}{n}\sum _{i=1}^{20} y_i=\frac{1}{20}\sum _{i=1}^{20} 2x_i=2\cdot \frac{1}{20}\sum _{i=1}^{20} x_i$

$\text{ e.. }\overline{y}=2\overline{x}\text{ or }\overline{x}=\frac{1}{2}\overline{y}$

$\text{ Substituting the values of }{x}_i\text{ and }x\text{ in (1), we get }$

$\sum _{i=1}^{20} {\left(\frac{1}{2}{y}_i-\frac{1}{2}\overline{y}\right)}^2=100\text{ , i.e., }\sum _{i=1}^{20} {\left(y_i-\overline{y}\right)}^2=400$

$\text{ Thus, the variance of new observations }=\frac{1}{20}\times 400=20$

  ALTIER  :     $$\begin{gathered} Variance= V \left(a X +b\right) = a^2 V\left(X\right) \\ so V\left(2 X\right) =2^2 V\left(X\right) =4\left(5\right)=20 \end{gathered}$$