Question

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

• A. $2^3\times 5$
• B. $2^2\times 5$
• C. $2\times 5$
• D. $2^4\times 5$

Answer 1

Answer: (B)

Let the observations be $x_1,x_2,....,x_{20}$ and $\bar{x}$ be their mean. Given that, variance = 5 and n = 20. We know that,
Variance ${\left(\sigma \right)}^2=\frac{1}{n}\sum _{i=1}^{20}(x_i-\bar{x})$
i.e. $5=\frac{1}{20}\sum _{i=1}^{20}(x_i-\bar{x}{)}^2$ or $\sum _{i=1}^{20}(x_i-\bar{x}{)}^2=100...(i)$
If each observation is multiplied by 2 and the new resulting observations are $y_i$, then
$y_i=2x_i$ i.e., $x_i=\frac{1}{2}{y}_i$
Therefore, $\bar{y}=\frac{1}{n}\sum _{i=1}^{20}{y}_i=\frac{1}{20}\sum _{i=1}^{20}2x_i=2.\frac{1}{20}$$\sum _{i=1}^{20}{x}_i$
i.e., $\bar{y}=2\bar{x}$ or $\bar{x}\frac{1}{2}\bar{y}$
On substituting the values of $x_i$ and $\bar{x}$ in eq. (i), we get
$\sum _{i=1}^{20}$${\left(\frac{1}{2}{y}_i-\frac{1}{2}\bar{y}\right)}^{{}^2}=100$ i.e. $\sum _{i=1}^{20}(y_i-\bar{y}{)}^2=400$
Thus, the variance of new observations
$=\frac{1}{20}\times 400=20=2^2\times 5$