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. $2^3 ? 5$
• B. $2^2 ? 5$
• C. $2 ? 5$
• D. $2^4 ? 5$

Answer 1

Answer: (B)

$2^2 ? 5$

Answer 2

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 $\left(\sigma\right)^{2}=\frac{1}{n} \displaystyle \sum_{i=1}^{20}(x_i-\bar{x})$ i.e. $5=\frac{1}{20}\displaystyle \sum_{i=1}^{20}(x_i-\bar{x})^2$ or $\displaystyle \sum_{i=1}^{20}(x_i-\bar{x})^2=100\,...(i)$ 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, $\bar{y}=\frac{1}{n}\displaystyle \sum_{i=1}^{20}y_i=\frac{1}{20}\displaystyle \sum_{i=1}^{20}2x_i=2.\frac{1}{20}$ $\displaystyle \sum_{i=1}^{20}x_i$ 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 e (i), we get $\displaystyle \sum_{i=1}^{20}$ $\left(\frac{1}{2}y_{i}-\frac{1}{2}\bar{y}\right)^{^2}=100$ i.e. $\displaystyle \sum_{i=1}^{20}(y_i-\bar{y})^2=400$ Thus, the variance of new observations $=\frac{1}{20}\times400=20=2^{2}\times5$