Question

Consider the first $10$ positive integers. If we multiply each number by $(-1)$ and then add $1$ to each number, the variance of the numbers so obtained is

• A. $8.25$
• B. $6.5$
• C. $3.87$
• D. $2.87$

Answer 1

Answer: (A)

First ten positive integers are $1,2,3,4,5,6,7,8,9$ and 10.
Sum of these numbers $\left(\Sigma x_j\right)=1+2+\dots +10=55$
Sum of squares of these numbers $\left(\Sigma x_i^2\right)=1^2+2^2+\dots +10^2$ $=385$
Standard deviation $(\sigma )=\sqrt{\frac{\sum x_i^2}{n}-{\left(\frac{1}{n}\Sigma x_i\right)}^2}$
$=\sqrt{\frac{385}{10}-(5.5{)}^2}=\sqrt{38.5-30.25}=\sqrt{8.25}$
$\therefore$ Variance $\left({\sigma }^2\right)=8.25$