Question

If $\sum _{i=1}^n(x_i-a)=nand\sum _{i=1}^n(x_i-a{)}^2=na,(n,a>1)$ then standard deviation of n observations $x_1,x_2,x_3,.......,x_n$is:

• A. $n\sqrt{(a-1)}$
• B. $\sqrt{(na-1)}$
• C. $(a-1)$
• D. $\sqrt{(a-1)}$

Answer 1

The correct option is D

$\sqrt{(a-1)}$

Simplify the given equations to determine the standard deviation.

Given : $\sum _{i=1}^n(x_i-a)=nand\sum _{i=1}^n(x_i-a{)}^2=na,(n,a>1)$

The standard deviation of n observations $x_1,x_2,x_3,.......,x_n$is

$$\begin{gathered} =\sqrt{\frac{\sum _{i=1}^n{(x_i-a)}^2}{n}-{\left(\frac{\sum _{i=1}^n(x_i-a)}{n}\right)}^2} \\ =\sqrt{\frac{na}{n}-{\left(\frac{n}{n}\right)}^2} \\ =\sqrt{(a-1)} \end{gathered}$$

Therefore, Option $\left(D\right)$ is the correct answer.