Question

A data consists of n observations:
$x_{1} , x_{2}, ......., x_{n}.$ If $ \displaystyle\sum^{n}_{i=1} \left(x_{i} +1\right)^{2} =9n $ and $\displaystyle\sum^{n}_{i=1} \left(x_{i} - 1\right)^{2} =5n $ , then the standard deviation of this data is :

• A. 5
• B. $\sqrt{5}$
• C. $\sqrt{7}$
• D. 2

Answer 1

Answer: (B)

$\sum\left(x_{i} +1\right)^{2} =9n $ .....(1)
$ \sum\left(x_{i} -1\right)^{2} =5n$ ......(2)
$ \left(1\right) +\left(2\right) \Rightarrow \sum\left(x_{1}^{2} +1\right)=7n $
$\Rightarrow \frac{\sum x_{i}^{2}}{n} =6 $
$ \left(1\right)-\left(2\right) \Rightarrow 4\sum x_{i} = 4n$
$ \Rightarrow \sum x_{i} =n $
$\Rightarrow \frac{\sum x_{i}}{n} = 1$
$ \Rightarrow $ variance $ = 6 - 1 = 5$
$ \Rightarrow $ Standard diviation $ = \sqrt{5} $