Question

A data consists of $n$ observation: $x_1,x_2,........,x_n.$ If $\sum _{i=1}^n(x_i+1{)}^2=9n$ and $\sum _{i=1}^n(x_i-1{)}^2=5n$, then the standard deviation of this data is:

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

Answer 1

Answer: (B)

$\sqrt{5}$

$\sum (x_i+1{)}^2=9n$.......$\left(1\right)$
$\sum (x_i-1{)}^2=5n$......$\left(2\right))$
$\left(1\right)+\left(2\right)\Rightarrow \sum (x_1^2+1)=7n$
$\Rightarrow \frac{\sum x_i^2}{n}=6$
$\left(1\right)-\left(2\right)\Rightarrow 4\sum x_i=4n$
$\Rightarrow \frac{\sum x_i}{n}=1$
$\Rightarrow$ variance $=6-1=5$
$\Rightarrow$ Standard deviation $=\sqrt{5}$