Question

If $\sum _{i=1}^9(x_i-5)=9$ and $\sum _{i=1}^9(x_i-5{)}^2=45,$ then the standard deviation of the $9$ items $x_1,x_2,\cdots ,x_9$ is

• A. $3$
• B. $9$
• C. $4$
• D. $2$

Answer 1

The correct option is D $2$

if you add or subtract the same amount from every term in the set, then the standard deviation doesn't change.

Let $x_i-5=y_i$
Then, $\sigma =\sqrt{\frac{\sum y_i^2}{9}-{\left(\frac{\sum y_i}{9}\right)}^2}$
$\Rightarrow \sigma =\sqrt{\frac{45}{9}-{\left(\frac{9}{9}\right)}^2}$
$\therefore \sigma =2$