Question

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

• A. $2$
• B. $3$
• C. $9$
• D. $4$

Answer 1

The correct option is A

$2$

Explanation for the correct option:

Step 1: Write variance expression.

$\begin{array}{rcl}Var& =& \left(\frac{1}{n}\right)\sum _{i=1}^n{x_i}^2-{\left[\left(\frac{1}{n}\right)\sum _{i=1}^n{x}_i\right]}^2\\ & =& \frac{1}{9}\times 45-{\left(\frac{1}{9}\times 9\right)}^2\\ & =& 5-1\\ & =& 4\end{array}$

Here, $Var$ is the variance

Step 2: Calculate the standard deviation.

$\begin{array}{rcl}S.D.& =& \sqrt{4}\\ S.D.& =& 2\end{array}$

Here, $S.D.$ is the standard deviation

Hence, option (a) is correct.