Question

The mean of $5$ observations is $5$ and their variance is $12.4$. If three of the observations are $1,2$ and $6$; then the mean deviation from the mean of the data is:

• A. $2.8$
• B. $2.6$
• C. $2.4$
• D. $2.5$

Answer 1

The correct option is A $2.8$

$\sum _{i=1}^5{x}_i=5\times 5=25$
$\text{Given},{\sigma }^2=12.4$
$\Rightarrow \frac{\sum _{i=1}^5{x}_i^2}{5}-(5{)}^2=12.4$
$\Rightarrow \sum _{i=1}^5{x}_i^2=187$
Three observations are $1,2$ and $6$.
$\Rightarrow x_4+x_5=16$
$\text{and}{x}_4^2+x_5^2=146$
Solving the two equations, we get
$x_4=5,x_5=11$ or $x_4=11,x_5=5$
$\therefore$ Mean deviation from the mean $=\frac{|1-5|+|2-5|+|6-5|+|5-5|+|11-5|}{5}$
$=\frac{14}{5}=2.8$