Question

The mean of the numbers $a,b,8,5,10$ is $6$ and their variance is $6.8$. If $M$ is the mean deviation of the numbers about the mean, then $25M$ is equal to:

• A. 60
• B. 55
• C. 50
• D. 45

Answer 1

Answer: (A)

${\sigma }^2=\frac{\sum _{i=1}^5{\left(x_i-\bar{x}\right)}^2}{n}$
Mean $=6$
$\frac{a+b+8+5+10}{5}=6$
$a+b=7$
$b=7-a$
$6.8=\frac{(a-6{)}^2+(b-6{)}^2+(8-6{)}^2+(5-6{)}^2+(10-6{)}^2}{5}$
$34=(a-6{)}^2+(7-a-6{)}^2+4+1+18$
$a^2-7a+12=0\Rightarrow a=4$ or $a=3$
$a=4a=3$
$b=3b=4$
$M=\frac{\sum _{i=1}^5\left|x_i-x\right|}{n}$
$M=\frac{|a-6|+|b-6|+|8-6|+|5-6|+|10-6|}{5}$
when $a=3,b=4$ when $a=4,b=3$
$M=\frac{3+2+2+1+4}{5}M=\frac{2+3+2+1+7}{5}$
$M=\frac{12}{5}M=\frac{12}{5}$
$25M=25\times \frac{12}{5}=60$