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 $25\, M$ is equal to:

• A. 60
• B. 55
• C. 50
• D. 45

Answer 1

Answer: (A)

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