Mathematics - Statistics Question with Solution | TestHub

Given,

The mean of the below data is $5$

$x$	$1$	$3$	$5$	$7$	$9$
Frequency $\left(f\right)$	$4$	$24$	$28$	$\alpha$	$8$

Now finding mean we get,

$\frac{4+72+28\times 5+7\alpha +72}{64+\alpha }=5$

$\Rightarrow \alpha =16$

So, total number of observation will be, $\sum f_i=80$

Now finding, mean deviation we get,

$\mathrm{M}.\mathrm{D}=\frac{\sum f_i\left|x_i-5\right|}{\sum f_i}$

$=\frac{4\times 4+24\times 2+0+16\times 2+8\times 4}{80}$$=\frac{8}{5}$

So, $m=\frac{8}{5}$

Now finding, variance we get, 

${\sigma }^2=\frac{\sum f_i{\left|x_i-5\right|}^2}{\sum f_i}$

$\Rightarrow {\sigma }^2=\frac{4\times 16+24\times 4+0+16\times 4+8\times 16}{80}$

$\Rightarrow {\sigma }^2=\frac{22}{5}$

Hence, the value of $\frac{3\alpha }{m+{\sigma }^2}=\frac{3\times 16}{\frac{8}{5}+\frac{22}{5}}=8$