Question

The daily expenditure of $100$ families are given below. Calculate $f_1$ and $f_2$ the mean daily expenditure is $Rs.188$.

Expenditure (in $Rs$)	$140-160$	$160-180$	$180-200$	$200-220$	$220-240$
Number of families	$5$	$25$	$f_1$	$f_2$	$5$

Answer 1

Expenditure in rupees	Number of families$\left(f_i\right)$	Class mark$\left(x_i\right)$	$f_i{x}_i$
$140-160$	$5$	$150$	$750$
$160-180$	$25$	$170$	$4250$
$180-200$	$f_1$	$190$	$190f_1$
$200-220$	$f_2$	$210$	$210f_2$
$220-240$	$5$	$230$	$1150$
Total	$\sum f_i=35+f_1+f_2$		$\sum f_i{x}_i=6150+190f_1+210f_2$

Sum of the frequencies$=100$

$\Rightarrow {\sum }_i{f}_i=100$

$\Rightarrow 35+f_1+f_2=100$

$\Rightarrow f_1+f_2=100-35=65$

$\Rightarrow f_2=65-f_1$ ........$\left(1\right)$

now, the mean of the given data is given by,

$\overline{x}=\frac{{\sum }_i{f}_i{x}_i}{{\sum }_i{f}_i}$

$\Rightarrow 188=\frac{6150+190f_1+210f_2}{100}$

$\Rightarrow 18800-6160=190f_1+210f_2$

$\Rightarrow 12650=190f_1+210(65-f_1)$ from $\left(1\right)$

$\Rightarrow 12650=190f_1-210f_1+13650$

$\Rightarrow 20f_1=1000$

$\Rightarrow f_1=50$

If $f_1=50$, then $f_2=65-50=15$

Thus, the value of $f_1$ is $50$ and $f_2$ is $15$.