Question

The following data gives the distribution of total monthly household expenditures of $200$ families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure.

$Expenditure$ (In Rs)	Number of families
$1000-1500$	$24$
$1500-2000$	$40$
$2000-2500$	$33$
$2500-3000$	$28$
$3000-3500$	$30$
$3500-4000$	$22$
$4000-4500$	$16$
$4500-5000$	$7$

Answer 1

Here, the maximum class frequency is $40$ and the class corresponding to it is $1500-2000$.

So, the modal class $=1500-2000$.

Thus, lower limit of the modal class $l=1500$

Frequency of the modal class, $f_1=40$

Frequency of the class preceding the modal class, $f_o=24$

Frequency of the class succeeding the modal class,$f_2=33$

and, class size $h=2000-1500=500$

Now, substituting these values in the formula of mode, we get

$Mode=l+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

$Mode=1500+\frac{40-24}{2\times 40-24-33}\times 500$

$Mode=1500+\frac{16}{80-57}\times 500$

$Mode=1500+\frac{8000}{23}=1500+347.83$

$Mode=1847.83$

Hence, the modal monthly expenditure of the families is $Rs.1847.83$.

Finding Mean:

$Expenditure$ $\left(inRs\right)$	$Numberoffamilies$ $\left(f_i\right)$	$Classmark$ $\left(x_i\right)$	$d_i=x_i-3250$	$u_i=\frac{d_i}{500}$	$f_i{u}_i$
$1000-1500$	$24$	$1250$	$-2000$	$-4$	$-96$
$1500-2000$	$40$	$1750$	$-1500$	$-3$	$-120$
$2000-2500$	$33$	$2250$	$-1000$	$-2$	$-66$
$2500-3000$	$28$	$2750$	$-500$	$-1$	$-28$
$3000-3500$	$30$	$3250$	$0$	$0$	$0$
$3500-4000$	$22$	$3750$	$500$	$1$	$22$
$4000-4500$	$16$	$4250$	$1000$	$2$	$32$
$4500-5000$	$7$	$4750$	$1500$	$3$	$21$
$Total$	$\sum f_i=200$				$\sum f_i{u}_i=-235$

Here, we have $\sum f_i=200,\sum f_i{u}_i=-235,h=500$ and Assumed mean $A=3250$.

Now, $\overline{x}=A+\left(\frac{\sum f_i{u}_1}{\sum f_i}\right)\times h$

$\Rightarrow$ $\overline{x}=3250+\left(\frac{-235}{200}\right)\times 500$

$\Rightarrow$ $\overline{x}=3250-\frac{235\times 5}{2}=3250-\frac{1175}{2}$

$\Rightarrow$ $\overline{x}=3250-587.50=2662.50$

Hence, the mean monthly expenditure is $Rs.2662.50$