Question

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

Expenditure	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

Step 1: Form the required table
Let us assume a mean $a=2750$
width of class interval $\left(h\right)=500$
The mid point $\left(x_i\right)=$(upper limit $+$lower limit) $÷$$2$

Class Interval	$f_i$	$x_i$	$d_i=x_i-a$	$u_i=\frac{d_i}{h}$	$f_i{u}_i$
$1000-1500$	$24$	$1250$	$-1500$	$-3$	$-72$
$1500-2000$	$40$	$1750$	$-1000$	$-2$	$-80$
$2000-2500$	$33$	$2250$	$-500$	$-1$	$-33$
$2500-3000$	$28$	$2750$	$0$	$0$	$0$
$3000-3500$	$30$	$3250$	$500$	$1$	$30$
$3500-4000$	$22$	$3750$	$1000$	$2$	$44$
$4000-4500$	$16$	$4250$	$1500$	$3$	$48$
$4500-5000$	$7$	$4750$	$2000$	$4$	$28$
	$\sum _{}{f}_i=200$				$\sum _{}{f}_i{u}_i=-35$

Step 2: Find the modal expenditure of the families

Modal class $=1500-2000$ [Since $f_i$ is greatest at this class interval]
lower limit $\left(l\right)=1500,f_m=40,f_1=24\&f_2=33$
Mode $=l+\frac{f_m-f_1}{2f_m-f_1-f_2}\times h$, where $f_m$ is the frequency of modal class, $f_1$ and $f_2$ is the frequency of previous and successive class respectively.
$$\begin{gathered} =1500+\frac{40-24}{80-24-33}\times 500 \\ =1500+347.83 \\ =1847.83 \end{gathered}$$

Step 3: Find the mean expenditure of the families

Mean $\left(\overline{x}\right)=a+\frac{\sum _{}{f}_i{u}_i}{\sum _{}{f}_i}\times h$
$$\begin{gathered} =2750+\frac{-35}{200}\times 500 \\ =2750-87.5 \\ =2662.5 \end{gathered}$$

Hence the modal and mean expenditure of the families are $\mathrm{Rs}.1874.83$ and $\mathrm{Rs}.2662.5$ respectively