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 (in C)	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

Mode: The class which has highest frequency.

In this case, class interval $1500-2000$ is the modal class.

Now,

Lower limit of modal class, $l=1500,h=500,f_1,=40,f_0=24,f_2=33$

We know that,

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

$=1500+\left(\frac{40-24}{2\left(40\right)-24-33}\right)\times 500$

$=1500+\left(\frac{16}{80-57}\right)500$$=1500+\left(\frac{16}{23}\right)\times 500$

$=1500+347.826$

Mode $=1847.83$

Lets take assumed mean, $A$ as $3250$.

Class interval $=500$

$N=200,\sum f_i{u}_i=-235$

$\overline{x}=A+h\frac{\sum f_i{u}_i}{N}$

$=3250+\frac{-235}{200}\times 500$

$=3250+\frac{-1175}{2}=2662.5$

Hence, the average expenditure is Rs $2662.50$.