Question

The following data gives the distribution of total monthly household expenditure of 200 families in a village. Find the modal monthly expenditure of the families. Also, find the mean $1500 - 2000$ 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

In this question, we have been asked to find the mode and mean. Starting with mode, mark the interval with the highest frequency as ${f_1}$, the interval before that as ${f_0}$ and after the modal class as ${f_2}$. After recognising the modal class, simply put the values in the formula to find the answer.
Next, find mean. First step is to write the midpoint of the class interval as ${x_i}$ Then, multiply the frequency with the midpoint to find ${f_i}{x_i}$. Find the sum of ${f_i}{x_i}$ and ${f_i}$, and put in the formula to find the mean.

Formula used: 1) Mode = $l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h$, where $l =$ lower limit of modal class, ${f_1} =$ frequency of the modal class, ${f_0} =$ frequency of class before modal class, ${f_2} =$ frequency of class after modal class and $h =$ class interval.
2) Mean = $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$

Complete step-by-step solution:
We are given a distribution table of monthly expenditure of 200 families. We have been asked to find the mode and mean.
Let us first start with the mode.

Expenditure (in Rs)	Number of families
$1000 - 1500$	$24$=${f_0}$
$1500 - 2000$	$40$=${f_1}$
$2000 - 2500$	$33$=${f_2}$
$2500 - 3000$	$28$
$3000 - 3500$	$30$
$3500 - 4000$	$22$
$4000 - 4500$	$16$
$4500 - 5000$	$7$

Since $1500 - 2000$ has the highest frequency, it is our modal class. The class before modal class will be ${f_o}$ and the class after that will be ${f_2}$.
By looking at the table, $l = 1500$, $h = 500$.
Now, let us put the values in the formula $l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times h$.
$\Rightarrow$ Mode = $1500 + \dfrac{{40 - 24}}{{2 \times 40 - 24 - 33}} \times 500$
Simplifying the equation,
$\Rightarrow$ Mode = $1500 + \dfrac{{16}}{{80 - 57}} \times 500$
$\Rightarrow$ Mode = $1500 + \dfrac{{16}}{{23}} \times 500$
$\Rightarrow$ Mode = $1500 + 347.82 = 1847.82$
Therefore, modal monthly expenditure of families is $1847.82$Rs.
Now, let us move towards finding mean. First step is to write the midpoint of the class interval as ${x_i}$ Then, multiply the frequency with the midpoint to find ${f_i}{x_i}$. Find the sum of ${f_i}{x_i}$ and ${f_i}$ and put in the formula to find the mean.

Expenditure (in Rs)	Number of families$({f_i})$	Mid-point$({x_i})$	${f_i}{x_i}$
$1000 - 1500$	$24$	$1250$	$30,000$
$1500 - 2000$	$40$	$1750$	$70,000$
$2000 - 2500$	$33$	$2250$	$74,250$
$2500 - 3000$	$28$	$2750$	$77,000$
$3000 - 3500$	$30$	$3250$	$97,500$
$3500 - 4000$	$22$	$3750$	$82,500$
$4000 - 4500$	$16$	$4250$	$68,000$
$4500 - 5000$	$7$	$4750$	$33,250$
$\sum {{f_i} = 200}$		$\sum {{f_i}{x_i} = 5,32,500}$

Mean = $\dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}$
Putting the values,
$\Rightarrow \bar X = \dfrac{{532500}}{{200}} = 2662.5$

$\therefore$ The mean monthly expenditure is $2662.5$ Rs.

Note: We have to remember that, the mean is the mathematical average of a set of two or more numbers. The arithmetic mean and the geometric mean are two types of mean that can be calculated. Summing the numbers in a set and dividing by the total number gives you the arithmetic mean. The students can use any other formula to find the mean – assumed mean method or shortcut method as per their choice. The formula used here is a direct and shorter method but it involves a lot of calculation.