Question

The following distribution shows the daily pocket allowance given the children of a multistorey building. The average pocket allowance is Rs. 18.00. Find out the missing frequency.

Class Interval	11 – 13	13 – 15	15 – 17	17 – 19	19 – 21	21 – 23	23 – 25
Frequency	7	6	9	13	?	5	4

Answer 1

Hint: First assume the missing frequency of class 19 – 21. Then take the mid values of each class as ${x_i}$ and frequency ${f_i}$. The mean value is equivalent to the fraction between the addition of a product of mid-value with frequency and the total frequency. Substitute the values in the mean formula and simplify to find the missing frequency

Complete step-by-step answer:
Given the mean for the given frequency distribution is Rs. 18.00.
Let the missing frequency be $x$.
The frequency distribution table for the given data is as follows:

Class	Frequency (${f_i}$)	Mid-value (${x_i}$)	${f_i}{x_i}$
11 – 13	7	12	84
13 – 15	6	14	84
15 – 17	9	16	144
17 – 19	13	18	234
19 – 21	$x$	20	$20x$
21 – 23	5	22	110
23 – 25	4	24	96
Total	$\sum {{f_i}} = 44 + x$		$\sum {{f_i}{x_i}} = 752 + 20x$

We know that the general formula to find the mean value is,
Mean $ = \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{x_i}} }}$
Now, we will substitute the value for the sum of the product of frequency and midpoint and the value for the sum of total frequency.
$ \Rightarrow 18 = \dfrac{{752 + 20x}}{{44 + x}}$
Cross-multiply the terms,
$ \Rightarrow 792 + 18x = 752 + 20x$
Move variable part on one side and constant part on another side,
$ \Rightarrow 20x - 18x = 792 - 752$
Subtract the like terms,
$ \Rightarrow 2x = 40$
Divide both sides by 2,
$\therefore x = 20$

Hence the missing frequency is 20.

Note: In such types of problems, the class will not be taken only mid-point should be taken because the interval cannot be multiplied to the frequency. If we don’t remember the formula, we can multiply each midpoint with frequency and add all of them then divide it with the sum of frequency.
In the mean formula, while computing $\sum {fx} $, don’t take the sum of $f$ and $x$ separately and then multiply them. It will be difficult. Students should carefully make the frequency distribution table; there are high chances of making mistakes while copying and computing data.