Question

If the mean of the following distribution is 27, find the value of p.

Class Interval	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
Frequency	8	$p$	12	13	10

Answer 1

Hint: First 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 solution:
Given the mean for the given frequency distribution is Rs. 18.00.
The frequency distribution table for the given data is as follows:

Class	Frequency (${f_i}$)	Mid-value (${x_i}$)	${f_i}{x_i}$
0 – 10	8	5	40
10 – 20	$p$	15	$15p$
20 – 30	12	25	300
30 – 40	13	35	455
40 – 50	10	45	450
Total	$\sum {{f_i}} = 43 + p$		$\sum {{f_i}{x_i}} = 1245 + 15p$

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 27 = \dfrac{{1245 + 15p}}{{43 + p}}$
Cross-multiply the terms,
$ \Rightarrow 1161 + 27p = 1245 + 15p$
Move variable part on one side and constant part on another side,
$ \Rightarrow 27p - 15p = 1245 - 1161$
Subtract the like terms,
$ \Rightarrow 12p = 84$
Divide both sides by 12,
$\therefore p = 7$

Hence the missing frequency is 7.

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.