Question

The mean of the following frequency distribution is 16, find the missing frequency.

Class:	0-4	4-8	8-12	12-16	16-20	20-24	24-28	28-32	32-36
Frequency:	6	8	17	23	16	15		4	3

Answer 1

We use the given information to calculate the mean by substituting the values from the data in the formulas. Assume missing frequency as a variable and substitute all values obtained in the formula of mean.

• Mean of any data is given by dividing the sum of the observations by the number of observations. If we have a grouped data then mean is given by
  $\overline x = \dfrac{{\sum\nolimits_{i = 1}^n {{f_i}{x_i}} }}{{\sum\nolimits_{i = 1}^n {{f_i}} }}$,
  Where ${x_i} =$(upper class limit-lower class limit)/2 and ${f_i}$ is the frequency of the class limit

Complete step-by-step solution:
We are given a mean of the given data is 16.
Since we are given a grouped data, we will find the frequency of each class and class mark for each respective class.
We form table of the given data which has class mark (${x_i}$) and then we find the frequency ${f_i}$ and then multiply each class mark with respective frequency (${f_i}{x_i}$).
Let us assume the missing frequency as ‘x’.

Class	Number of patients(${f_i}$)	Class mark (${x_i}$)	${f_i}{x_i} = {f_i} \times {x_i}$
0-4	6	$\dfrac{{0 + 4}}{2} = 2$	$6 \times 2 = 12$
4-8	8	$\dfrac{{4 + 8}}{2} = 6$	$8 \times 6 = 48$
8-12	17	$\dfrac{{8 + 12}}{2} = 10$	$17 \times 10 = 170$
12-16	23	$\dfrac{{12 + 16}}{2} = 14$	$23 \times 14 = 322$
16-20	16	$\dfrac{{16 + 20}}{2} = 18$	$16 \times 18 = 288$
20-24	15	$\dfrac{{20 + 24}}{2} = 22$	$15 \times 22 = 330$
24-28	x	$\dfrac{{24 + 28}}{2} = 26$	$x \times 26 = 26x$
28-32	4	$\dfrac{{28 + 32}}{2} = 30$	$4 \times 30 = 120$
32-36	3	$\dfrac{{32 + 36}}{2} = 34$	$3 \times 34 = 102$
Total:	$\sum {{f_i} = 92 + x}$		$\sum {{f_i}{x_i} = 1392 + 26x}$

We substitute the values in formula for mean
Mean: $\overline x = \dfrac{{1932 + 26x}}{{92 + x}}$
Since we are given the value of mean as 16, substitute $\overline x = 16$
$\Rightarrow 16 = \dfrac{{1932 + 26x}}{{92 + x}}$
Cross multiply the values from RHS to LHS of the equation
$\Rightarrow 16 \times 92 + 16x = 1932 + 26x$
Calculate the product on LHS
$\Rightarrow 1472 + 16x = 1392 + 26x$
Bring all constants on one side of the equation
$\Rightarrow 26x - 16x = 1472 - 1392$
$\Rightarrow 10x = 80$
Divide both sides by 10
$\Rightarrow \dfrac{{10x}}{{10}} = \dfrac{{80}}{{10}}$
Cancel same factors from numerator and denominator
$\Rightarrow x = 8$

$\therefore$The value of missing frequency is 8

Note: Students are likely to make mistakes in the calculation part of the table as they directly make columns for ${f_i}{x_i}$ and don’t calculate ${x_i}$. Keep in mind we need to calculate the class mark as well. Also, many students directly apply the formula of mean as sum of observations divided by number of observations, this is wrong as the data given is not normal data it is grouped data. Always change the sign of values when shifting one value from one side of the equation to another.