Question

The mode of the following data is $67$. Find the missing frequency $x$.

Class	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$
Frequency	$5$	$x$	$15$	$12$	$7$

Answer 1

Find the missing frequency $x$.

Given,

The value of mode is $67$.
The value of Modal class varies from$60-70.$
Mode $=l+\left(f_1-f_0\right)h/\left(2f_1-f_0-f_2\right)$
Here $I=$ lower limit of modal class $=60.$
$h=$size of class interval $=10$.
Frequency of modal class$\left({\mathrm{f}}_1\right)$ is $15$
Frequency of class preceding modal class $\left({\mathrm{f}}_0\right)$is $x$.
Frequency of class succeeding modal class $f_1$is $12$
$$\begin{gathered} 67=60+(15-x)10/(18-x) \\ \Rightarrow 7=(15-x)10/(18-x) \\ \Rightarrow 7(18-x)=10(15-x) \\ \Rightarrow 126-7x=150-10x \\ \Rightarrow 3x=24 \\ \Rightarrow x=\frac{24}{3} \\ \Rightarrow x=8 \end{gathered}$$
Therefore, the value of $x$ is $8$ .