Find the mode of the following data class:

Marks	$0-20$	$20-40$	$40-60$	$60-80$	$80-100$	$100-120$	$120-140$
Frequency	$6$	$8$	$10$	$12$	$6$	$5$	$3$

Find the mode of the given data:

We are given a continuous set of data.

For the data, class interval with highest frequency is $60-80$ with a frequency of $12$.

We know that, $\mathrm{mode}=l+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)h$, where

$l=60$, lower limit of modal class

$f_1=12$ , frequency of the modal class

$f_0=10$ , frequency of the class preceding modal class

$f_2=6$, frequency of the class succeeding modal class

$h=20$, size of the class interval

So, the mode for modal class $60-80$ will be,

$$\begin{gathered} \Rightarrow \mathrm{mode}=60+\left\{\frac{12-10}{(2\times 12)-10-6}\right\}\times 20 \\ \Rightarrow \mathrm{mode}=60+\left(\frac{2}{24-16}\right)\times 20 \\ \Rightarrow \mathrm{mode}=60+\frac{2\times 20}{8} \\ \Rightarrow \mathrm{mode}=60+5 \\ \Rightarrow \mathrm{mode}=65 \end{gathered}$$

Hence, the mode of the given data is $65$.