Question

Find the mode of the following frequency distribution.

Class	0 – 100	100 – 200	200 – 300	300 – 400	400 – 500	500 – 600
Frequency	64	62	77	62	66	54

Answer 1

Hint: We use the fact that if the data is in groups or class intervals, the modal interval corresponds to the highest frequency. We then use the formula for the mode $l + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}$. Substitute the values in this formula and do simplification to find the mode.

Complete step-by-step solution:
Let us write the formula of mode for grouped data.
Mode for grouped data is given as,
Mode $ = l + \left( {\dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}}} \right) \times h$
Where $l$ is the lower modal class,
$h$ is the size of the class interval,
${f_1}$ is the frequency of modal class,
${f_0}$ is the frequency of the class preceding the modal class,
${f_2}$ is the frequency of the class succeeding the modal class,
The modal class is the interval with the highest frequency.
$ \Rightarrow $Modal class $ = 200 - 300$
The lower limit of the modal class is,
$ \Rightarrow l = 300$
The class-interval is,
$ \Rightarrow h = 300 - 200 = 100$
The frequency of the modal class is,
$ \Rightarrow {f_1} = 77$
The frequency of the class preceding the modal class is,
$ \Rightarrow {f_0} = 62$
The frequency of the class succeeding modal class is,
$ \Rightarrow {f_2} = 62$
Substitute these values in the mode formula,
$ \Rightarrow $ Mode $ = 200 + \dfrac{{77 - 62}}{{2\left( {77} \right) - 62 - 62}} \times 100$
Simplify the terms,
$ \Rightarrow $ Mode $ = 200 + \dfrac{{15}}{{154 - 124}} \times 100$
Subtract the values in the denominator and multiply the terms in the numerator,
$ \Rightarrow $ Mode $ = 200 + \dfrac{{1500}}{{30}}$
Divide the numerator by denominator,
$ \Rightarrow $ Mode $ = 200 + 50$
Add the terms,
$\therefore $ Mode $ = 250$

Hence, the mode is 250.

Note: You may mistake the mode formula with that of the median. The Median formula is given as $l + \left( {\dfrac{{\dfrac{n}{2} - cf}}{f}} \right) \times h$. In this question, we wrote mode for grouped data. To find the mode for ungrouped data, we will find the observation which occurs the maximum number of times.