Question

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below. Find the mode of the data.

Number of cars	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70	70 – 80
Frequency	7	14	13	12	20	11	15	8

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 answer:
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 $ = 40 - 50$
The lower limit of the modal class is,
$ \Rightarrow l = 40$
The class-interval is,
$ \Rightarrow h = 50 - 40 = 10$
The frequency of the modal class is,
$ \Rightarrow {f_1} = 20$
The frequency of the class preceding the modal class is,
$ \Rightarrow {f_0} = 12$
The frequency of the class succeeding modal class is,
$ \Rightarrow {f_2} = 11$
Substitute these values in the mode formula,
$ \Rightarrow $ Mode $ = 40 + \dfrac{{20 - 12}}{{2\left( {20} \right) - 12 - 11}} \times 10$
Simplify the terms,
$ \Rightarrow $ Mode $ = 40 + \dfrac{8}{{40 - 23}} \times 10$
Subtract the values in the denominator and multiply the terms in the numerator,
$ \Rightarrow $ Mode $ = 40 + \dfrac{{80}}{{17}}$
Divide the numerator by denominator,
$ \Rightarrow $ Mode $ = 40 + 4.71$
Add the terms,
$\therefore $ Mode $ = 44.71$

Hence, the mode is 44.71.

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.