Question

A student noted the number of cars passing through the spot on the road for 200 periods each of 10 minutes and summarized 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	9	13	21	12	15	4	12

Answer 1

Hint: Here we have to find the mode of the following frequency data. For that, we will first find the modal class. Then we will find the lower limit of the modal class, frequency of the modal class, frequency of the class before the modal class, frequency of the class after the modal class. Then we will find the size of the class interval. We will put all the data in the formula of mode.

Formula Used:
We will use the formula of the mode which is given by
\[Mode = l + \dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}} \times h\] , where, \[l\] is the lower limit of modal class, \[h\] is the size of the class intervals, \[f\] is the modal class, \[{f_1}\] is the frequency of the class preceding the modal class, \[{f_2}\] is the frequency of the class succeeded in the modal class.

Complete step-by-step answer:
We will find the modal class. Modal class is the class of highest frequency.
Here the modal class is \[30 - 40\] .
From the given data, we have
\[\begin{array}{l}l = 30\\f = 21\\{f_1} = 13\\{f_2} = 12\\h = 10\end{array}\]
Now, we will substitute all the values in the formula of mode, \[mode = l + \dfrac{{f - {f_1}}}{{2f - {f_1} - {f_2}}} \times h\].
\[ \Rightarrow {\rm{mode}} = 30 + \dfrac{{21 - 13}}{{2 \times 21 - 13 - 12}} \times 10\]
On further simplification, we get
\[ \Rightarrow {\rm{mode}} = 30 + \dfrac{8}{{17}} \times 10\]
On simplifying the terms, we get
\[ \Rightarrow {\rm{mode}} = 34.705\]
This is the required mode of the given data.

Note: The value of mode is equal to the value of mean and median for the normal distribution. Mode can be used to describe qualitative terms e.g. consumer preferences, brand preference etc. The mode can also be used when the data are categorical or nominal, such as gender, religious preference or political affiliation.