Question

Choose the correct option for the following question given below:
Represent the following data using histogram, hence find the mode.

Height of the students(cm)	\[140 - 144\]	\[145 - 149\]	\[150 - 154\]	\[155 - 159\]
Number of students	\[2\]	\[12\]	\[10\]	\[4\]

A. \[146cm\]
B. \[147cm\]
C. \[149cm\]
D. \[150cm\]

Answer 1

Hint: Find the maximum frequency from the given number of students. Conclude the group of maximum frequency, lower class boundary of the class, frequency of the group, i.e., maximum frequency, the frequency of the preceding class and the frequency of the succeeding class plus the width of the class. Substitute all these values in the formula of the mode and simplify the equation to find the mode. Draw a histogram based on the given data.

Complete step-by-step solution:
Complete step by step process:
By considering all the groups, and the given frequency of the respective groups draw a histogram. The histogram follows as given in the below graph:

From the given number of students, out of all the students find the maximum frequency, i.e. the highest number of students having a certain height.
Maximum frequency \[ = 12\]
The group of this maximum frequency \[ = 145 - 149\]
Now, we label the terms finding the following terms as given below;
\[L = \] lower class boundary of the modal group \[ = 145\]
\[{f_1} = \] frequency of the modal group \[ = 12\]
\[{f_0} = \] frequency of the class preceding the modal class \[ = 2\]
\[{f_2} = \] frequency of the class succeeding the modal class \[ = 10\]
\[w = \] class width \[ = 4\]
Mode \[ = L + \dfrac{{{f_1} - {f_0}}}{{2{f_1} - {f_0} - {f_2}}} \times w\]
Substituting the values in the above equation, we get;
$\Rightarrow$Mode \[ = 145 + \dfrac{{12 - 2}}{{2 \times 12 - 2 - 10}} \times 4\]
Simplifying the equation, we get;
$\Rightarrow$Mode \[ = 145 + 3.333\]
Adding the terms, we get;
$\Rightarrow$Mode \[ = 148.333\]
Rounding off to the nearest integer of the given options, we get;
Hence,
\[ \Rightarrow \] Mode\[ = 147\]

$\therefore $ The correct option is B.

Note: Simple mathematics is used to solve this problem. The formula for mode is known, inferring the required data from the given question and substituting the values in the formula of mode is needed. To draw the histogram, you need to take the lower intervals of every class in the horizontal axis and the frequency in the vertical axis respectively. You have to draw a bar from the lower interval of one class to the lower interval of the other class based on the frequency.