Question

Represent the following data using a histogram and then, find the mode.

Heights of students (cm)	140-144	145-149	150-154	155-159
Number of students	2	12	10	4

Answer 1

Here, the classes are discontinuous. So we will convert them into continuous classes .We have to extend the classes equally at both the ends.
This can be done by extending the classes by half of the gap between two consecutive classes .

Extended classes ( Heights of students in cm)	139.5-144.5	144.5-149.5	149.5-154.5	154.5-159.5
Number of students	2	12	10	4

Now, we will draw the heights of the students on the X-axis and the number of students on the Y-axis.
We have chosen the following scale for this purpose.
On the X -axis:
1 big division = 5 cm
On the Y- axis
1 big division = 2 students
We will find the mode through the following steps:
(i) Draw the histogram to represent the following data.
(ii) Locate the modal class , pre-modal class and post-modal class.
(iii) Join the right-hand top vertices of modal class (C) and pre-modal class (A) by a straight line .
(iv) Join the left-hand top vertices of modal class (B) and post-modal class (D) by a straight line .
(v) Let the point of intersection of AC and BD be E.
(vi) Draw a segment EF perpendicular to the X-axis .
(vii) The foot of the perpendicular, F, is the mode .

Here is the histogram showing the mode graphically .


From the graph, we find that the value of mode is 148 cm.

The mode can also be verified using the standard formula.
Here L=144.5, f_m =12, f₁ =2, f₂ =10 , h=5

$$\begin{gathered} \mathrm{Therefore},\mathrm{mode}=L+\left[\frac{f_m-f_1}{2f_m-f_1-f_2}\right]h \\ =144.5+\left[\frac{12-2}{2\times 12-2-10}\right]\times 5 \\ =144.5+\frac{50}{12} \\ =148.6 \end{gathered}$$