Question

The marks scored by students in mathematics in a certain examination are given below.

Marks scored	1-20	21-40	41-60	61-80	81-100
Number of students	3	8	19	18	6

Draw the histogram and find the mode graphically.

Answer 1

Here, the classes are discontinuous. So, first we will make the classes continuous by extending them equally at both the ends.
This can be done by extending the classes by half of the gap between two consecutive classes.

Extended classes (Marks scored)	0.5-20.5	20.5-40-5	40.5-60.5	60.5-80.5	80.5-100.5
Number of students	3	8	19	18	6

Using the following steps, we will draw the histogram and also find the mode.
(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 the modal class (C) and pre-modal class (A) by a straight line.
(iv) Join the left-hand top vertices of the 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 the mode is 58 marks.


We can verify the mode by using the formula also.
Here, L=40.5, f_m =19, f₁ =8, f₂ =18 , h=20

$$\begin{gathered} \mathrm{Therefore},\mathrm{mode}=L+\left[\frac{f_m-f_1}{2f_m-f_1-f_2}\right]h \\ =40.5+\left[\frac{19-8}{2\times 19-8-18}\right]\times 20 \\ =40.5+\frac{220}{12} \\ =58.83 \end{gathered}$$