Question

Following are the marks obtained by students in mathematics. Draw a histogram and find the mode .

Marks	0-19	20-39	40-59	60-79	80-99
No. of students	4	8	32	16	8

Answer 1

The given data is in discontinuous form. First of all, we will convert the data into continuous form .
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	-0.5-19.5	19.5-39.5	39.5-59.5	59.5-79.5	79.5-99.5
Frequencies	4	8	32	16	8

We will find the mode with the help of the following steps:-
(i) Draw the histogram to represent the 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 perpendicular, F, is the mode.

Here is the histogram showing the mode graphically.



The mode can also be found using the formula for calculating mode.
Here, L = 39.5, f_m = 32, f₁ = 8, f₂ = 16 , h = 20
$$\begin{gathered} \mathrm{Therefore},\mathrm{mode}=L+\left[\frac{f_m-f_1}{2f_m-f_1-f_2}\right]h \\ =39.5+\left[\frac{32-8}{2\times 32-8-16}\right]\times 20 \\ =39.5+\frac{480}{40} \\ =51.5 \end{gathered}$$