Question

Draw a frequency polygon for the following data using histogram.

Marks	10-20	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
Number of students	4	6	8	5	7	4	9	5	7

Answer 1

Hint:
We will draw a histogram for the given data. We will represent the frequency (number of students) for each class of marks using a bar. Then we will locate the midpoint of the upper horizontal side of each bar. Then we will join all the mid-points using line segments one after the other and this will give us the required frequency polygon.

Complete step by step solution:
We will make a histogram to represent the data in the table. On the X-axis, we will take all the class intervals (marks of students) and on the Y-axis, we will take the frequency (number of students).
There are no students who have got marks between 0 and 10, so we will make a class interval 0-10 (because the values on the x-axis start from 0.If we want to take values from 10, we will have to draw a kink)and take its frequency as 0.

Now we will find the midpoint of the upper horizontal side of each bar of the histogram. The midpoint of a class is given by \[M = \dfrac{{U + L}}{2}\] where M is the midpoint, U is the upper class limit and L lower class limit.

Marks	Midpoint of the bar	Height of the bar (Frequency)
0-10	\[\dfrac{{10 + 0}}{2} = 5\]	0
10-20	\[\dfrac{{20 + 10}}{2} = 15\]	4
20-30	\[\dfrac{{30 + 20}}{2} = 25\]	6
30-40	\[\dfrac{{40 + 30}}{2} = 35\]	8
40-50	\[\dfrac{{50 + 40}}{2} = 45\]	5
50-60	\[\dfrac{{60 + 50}}{2} = 55\]	7
60-70	\[\dfrac{{70 + 60}}{2} = 65\]	4
70-80	\[\dfrac{{80 + 70}}{2} = 75\]	9
80-90	\[\dfrac{{90 + 80}}{2} = 85\]	5
90-100	\[\dfrac{{100 + 90}}{2} = 95\]	7

We will represent the data in the above table using a frequency polygon. We will represent the mid-points on the x-axis and the frequency on the y-axis.

Note:
Here, we have drawn a histogram because it becomes much easier to show different data of different ranges through bars. The taller bar represents more value in a particular range. We can also directly draw the polygon without making the histogram. We will directly find the mid-points of the given intervals and plot their corresponding frequencies on the y-axis. We will represent the mid-points on the x-axis.