Theory:

A frequency polygon is a line graph, and it is obtained by joining the mid-points of the adjacent rectangles of the histogram.
Let us look at an example of a frequency polygon.
 
5.svg
 
The line graph forms a closed surface with many sides, hence the name frequency 'polygon'.
Types of frequency polygon
Frequency polygons are of two types, namely:
 
1. Frequency polygon with histogram
 
2. Frequency polygon without histogram.
 
When the line graph is over a histogram, then it is a frequency polygon with a histogram.
 
When the line graph alone is visible while the histogram is not, it is a frequency polygon without a histogram.
 
Consider the following data table, and let us try to construct a frequency table with histogram, and a frequency table without a histogram using the same data table.
 
Problem:
 
A school management decides to conduct a sports meet for the academic year. The management discusses with the sports teachers of the school and comes up with an action plan.
 
According to the action plan, the number of students in the school will be classified considering their ages and competing against students of the same age group.
 
The data table is as follows:
 
Age groups
\(5\) - \(8\)
\(8\) - \(11\)
\(11\) - \(14\)
\(14\) - \(17\)
Number of students
\(75\)
\(89\)
\(70\)
\(100\)
 
Construct the following using the given data table:
 
1. A frequency polygon using a histogram
 
2. A frequency polygon without using a histogram
A frequency polygon using a histogram
Step \(1\): Find the mid-point of each of the intervals, and make a tabular column.
 
Age groups
Mid-point
Number of students
\(5\) - \(8\)
\(6.5\)
\(75\)
\(8\) - \(11\)
\(9.5\)
\(89\)
\(11\) - \(14\)
\(12.5\)
\(70\)
\(14\) - \(17\)
\(15.5\)
\(100\)
 
Step \(2\): Draw a histogram of the given data.
 
1276_7.svg
 
Step \(3\): Mark the mid-points on each of the histogram's frequency rectangles.
 
1276_8.svg
 
Step \(4\): Join the mid-points with a straight line.
 
1276_9.svg
 
Step \(5\): If the class intervals before the first rectangle and after the last rectangles are unknown, always assume equivalent class intervals both preceding and succeeding the rectangles.
 
The assumed intervals are called imagined class intervals.
 
Now, calculate the mid-point of the imagined class intervals and extend a line up to the imagined class intervals' mid-points.
 
1276_10.svg
A frequency polygon without using a histogram
Step \(1\): Find the mid-point of each of the intervals, and make a tabular column similar to creating frequency polygons without histograms.
 
Age groups
Mid-point
Number of students
\(5\) - \(8\)
\(6.5\)
\(75\)
\(8\) - \(11\)
\(9.5\)
\(89\)
\(11\) - \(14\)
\(12.5\)
\(70\)
\(14\) - \(17\)
\(15.5\)
\(100\)
 
Step \(2\): Mark the mid-points on the graph.
 
1276_13.svg
 
Step \(3\): Join the mid-points with a straight line.
 
1276_14.svg
 
Step \(4\): If the class intervals before the first rectangle and after the last rectangles are unknown, always assume equivalent class intervals both preceding and succeeding the rectangles.
 
The assumed intervals are called imagined class intervals.
 
Now, calculate the mid-point of the imagined class intervals and extend a line up to the imagined class intervals' mid-points.
 
1276_11.svg
 
The required frequency polygon is now constructed.