Frequency Polygon: Definition, Steps and Solved Examples

A frequency polygon is a graphical representation used to understand the distribution of a dataset. It involves plotting the frequency of data points and connecting them with straight lines to form a polygon. This method is particularly useful in visualizing the shape of a distribution and identifying patterns, trends, or outliers. To create a frequency polygon, you need to follow specific steps: organize the data into a frequency distribution table, plot the midpoints of each class interval on a graph, and then connect these points with straight lines. Key terms related to frequency polygons include class intervals, midpoints, and frequency. Frequency polygons are commonly used in competitive exams such as JEE, NEET, SSC, CAT, MAT, GMAT, GRE, and various professional certification tests where data interpretation and statistical analysis are assessed.

Definition of Frequency Polygon

A frequency polygon is a graphical representation of a frequency distribution, showcasing how data points are spread across different intervals. It is constructed by plotting points above the midpoints of each class interval, corresponding to the frequency of each interval, and then connecting these points with straight lines. Important terms related to frequency polygons include:

• Class Interval: The range of data values divided into intervals.
• Midpoint: The central value of each class interval, calculated as the average of the upper and lower boundaries.
• Frequency: The number of data points within each class interval.
• Frequency Distribution Table: A table that displays the frequency of data points for each class interval.
• Continuous Data: Data that can take any value within a given range.

Steps to Draw Frequency Polygon

To draw a frequency polygon, follow these steps:

• Arrange your data into a frequency distribution table, listing class intervals and their corresponding frequencies.
• Calculate the midpoint of each class interval. The midpoint is the average of the upper and lower boundaries of the interval.
• On a graph, plot points where the x-axis represents the midpoints and the y-axis represents the frequencies of each class interval.
• Connect the plotted points with straight lines. To close the polygon, extend lines to the x-axis at the midpoints of the intervals immediately before the first class and after the last class, which are often set to zero frequency.
• Ensure the x-axis and y-axis are properly labeled with appropriate titles and scales.

By following these steps, you can effectively create a frequency polygon that visually represents the distribution of your data.

Also Read: Find the Area of the Shaded Region: Square, Rectangle, Circle and Triangle

Properties of Frequency Polygon With Formulas

A frequency polygon has several key properties and associated formulas that help in understanding the distribution of data. Here are the important properties along with relevant formulas.

1. Visual Representation: A frequency polygon provides a clear visual representation of the distribution of data, making it easy to identify trends, patterns, and outliers.
2. Comparison: It is useful for comparing multiple datasets on the same graph, allowing for easy comparison of different distributions.
3. Midpoints: Points are plotted at the midpoints of each class interval, providing a precise representation of the data distribution within each interval.
4. Connection of Points: Points are connected by straight lines, forming a polygon that helps to visualize the shape and spread of the data.
5. Closure to Axes: To complete the polygon, lines are often drawn from the first and last points to the x-axis, creating a closed figure.

Formulas

Midpoint Calculation:

Midpoint = Lower Boundary + Upper Boundary / 2

This formula calculates the midpoint for each class interval, where the lower and upper boundaries are the limits of the class interval.

Frequency Density (if class intervals are of different widths):

Frequency Density = Frequency / Class Width

This formula adjusts the frequency for class intervals of different widths, ensuring that the area under the frequency polygon accurately represents the data.

Cumulative Frequency (for cumulative frequency polygons or ogives):

Cumulative Frequency = ∑Frequency of All Previous Classes

This formula calculates the cumulative frequency for each class interval by summing the frequencies of all previous intervals.

Example of Creating a Frequency Polygon

Several examples for creating a frequency polygon are mentioned below:

1. Data Organization:
   • Class Intervals: 0-10, 10-20, 20-30, 30-40
   • Frequencies: 5, 10, 7, 3
2. Midpoint Calculation:
   • Midpoints: 5, 15, 25, 35
3. Plot Points:
   • (5, 5), (15, 10), (25, 7), (35, 3)
4. Connect Points:
   • Connect the points with straight lines.
5. Closure to Axes:
   • Extend lines from (5, 5) to (0, 0) and from (35, 3) to (40, 0) to complete the polygon.

Also Read: 10+ Questions of Venn Diagrams

Properties of Frequency Polygon With Solved Examples

Here, we will explore the properties of Frequency Polygon formulas and provide some solved examples.

Example 1: Heights of Students

Height (cm)	Number of Students
140-145	5
145-150	12
150-155	20
155-160	18
160-165	8

Solution:

1. Calculate the midpoints: 142.5, 147.5, 152.5, 157.5, 162.5.
2. Plot the points (142.5, 5), (147.5, 12), (152.5, 20), (157.5, 18), (162.5, 8).
3. Connect the points with line segments.
4. Extend the line segments to meet the x-axis at 137.5 and 167.5.

Example 2: The frequency polygon of a frequency distribution is shown below.

Answer the following about the distribution from the histogram.

(i) What is the frequency of the class interval whose class mark is 15?

(ii) What is the class interval whose class mark is 45?

(iii) Construct a frequency table for the distribution.

Solution:

(i) 18

(ii) 40 – 50

(iii) As the class marks of consecutive overlapping class intervals are 5, 15, 25, 35, 45, 55 we find the class intervals are 0 – 10, 10 – 20, 20 – 30, 30 – 40, 40 – 50, 50 – 60. Therefore, the frequency table is constructed as below.

Class Interval	Frequency
0 – 10	10
10 – 20	18
20 – 30	14
30 – 40	26
40 – 50	8
50 – 60	18

Example 3: The following frequency polygon displays the weekly incomes of laborers of a factory.

Answer the following.

(i) Find the class interval whose frequency is 25.

(ii) How many labourers have a weekly income of at least $ 500 but not more than $ 700?

(iii) What is the range of weekly income of the largest number of labourers?

(iv) Prepare the frequency distribution table.

Solution:

(i) The frequency 25 corresponds to the class mark 800.

The common width of class intervals = 400 – 200 = 200

So, the class interval is (800 – 200/2) – (800 + 200/2), i.e., 700-900

(ii) The number of labourers has to fall in the class interval 500 – 700 whose class mark is 600. The frequency corresponding to the class mark 600 is 20. Hence, the required number of labourers is 20.

(iii) The largest number of labourers belong to the class interval whose class mark is 400. The corresponding class interval is (400 – 200/2) – (400 + 200/2), i.e., (300 – 500). So, the largest numbers of labourers have a weekly income of at least $ 300 but less than $ 500.

(iv) 

Weekly Income (in $)	No. of Labours
100 – 300	30
300 – 500	40
500 – 700	20
700 – 900	25

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