Frequency polygons are a graphical representation of the frequency distribution of a dataset. They help visualise the shape of the data distribution and are particularly useful in comparing multiple datasets. Understanding frequency polygons is crucial for students as it enhances their ability to analyze and interpret data effectively.

In this article, we will learn about the Frequency Polygon, What it is, steps to construct it, Formulas and related concepts, Practice questions with solutions and a worksheet for more practice.

What is a Frequency Polygon?

A frequency polygon is a graph used in statistics to visualize data distribution. It resembles a line graph but instead of plotting individual data points, it connects the midpoints of class intervals in a histogram with straight-line segments. This creates a closed loop, unlike a histogram which has bars instead of lines.

Frequency polygons are an effective way to visualize and compare data distributions, making it easier to understand and interpret statistical information. By connecting midpoints of class intervals, they provide a clear picture of how data points are spread across different intervals.

Steps to Construct a Frequency Polygon

Various steps to Construct a Frequency Polygon include:

Step 1: Determine Class Intervals and Frequencies: Divide the data into intervals and count the frequencies for each interval.

Step 2: Find Midpoints: Calculate the midpoint for each class interval.

Step 3: Plot Points: Plot the midpoints on the x-axis and the corresponding frequencies on the y-axis of the graph.

Step 4: Connect Points: Connect the plotted points with straight lines to form the frequency polygon.

Example: Given the following data on the number of hours studied by students:

• Class Intervals: 0-10, 10-20, 20-30, 30-40, 40-50
• Frequencies: 4, 7, 10, 6, 3

Calculate the midpoints:

• Midpoints: 5, 15, 25, 35, 45

Plot the points: (5, 4), (15, 7), (25, 10), (35, 6), (45, 3) and connect them with lines to create the frequency polygon.

Frequency Polygon-Related Formulas/Concepts

Class Intervals: Class intervals are the ranges into which the entire data set is divided. Every interval has an upper boundary and lower boundary.

Frequency: Frequency refers to the number of data points that fall within each class interval.

Midpoint of Class Interval

The midpoint (or class mark) of a class interval is calculated by taking the average of the upper and lower boundaries of that interval.

Midpoint = (Lower boundary + Upper boundary) /2

Relative Frequency

Relative Frequency is the fraction or percentage of the total data points that fall within each class interval. It is calculated by dividing the frequency of the interval by the total number of data points.

Relative Frequency = Frequency/Total number of Data Points

Cumulative Frequency

Cumulative frequency is the sum of the frequencies for all class intervals up to and including the current interval.

Cumulative Frequency = ∑ Frequency of Current and Previous Intervals

Frequency Polygon Practice Questions

Below are the practice questions on Frequency Polygon are as follows:

Question 1: The following table shows the number of hours students in a class studied for a math exam. Construct a frequency polygon for this data.

Hour Studied	Frequency
1-2	5
3-4	8
5-6	12
7-8	7
9-10	3

Solution:

Calculate Class Marks:

First, find the class mark for each interval.

• Class Mark (1-2) = (2 + 1) / 2 = 1.5
• Class Mark (3-4) = (4 + 3) / 2 = 3.5
• Class Mark (5-6) = (6 + 5) / 2 = 5.5
• Class Mark (7-8) = (8 + 7) / 2 = 7.5
• Class Mark (9-10) = (10 + 9) / 2 = 9.5

Plot the Points:

On a graph, plot the class marks on the horizontal axis and the frequencies on the vertical axis. So you’ll have points at (1.5, 5), (3.5, 8), (5.5, 12), (7.5, 7), and (9.5, 3).

Connect the Points:

Connect these points using straight line segments.

Question 2: The weights (in kg) of 20 athletes are grouped into the following classes. Draw the frequency polygon and describe the distribution of weights.

Weight (kg)	Frequency
50-55	3
56-60	7
61-65	5
66-70	3
71-75	2

Solution:

Calculate Class Marks:

• Weight (kg): 50-55 -> Class Mark: (50 + 55) / 2 = 52.5
• Weight (kg): 56-60 -> Class Mark: (56 + 60) / 2 = 58
• Weight (kg): 61-65 -> Class Mark: (61 + 65) / 2 = 63
• Weight (kg): 66-70 -> Class Mark: (66 + 70) / 2 = 68
• Weight (kg): 71-75 -> Class Mark: (71 + 75) / 2 = 73

Plot the Points:

Plot points at (52.5, 3), (58, 7), (63, 5), (68, 3), and (73, 2).

Connect the Points:

Connect these points with line segments. You’ll notice the polygon slants slightly to the right, indicating a possible positive skew (more athletes on the heavier side).

Question 3: The following table shows the travel time (in minutes) to work for a group of employees. Construct a frequency polygon.

Travel Time (Minutes)	Frequency
20-29	10
30-39	15
40-49	8
50-59	4
60-69	3

Solution:

Calculate class marks:

• Travel Time (minutes): 20-29 -> Class Mark: (20 + 29) / 2 = 24.5
• Travel Time (minutes): 30-39 -> Class Mark: (30 + 39) / 2 = 34.5
• Travel Time (minutes): 40-49 -> Class Mark: (40 + 49) / 2 = 44.5
• Travel Time (minutes): 50-59 -> Class Mark: (50 + 59) / 2 = 54.5
• Travel Time (minutes): 60-69 -> Class Mark: (60 + 69) / 2 = 64.5

Plot points at (24.5, 10), (34.5, 15), (44.5, 8), (54.5, 4), and (64.5, 3).

Connect the points with line segments. The polygon should start and end at 19.5 or 69.5 minutes.

Question 4: The ages (in years) of students in a history class are grouped as follows. Draw the frequency polygon and comment on the distribution.

Ages(years)	Frequency
18-20	5
21-23	8
24-26	12
27-29	7
30-32	3

Solution:

Calculate class marks:

• Age (years): 18-20 -> Class Mark: (18 + 20) / 2 = 19
• Age (years): 21-23 -> Class Mark: (21 + 23) / 2 = 22
• Age (years): 24-26 -> Class Mark: (24 + 26) / 2 = 25
• Age (years): 27-29 -> Class Mark: (27 + 29) / 2 = 28
• Age (years): 30-32 -> Class Mark: (30 + 32) / 2 = 31

Plot points at (19, 5), (22, 8), (25, 12), (28, 7), and (31, 3).

Connect the points:

The polygon might be symmetrical, suggesting a normal distribution. However, a larger sample size would be ideal for confirmation.

Question 5 The following table shows the number of books borrowed by library members in a week. Construct a frequency polygon.

Books Bowrroed	Frequency
1-2	12
3-4	18
5-6	15
7-8	7
9-10	3

Solution:

Calculate class marks:

• Books Borrowed: 1-2 -> Class Mark: (1 + 2) / 2 = 1.5
• Books Borrowed: 3-4 -> Class Mark: (3 + 4) / 2 = 3.5
• Books Borrowed: 5-6 -> Class Mark: (5 + 6) / 2 = 5.5
• Books Borrowed: 7-8 -> Class Mark: (7 + 8) / 2 = 7.5
• Books Borrowed: 9-10 -> Class Mark: (9 + 10) / 2 = 9.5

Plot points at (1.5, 12), (3.5, 18), (5.5, 15), (7.5, 7), and (9.5, 3).

Connect the points:

The polygon might be bell-shaped, indicating a possible normal distribution.

Question 6 (Cumulative Frequency Polygon):

The table shows the waiting time (in minutes) at a doctor’s clinic. Convert the data into a cumulative frequency table and draw the corresponding cumulative frequency polygon.

Watching Time (min)	Frequency
10-14	5
15-19	8
20-24	12
25-29	7
30-34	3

Solution (Cumulative Frequency):

Step 1: Calculate Cumulative Frequency

Cumulative frequency refers to the total number of observations that fall less than or equal to a specific class. Here’s how to calculate it:

Waiting Time	Frequency	Cumulative Frequency
10-14	5	5
15-19	8	5+8=13
20-24	12	13+12=25
25-29	7	25+7=32
30-34	3	32+3=35

Step 2: Draw Cumulative Frequency Polygon

• Plot the upper class limits (14, 19, 24, 29, 34) on the horizontal axis and the corresponding cumulative frequencies (5, 13, 25, 32, 35) on the vertical axis.
• Connect the points with line segments to create the cumulative frequency polygon.

Question 7 Theslepttest scores in chemistry for a class are grouped into these categories. Draw the frequency polygon and describe the distribution.

Test Score (%)	Frequency
50-59	8
60-69	12
70-79	15
80-89	7
90-99	3

Solution:

Class mark for the test scores in chemistry can be calculated as follows:

• Test Scores: 50-59 -> Class Mark: (50 + 59) / 2 = 54.5
• Test Scores: 60-69 -> Class Mark: (60 + 69) / 2 = 64.5
• Test Scores: 70-79 -> Class Mark: (70 + 79) / 2 = 74.5
• Test Scores: 80-89 -> Class Mark: (80 + 89) / 2 = 84.5
• Test Scores: 90-99 -> Class Mark: (90 + 99) / 2 = 94.5

Frequency Polygon:

Now that you have the frequencies, you can plot the frequency polygon:

• On a graph, mark the class marks (54.5, 64.5, 74.5, 84.5, 94.5) on the horizontal axis (x-axis).
• Plot the corresponding frequencies (8, 12, 15, 7, 3) on the vertical axis (y-axis).
• Connect the points with straight line segments to create the frequency polygon.

Question 8: The following data shows the number of hours slept by a group of teenagers. Construct a frequency polygon.

Sleep Time (Hour)	Frequency
6-7	8
8-9	12
10-11	7
12-13	3

Solution:

Calculate the Class Marks:

• Sleep Time (hours): 6-7 -> Class Mark: (6 + 7) / 2 = 6.5
• Sleep Time (hours): 8-9 -> Class Mark: (8 + 9) / 2 = 8.5
• Sleep Time (hours): 10-11 -> Class Mark: (10 + 11) / 2 = 10.5
• Sleep Time (hours): 12-13 -> Class Mark: (12 + 13) / 2 = 12.5

Frequency Polygon:

Now you can construct the frequency polygon:

• Plot the class marks we just calculated (6.5, 8.5, 10.5, 12.5) on the horizontal axis (sleep time).
• Plot the corresponding frequencies (8, 12, 7, 3) on the vertical axis (number of teenagers).
• Connect the points with line segments to create the polygon.

Frequency Polygon: Worksheet

Question 1: The following table shows the number of pencils owned by students in a class. Construct a frequency polygon.

Pencils Owned	Frequency
1-2	7
3-4	12
5-6	8
7-8	5
9-10	3

Question 2: The waiting times (in minutes) at a coffee shop are grouped as follows. Draw the frequency polygon and describe the distribution.

Watching Time (minutes)	Frequency
5-9	10
10-14	15
15-19	8
20-24	4
25-29	3

Question 3: The following data shows the daily expenses (in dollars) of a group of people. Construct a frequency polygon.

Daily Expense	Frequency
20-30	8
31-40	12
41-50	7
51-60	5
61-70	3

Question 4: The number of pages read by a book club in a month is recorded. Draw the frequency polygon and analyze the distribution.

Pages Read	Frequency
50-99	4
100-149	8
150-199	10
200-249	5
250-299	3

Question 5 The table shows the travel distance (in kilometres) for a group of commuters. Construct a frequency polygon.

Travel Distance (km)	Frequency
10-19	5
20-29	12
30-39	8
40-49	7
50-59	3

Question 6 (Cumulative Frequency Polygon): The table shows the number of hours spent studying for a math test. Convert the data into a cumulative frequency table and draw the corresponding cumulative frequency polygon.

Study Time	Frequency
1-2	8
3-4	10
5-6	7
7-8	3
9-10	2

Question 7: The following data shows the shoe sizes of a group of athletes. Draw the frequency polygon and describe the distribution.

Shoe size	Frequency
36-38	4
39-41	7
42-44	12
45-47	5
48-50	2

Question 8: The heights (in centimetres9) of students in a biology class are recorded. Draw the frequency polygon and analyze the distribution.

Height (cm)	Frequency
150-159	3
160-169	8
170-179	10
180-189	7
190-199	2

Question 9: Following table shows the amount of rainfall (in millimetres) recorded in a week. Construct a frequency polygon.

Rainfall (mm)	Frequency
0-4	5
5-9	8
10-14	12
15-19	3
20-24	2

Question 10: The ages (in years) of members of a sports club are grouped into these categories. Draw the frequency polygon and describe the distribution.

Age (years)	Frequency
20-24	7
25-29	10
30-34	8
35-39	5
40-44	2

Also Read:

• Frequency Polygon in Statistics
• Cumulative Frequency
• How to create a frequency polygon in Excel?

Frequency Polygon – FAQs

What is another name for a frequency polygon?

A frequency polygon is also referred to as a cumulative frequency polygon. This type of graph is created on graph paper by plotting either the lower or upper limits of class intervals on the x-axis and their corresponding cumulative frequencies on the y-axis.

What distinguishes a histogram from a frequency polygon?

A frequency polygon is represented by a line connecting points, forming a curve. In contrast, a histogram displays data using adjoining rectangular bars with no gaps between them.

Is it possible to create a frequency polygon without a histogram?

Yes, a frequency polygon can be drawn independently of a histogram. This is done by plotting the midpoints of each class interval, with the height of each point determined similarly to a histogram. These points are then connected to form the frequency polygon.

How do you construct a frequency polygon?

• Determine the midpoint of each class interval.
• Plot the class frequency at each midpoint.
• Connect these points with straight lines to form the polygon.

What are the components of a frequency polygon?

A frequency polygon is a curve plotted on a graph with the x-axis representing data values and the y-axis representing the frequency of each category. It can be used as an alternative to a histogram.