Frequency Polygons in Statistics
Last Updated : 23 Jul, 2025
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Frequency Polygons in Statistics: A frequency polygon is a type of line graph
where the frequencies of classes are plotted against their midpoints. This graphical
representation closely resembles a histogram and is typically used for comparing
data sets or showing cumulative frequency distributions. It uses a line graph to
represent quantitative data.
Frequency polygons are one of the great methods to represent statistical data so
that it can be read easily. In statistics, we deal with lots of data, and reading it
quickly is necessary for solving statistical problems effectively.
Frequency polygons help us to achieve the same result. In this article, we will learn
about frequency polygons, their formula, examples, and others in detail.
Frequency Polygons
Table of Content
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What is a Frequency Polygon in Statistics?
Frequency Polygon Graph
Cumulative Frequency Polygon
How to Draw Frequency Polygon?
Histogram and Frequency Polygons
Frequency Polygons Examples
Frequency Polygons Are Used For - Applications
What is a Frequency Polygon in Statistics?
A visual representation of the frequency distribution of continuous data is the
frequency polygon. Karl Pearson, an English statistician, made the initial
presentation of it in the late 19th century. Based on the research of earlier statisticians
like Francis Galton and Adolphe Quetelet, he put forth the idea. Its value comes from
its capacity to graphically portray data, making understanding and analysis easier.
They, therefore, serve as an important tool in statistics, assisting researchers in
finding patterns and trends in huge data sets.
They are useful for showing data sets with numerous observations or values, which
is one of their key features. The polygon represents the data distribution succinctly
and clearly by splitting the data into equal intervals and charting the frequency of
each container. As they take the shape of points that sit beyond the typical range of
the distribution, they are also useful in finding outliers or abnormalities in the data
set. Once more, this facilitates identifying and examining odd or unexpected data
points.
Frequency Polygon Definition
Frequency polygons are a type of graphical data distribution that aids in recognizing
the data by giving it a particular form. Although frequency polygons and histograms
are quite similar, they are more effective when comparing two or more sets of data.
As a line graph, the graph primarily shows data from the cumulative frequency
distribution.
Formula to Find Midpoint of Frequency Polygons
If you want to plot a frequency polygon graph, you must figure out the midpoint or
class mark for each of the class intervals. The following is the formula to achieve
that:
Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2
Frequency Polygon Graph
Frequency Polygon Graph is the graphical representation of the data given in the
form of class interval and frequency. Let's consider an example for better
understanding:
Example: Plot the graph of the Frequency Polygon for the following data
which represents the number of goals scored in a match in a league
throughout the season:
Goals Scored Frequency
0
3
1
7
2
8
3
12
Goals Scored Frequency
4
2
5
5
Answer:
For the given data, we can plot the frequency polygon by representing the goals
scored on the vertical axis and frequency on the horizontal axis, as follows:
Cumulative Frequency Polygon
In statistics, the cumulative frequencies of a dataset are shown graphically using
cumulative frequency polygons. The dots at the upper-class borders are plotted
against the corresponding cumulative frequencies to create the graph. The graph
illustrates the visual accumulation of data across time or intervals. The polygon
created by the line connecting the points aids in the visualization of the data's trends
and patterns.
It helps analysts grasp the structure and central tendency of cumulative frequency
distributions by clearly showing them. Researchers and analysts may get important
insights into the distribution of the data and make wise judgments based on the trends
seen by employing cumulative frequency polygons.
How to Draw Frequency Polygon?
Following the procedures listed below, construct a histogram before beginning to
draw frequency polygons:
 Step 1: First, select the class interval and then indicate the values on the axes.
 Step 2: Label the horizontal axes with the midpoint of each interval.
 Step 3: Label the vertical axes with the class frequency.
 Step 4: Mark a point at the height in the centre of each class interval according
to the frequency of each class interval.
 Step 5: Use the line segment to join these spots.
 Step 6: The representation that was obtained is a frequency polygon.
Read More: Bar Graphs and Histograms
Histogram and Frequency Polygons
The difference between the frequency polygon and histogram is discussed in the
table below:
Frequency Polygons
Histogram
A frequency polygon graph allows for
more visually accurate data comparison.
In a histogram graph, data comparison
is not visually pleasing.
The midpoint of the frequencies is utilized
in a frequency polygon graph.
The frequencies in a histogram are
uniformly distributed throughout the
class intervals.
The data of a certain class interval is
represented by the correct points in a
frequency polygon graph.
The height of the bars in a histogram
simply represents the amount of data.
A line segment used to represent a curve is
called a frequency polygon graph.
A histogram is a graph that presents
data as a series of uninterrupted,
rectangular bars.
Frequency Polygons Examples
Example 1: The frequency data below was used to create a frequency polygon.
Class Interval Frequency
49.5-59.5
5
59.5-69.5
10
69.5-79.5
30
Class Interval Frequency
79.5-89.5
40
89.5-99.5
15
Solution:
By first determining the classmark using the equation Classmark = (Upper Limit +
Lower Limit) / 2, we may build a frequency polygon without a histogram.
Additionally, by combining the previous and next frequencies, we can get the
cumulative frequency of each class interval.
Class Interval:
 (59.5 + 49.5/2) = 54.5
 (69.5 + 59.5/2) = 64.5
 (79.5 + 69.5/2) = 74.5
 (89.5 + 79.5/2) = 84.5
 (99.5 + 89.5/2) = 94.5
Class Interval Lower Bound
Upper Bound
Classmark
Frequency
49.5-59.5
49.5
59.5
54.5
5
59.5-69.5
59.5
69.5
64.5
10
69.5-79.5
69.5
79.5
74.5
30
79.5-89.5
79.5
89.5
84.5
40
89.5-99.5
89.5
99.5
94.5
15
We note the before and after classmarks as well while plotting the graph. The
before in this instance is 44.5, while the after is 104.5. The scores are shown on the
x-axis, while the frequency is indicated on the y-axis. Consequently, the frequency
polygons graph will seem like follows:
Example 2: Assume that a class of 65 students' weights are distributed as
follows: 15 - 25, 25 - 35, 35 - 45, and 45 - 55. How many grade points would
there be for each weight category?
Solution:
Formula used to get the classmark for a Frequency Polygon Graph is:
Classmark = (Upper Limit + Lower Limit) / 2
Hence,
 Class interval 15-25 = (15 + 25)/2 = 20
 Class interval 25-35 = (25 + 35)/2 = 30
 Class interval 35-45 = (35 + 45)/2 = 40
 Class interval 45-55 = (45 + 55)/2 = 50
Frequency Polygons Are Used For - Applications
Here are some of the key applications of frequency polygons:
1. Comparing Distributions: Frequency polygons allow for easy comparison
between different datasets. By overlaying multiple polygons on the same graph,
you can visually assess differences in shapes, spreads, and central tendencies of
the distributions.
2. Identifying Skewness and Kurtosis: The shape of a frequency polygon can
provide insights into the skewness (asymmetry) and kurtosis (peakedness) of
the data distribution. This is useful for understanding the nature of the dataset,
whether it is normally distributed, positively skewed, negatively skewed, etc.
3. Analyzing Trends: In time series data, frequency polygons can help in
visualizing trends over time. This makes it easier to identify patterns such as
seasonal variations or cyclic changes.
4. Educational Tool: In teaching statistics, frequency polygons serve as a clearer
alternative to histograms for some students. They can help in understanding
how frequencies change between intervals.
5. Cumulative Frequency Analysis: Although not a cumulative frequency graph
itself, frequency polygons can be adapted to show cumulative frequencies,
providing a visual representation of data accumulation over intervals.
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How to Make a Frequency Polygon in Excel?
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Summary - Frequency Polygons in Statistics
Frequency polygons are graphical tools used in statistics to represent the
distribution of data points. They are essentially line graphs that plot class
frequencies against class midpoints, making them very similar to histograms but
offering a clearer visual comparison between multiple datasets. These graphs are
particularly valuable for analyzing trends, comparing distributions, and identifying
characteristics such as skewness and kurtosis in the data. By providing a visual
summary of data, frequency polygons serve as an effective educational tool to
enhance understanding of statistical distributions and are useful in diverse
applications from business analytics to academic research.
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