Frequency Polygons
Frequency polygons are a graphical representation of data distribution that helps in understanding the data through a specific shape. Frequency polygons are very similar to histograms but are helpful and useful while comparing two or more data. The graph mainly showcases cumulative frequency distribution data in the form of a line graph. Let us learn about the frequency polygons graph, the steps in creating a graph, and solve a few examples to understand the concept better.
Definition of Frequency Polygons
Frequency Polygons can be defined as a form of a graph that interprets information or data that is widely used in statistics. This visual form of data representation helps in depicting the shape and trend of the data in an organized and systematic manner. Frequency polygons through the shape of the graph depict the number of occurrence of class intervals. This type of graph is usually drawn with a histogram but can be drawn without a histogram as well. While a histogram is a graph with rectangular bars without spaces, a frequency polygon graph is a line graph that represents cumulative frequency distribution data. Frequency polygons look like the image below:
Steps to Construct Frequency Polygons
The curve in a frequency polygon is drawn on an xaxis and yaxis. As a regular graph, the xaxis represents the value in a dataset and the yaxis shows the number of occurrences of each category. While plotting a frequency polygon graph, the most important aspect is the midpoint which is called the class interval or class marks. The frequency polygon curve can be drawn with or without a histogram. For drawing with a histogram, we first draw rectangular bars against the class intervals and join the midpoints of the bars to get the frequency polygons. Here are the steps to drawing a frequency polygon graph without a histogram:
 Step 1: Mark the class intervals for each class on an xaxis while we plot the curve on the yaxis.
 Step 2: Calculate the midpoint of each of the class intervals which is the classmarks. (The formula is mentioned in the next section)
 Step 3: Once the classmarks are obtained, mark them on the xaxis.
 Step 4: Since the height always depicts the frequency, plot the frequency according to each class mark. It should be plotted against the classmark itself and not on the upper or lower limit.
 Step 5: Once the points are marked, join them with a line segment similar to a line graph.
 Step 6: The curve that is obtained by this line segment is the frequency polygon.
Formula to Find the Frequency Polygons Midpoint
While plotting a frequency polygon graph we require to calculate the midpoint or the classmark for each of the class intervals. The formula to do so is:
Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2
Difference Between Frequency Polygons and Histogram
Even though a frequency polygon graph is similar to a histogram and can be plotted with or without a histogram, the two graphs are yet different from each other. The two graphs have their own unique properties that show the difference visually. The differences are:
Frequency Polygons  Histograms 
A frequency polygon graph is a curve that is depicted by a line segment.  A histogram is a graph that depicts data through rectangularshaped bars with no spaces between them. 
In a frequency polygon graph, the midpoint of the frequencies is used.  In a histogram, the frequencies are evenly spread over the class intervals. 
The accurate points in a frequency polygon graph represent the data of the particular class interval.  The height of the bars in a histogram only depicts the quantity of the data. 
Comparison of data is visually more accurate in a frequency polygon graph.  Comparison of data is not visually appealing in a histogram graph. 
Related Topics
Listed below are a few topics that are related to frequency polygons, take a look.
Examples on Frequency Polygons

Example 1: Construct a frequency polygon without a histogram using the data given below.
Test Scores Frequency 49.5  59.5 10 59.5  69.5 3 69.5  79.5 7 79.5  89.5 15 89.5  99.5 5 Solution:
To construct a frequency polygon without a histogram we first find the classmark by using the formula Classmark = (Upper Limit + Lower Limit) / 2. And we will find the cumulative frequency of each class interval as well by adding the next frequency and previous frequency together.
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
Test Scores Frequency Classmark 49.5  59.5 3 54.5 59.5  69.5 5 64.5 69.5  79.5 7 74.5 79.5  89.5 10 84.5 89.5  99.5 15 94.5 While plotting the graph, we also mark the before and after classmark as well. In this case, the before is 44.5 and the after is 104.5. The scores are plotted on the xaxis and the frequency is plotted on the yaxis. Hence, the frequency polygons graph will look like this:

Example 2: In a city, the weekly observations made in a study on the cost of a living index are given in the following table: Draw a frequency polygon for the data below with a histogram.
Cost of Living Index Number of weeks 140  150 2 150  160 8 160  170 14 170  180 20 180  190 10 190  200 6 Total 60 Solution: To plot a frequency polygon with a histogram, we need to follow these steps to construct a histogram:
 The cost of living index is represented on the xaxis.
 The number of weeks is represented on the yaxis.
 Now rectangular bars of widths equal to the class size and the length of the bars corresponding to a frequency of the class interval are drawn.
To calculate the midpoint, we use the formula Classmark = (Upper Limit + Lower Limit) / 2
Classmark = (150 + 140)/2 = 145, (160 + 150)/2 = 155 and so on.
Cost of Living Index Number of weeks Classmark 140  150 2 145 150  160 8 155 160  170 14 165 170  180 20 175 180  190 10 185 190  200 6 195 Total 60 While plotting the graph, we also mark the before and after classmark as well. In this case, the before is 135 and the after is 205. ABCDEFGH represents the given data graphically in form of frequency polygon i.e. those are the midpoints. Hence, the frequency polygons graph will look like this:

Example 3: If the weight range for a class of 45 students is distributed by 35  45, 45  55, 55  65, 65  75. What would be the class marks for each weight range?
Solution:
To calculate the classmark for a frequency polygon graph, we use the formula, Classmark = (Upper Limit + Lower Limit) / 2.
Hence,
Class interval 35  45 = (45 + 35)/2 = 40
Class interval 45  55 = (55 + 45)/2 = 50
Class interval 55  65 = (65 + 55)/2 = 60
Class interval 65  75 = (75 + 65)/2 = 70
FAQs on Frequency Polygons
What is Frequency Polygons?
A frequency polygon is a type of line graph where the class frequency is plotted against the class midpoint and the points are joined by a line segment creating a curve. The curve can be drawn with and without a histogram. A frequency polygon graph helps in depicting the highs and lows of frequency distribution data. To obtain the curve for a frequency polygon, we need to find the classmark or midpoint from the class intervals.
How Do You Construct a Frequency Polygons?
A frequency polygon can be constructed with and without a histogram. The steps to construct a frequency polygon without a histogram are:
 Mark the class intervals for each class on an xaxis while we plot the curve on the yaxis.
 Calculate the midpoint of each of the class intervals which is the classmarks.
 Mark the classmarks on the xaxis.
 Since the height always depicts the frequency, plot the frequency according to each class mark. It should be plotted against the classmark itself and not on the upper or lower limit.
 Once the points are marked, join them with a line segment similar to a line graph.
 The curve that is obtained by this line segment is the frequency polygon.
What is the Difference Between Histogram and Frequency Polygons?
A frequency polygon graph is the improved version of a histogram. A histogram is a bar graph with rectangleshaped bars depicting the data whereas a frequency polygon is a line graph where a curved line depicts the data. A frequency polygon is more widely used when distributive data needs to be compared since in a histogram the comparison will not be clear.
Why Do We Use Frequency Polygons?
Frequency polygons graphs are used in comparing a set of data as it is clear and more readable. These graphs are also widely used for depicting cumulative frequency distribution.
What are the Characteristics of Frequency Polygons?
A frequency polygon graph is considered as a closed dimensional figure of a line segment joining the midpoints of the given class intervals. The graph can either be drawn with a histogram or without a histogram. The first point is on the xaxis where y = 0 and is placed in the middle of the interval which precedes the first class interval.
What is the Similarity Between Frequency Polygons and Line Graphs?
A frequency polygon is a type of line graph where a line segment curves to join the midpoints of all the class intervals. The shape of the curved line helps in providing accurate data. Both a line graph and frequency polygon graph are widely used when data is required to be compared.
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