Exploring Frequency Polygons

Frequency polygons are a fundamental graphical tool in statistics. They represent the frequency distribution of continuous data. In the late 19th century, the frequency polygon was first introduced by the English statistician Karl Pearson.

A frequency polygon shows the distribution of a dataset across different class intervals. It helps to simplify and organize large datasets for easier understanding. It represents data using a line graph that connects the midpoints of the other intervals.

For example,
If a teacher records the marks of students and groups them into class intervals:

 

Here,
First, find the midpoints of each interval
 

• 0 - 10 → midpoint = 5
• 10 - 20 → midpoint = 15
• 20 - 30 → midpoint = 25
• 30 - 40 → midpoint = 35
• 40 - 50 → midpoint = 45

Now, plot each midpoint against its frequency

\((5, 3), (15, 7), (25, 12), (35,9), (45, 4)\)
 

Connect the points with straight lines. The resulting line graph is the frequency polygon, showing how students marks are distributed across the intervals.
 

Now, let us learn how to draw a frequency polygon. Like a regular graph, a frequency polygon has two axes, one is the x-axis, and the next is the y-axis. On both the x-axis and y-axis, the frequency polygon’s curve is depicted. In the x-axis, the various values in a dataset are represented, while the number of occurrences in each category is represented by the y-axis.

While drawing a frequency polygon, the most important aspect to consider is the class intervals. We can depict a frequency polygon with or without a histogram. Histograms are rectangular bars that represent the information or values in a dataset. So, if we draw a frequency polygon with a histogram, we first draw the rectangular bars for each class interval.

Later, connect the bar’s midpoints to form a frequency polygon. Below are some steps, that we should follow while drawing a frequency polygon without a histogram:

 
Step 1: Mark the class intervals on the x-axis and plot the frequencies on the y-axis. 

Step 2: Calculate the midpoints of each class interval, or the class marks.

Step 3: Plot the class marks on the x-axis. 

Step 4: Plot the frequency based on each class mark on the height. Be sure to mark it on the same class mark, not at the higher or lower limit of the interval.  

Step 5: After marking the points, connect them with line segments. These are similar to a line graph. 

Step 6: After this, we get a curve known as the frequency polygon. 

When drawing a frequency polygon, we need a critical value for each class interval, the midpoint, also called the class mark. This midpoint tells us the middle value of a class interval and helps us place each point correctly on the graph.


Formula for midpoint

\(\text{Midpoint (Class mark)} = \frac{\text{Upper limit} + \text{Lower limit}}{2} \)

For example,

If we have a class interval, 20-30
Lower limit = 20
Upper limit = 30
By using the formula,

\(\text{Midpoint} = \frac{20 + 30}{2} = \frac{50}{2} = 25 \)


So, 25 is the midpoint of the interval 20 - 30.
This midpoint represents the entire class interval on the frequency polygon.

Data presented in the form of class intervals and frequencies is graphically represented by a frequency polygon graph. It allows us to analyze and compare large datasets while identifying patterns and trends in data distribution. For a better understanding, we can consider an example. Here is a frequency table that shows the goals scored by students in a match: 

For the above frequency table, we can plot the frequency polygon by marking the frequency on the x-axis and the goals scored on the y-axis. 

The cumulative frequency polygon depicts the cumulative frequencies of a given dataset. In this graph, the dots are plotted at the upper-class borders against the corresponding cumulative frequencies. After that, the dots are connected by a line and result in a cumulative frequency polygon. The accumulation of data over time or intervals is illustrated by these graphs. Now, let us consider an example. Here is a frequency table showing the marks scored by students for the English exam:

For the above frequency table, we can plot the cumulative frequency polygon by marking the frequency on the y-axis and the marks scored on the x-axis. When we calculate the cumulative frequency, we add up the 

frequencies \( ( 3 + 5 = 8)\).   

A frequency polygon is a graphical representation of data that is represented in the form of class intervals and frequencies. It is similar to histograms, but a frequency polygon uses line segments to join the midpoints of each class interval. The main differences between these two graphs are listed below: 

• A frequency polygon is a curve represented by a line segment, while a histogram uses rectangular bars with no spaces between them to depict the data distribution. 
  
   
• The frequency polygons use the midpoints of each class interval and connect the points to form a polygon. In contrast, the frequencies are uniformly distributed over the class intervals of a histogram.
  
   
• The data of specific class intervals are represented by the accurate points in a frequency polygon. The height of the bars in a histogram represents the frequency or quantity of data. 
  
   
• In a frequency polygon, the visual comparison of data is more accurate and easier. Whereas, comparing data is not visually appealing, it is more accurate and easier. Whereas, the data comparison in a histogram is not visually pleasing.

A frequency polygon is drawn using the x-axis and y-axis, just like a regular graph.

• The x-axis represents the values or class intervals from the dataset.
• The y-axis represents the frequencies, that are the number of occurrences.

The most important part of creating a frequency polygon is finding the midpoints of each class interval. A frequency polygon can be drawn with or without a histogram. When drawn with a histogram, you simply connect the midpoints at the top of each bar.

Here, we focus on drawing a frequency polygon without a histogram. 

Step 1: Write the class intervals on the x-axis, and mark the corresponding frequencies on the y-axis.

Step 2: Calculate the midpoint for each interval using,
\(\text{Midpoint} = \frac{\text{Upper limit} + \text{Lower limit}}{2} \)

Step 3: Plot these midpoints along the x-axis.

Step 4: For each midpoint, plot its frequency directly above it on the y-axis.

Step 5: Connect all the plotted points using straight line segments, just like drawing a line graph.

Step 6: The connected line forms the frequency polygon, representing the frequency distribution clearly and neatly.

Frequency polygons are a complex mathematical concept. In this section, we will discuss some important tips and tricks to master frequency polygons.

• Include an additional class interval with zero frequency on both sides. This helps the polygon look neat and complete.
  
   
• Use graph paper and place each point exactly according to the class marks and their frequencies. Accurate plotting prevents the graph from becoming uneven or misleading.
  
   
• Connect the plotted points with a ruler to keep the polygon tidy, clear, and easy to read.
  
   
• Try drawing multiple frequency polygons on the same graph. This is a great way to compare patterns in different datasets, such as scores, temperatures, or sales.
  
   
• Peaks indicate the highest frequency, while rising and falling slopes reveal trends in the data.
  
   
• Children should practice plotting class marks and frequencies on graph paper with precision, while parents and teachers can guide them to avoid mistakes and keep the graph neat.
  
   
• Teachers can encourage children to use a ruler or straightedge to join the plotted points. This makes the frequency polygon clean, clear, and easy to read.
  
   
• Parents and teachers can help children draw two or more frequency polygons on the same graph. This builds understanding by comparing patterns, such as test performance or daily habits.

A frequency polygon is a graphical representation of data using line segments, which form a curve. In various fields, the frequency polygons are commonly used to interpret data and identify trends and patterns. Here are some of the real-world applications of frequency polygons: 
 

• To evaluate and analyze the performance of students in a school or college, educational institutions use frequency polygons. For example, teachers can figure out the marks scored by students for different subjects to understand the performance of each student. 
   
• Business and companies can track their sales and profit by using frequency polygons. For instance, companies can track the number of items sold over a time period to figure out the sales performance. 
   
• In the field of medical research and healthcare, medical professionals can use frequency polygons to keep a record to analyze the data of patients, their disease, and recovery rates. 
   
• Weather forecasters use the frequency polygons to track the temperature or weather variations over time. For instance, they can plot the total number of rainfalls that happened in a year to observe the climate changes.
   
•  Economists use frequency polygons to analyze income distribution, inflation rates, or stock market trends over time. It helps in comparing different economic periods or identifying growth patterns in financial data.