How to Draw a Frequency Polygon with Formula and Solved Examples
The concept of frequency polygon plays a key role in mathematics and is widely used in statistics to represent data visually for both classroom learning and exams. Understanding frequency polygons makes interpreting, comparing, and analyzing grouped data much easier for students.
What Is Frequency Polygon?
A frequency polygon is a special type of line graph that shows the distribution of frequency for a set of grouped or ungrouped data by joining points at the class mid-values with straight lines. You’ll find this concept applied in areas such as statistics, graphical data representation, and even in competitive exam data analysis.
Key Formula for Frequency Polygon
Here’s the standard formula for finding the class mark (midpoint), which you use when plotting a frequency polygon:
\( \text{Class Mark} = \frac{\text{Upper Limit} + \text{Lower Limit}}{2} \)
Cross-Disciplinary Usage
Frequency polygons are useful in Maths, but also play an important role in Physics, Computer Science, Economics, and social sciences. For students preparing for board exams or Olympiads, visualizing data using frequency polygons helps in data-driven reasoning and quick pattern recognition. You might encounter questions involving frequency polygons in exams like CBSE, ICSE, and IGCSE as well.
Step-by-Step Illustration
- Prepare the frequency distribution table.
List class intervals and corresponding frequencies. Example:Class Interval Frequency 10–20 5 20–30 8 30–40 12 40–50 7
- Find the class marks for each interval.
Calculate class marks:
10–20: (10+20)/2 = 15
20–30: (20+30)/2 = 25
30–40: (30+40)/2 = 35
40–50: (40+50)/2 = 45
- Plot the class marks (x-axis) against frequencies (y-axis).
On graph paper, label the x-axis as "class mark" and y-axis as "frequency." Then, plot the points: (15,5), (25,8), (35,12), (45,7).
- Anchor the polygon at the baseline.
Add one class mark before the first and after the last interval with frequency zero, i.e., (5,0) and (55,0), to make the polygon closed at both ends.
- Join the points with straight lines.
Connect all these points in order using straight lines to complete your frequency polygon.
Speed Trick for Frequency Polygon Drawing
Many students get confused between histograms and frequency polygons. Here’s a quick guide to avoid mistakes and draw a frequency polygon efficiently:
- Don’t draw bars — just plot the midpoints of each class interval directly.
- Always use the class mark, not just class boundaries, for plotting.
- Start and end your polygon at the baseline for clear enclosure.
Tip: When two data sets have the same class intervals, you can plot frequency polygons for both in the same graph and easily compare their shapes, peaks (modes), and spreads.
Example Problem: Constructing a Frequency Polygon
Let’s solve a common board-style question:
Question: The marks scored by 30 students in a test are grouped as shown below. Draw the frequency polygon.
| Marks | No. of Students |
|---|---|
| 0–10 | 4 |
| 10–20 | 7 |
| 20–30 | 12 |
| 30–40 | 5 |
| 40–50 | 2 |
Solution Steps:
1. Find class marks:0–10: (0+10)/2 = 5
10–20: (10+20)/2 = 15
20–30: (20+30)/2 = 25
30–40: (30+40)/2 = 35
40–50: (40+50)/2 = 45
2. Plot the points (marks, frequency):
(5,4), (15,7), (25,12), (35,5), (45,2).
3. Add extra points at both ends:
Add (–5,0) before and (55,0) after to anchor.
4. Join points in order by straight lines.
Your frequency polygon is complete.
Types and Features of Frequency Polygon
| Type | Feature |
|---|---|
| Simple Frequency Polygon | Created from a single frequency distribution |
| Cumulative Frequency Polygon (Ogive) | Plots cumulative frequency (increasing total) |
| Multiple Frequency Polygons | Compare two or more data sets on the same graph |
- Shows the shape and spread of data clearly.
- Helps to compare distributions visually.
- The peak point shows the mode (most frequent value).
Advantages and Disadvantages
| Advantages | Disadvantages |
|---|---|
| Easy comparison Visualizes large datasets Identifies mode and trends |
Approximate for small datasets Less precise than bar graphs for exact values Needs careful plotting of midpoints |
Try These Yourself
- Draw a frequency polygon for the data: 5, 8, 12, 7 (class intervals: 10–20, 20–30, 30–40, 40–50).
- Given class intervals 0–5, 5–10, 10–15 with frequencies 2, 6, 7, find and plot their mid-values.
- Compare a frequency polygon with a histogram for the same data. What differences do you see?
- Find the mode from a drawn frequency polygon curve.
Frequent Errors and Misunderstandings
- Plotting frequencies at upper or lower limits instead of class marks.
- Forgetting to anchor the polygon ends at frequency zero.
- Confusing frequency polygons with line graphs or histograms.
- Not using equal class intervals – this makes the polygon misleading.
Relation to Other Concepts
The idea of frequency polygon is closely connected with histogram, cumulative frequency distribution (ogive), and line graph. Mastering frequency polygons helps in understanding the broader topic of representing and interpreting statistical data.
Classroom Tip
A simple way to remember frequency polygons: "Midpoints meet, lines connect, ends touch baseline." Vedantu’s teachers regularly explain this using colorful class activities so students can visualize the concept better.
We explored frequency polygons—from the definition, formula, step-by-step construction, examples, common mistakes, and their connection to other data representations. Continue practicing with Vedantu to become confident in using frequency polygons for any exam or real-life data comparison.
Explore more about: Histogram, Cumulative Frequency Distribution, Line Graph, Statistics.
FAQs on Frequency Polygons in Statistics Explained Clearly
1. What is a frequency polygon in statistics?
A frequency polygon is a line graph that shows the distribution of data by joining points representing class frequencies at their midpoints. It is constructed by plotting:
- The class midpoints on the horizontal axis
- The corresponding frequencies on the vertical axis
2. How do you construct a frequency polygon step by step?
To construct a frequency polygon, plot class midpoints against their frequencies and join them with straight lines. Follow these steps:
- Prepare a grouped frequency table
- Calculate each class midpoint using: midpoint = (lower limit + upper limit) / 2
- Draw axes and mark midpoints on the x-axis
- Plot corresponding frequencies on the y-axis
- Join the plotted points with straight lines
- Optionally, connect the first and last points to the x-axis to close the polygon
3. What is the formula for calculating the class midpoint in a frequency polygon?
The formula for the class midpoint is (Lower limit + Upper limit) ÷ 2. This midpoint represents the center of each class interval. For example, for the class 10–20:
- Midpoint = (10 + 20) ÷ 2 = 15
4. What is the difference between a frequency polygon and a histogram?
The main difference is that a histogram uses bars, while a frequency polygon uses connected line segments. Key differences include:
- Histogram: Rectangular bars with no gaps
- Frequency polygon: Points joined by straight lines
- Histogram: Better for showing individual class frequencies
- Frequency polygon: Better for comparing multiple distributions
5. Why do we close a frequency polygon at both ends?
A frequency polygon is closed at both ends to clearly show the complete data distribution. This is done by:
- Adding an imaginary class before the first class and after the last class
- Assigning them a frequency of 0
- Joining these points to the first and last plotted points
6. Can you give an example of a frequency polygon with numbers?
A frequency polygon can be drawn by plotting midpoints against frequencies and joining them. Example frequency table:
- 0–10: 5
- 10–20: 8
- 20–30: 6
Step 2: Plot points (5,5), (15,8), (25,6)
Step 3: Join these points with straight lines
The resulting line graph is the frequency polygon.
7. What are the advantages of a frequency polygon?
A frequency polygon is useful because it clearly shows the shape of a distribution and allows easy comparison. Its advantages include:
- Displays trends and patterns clearly
- Useful for comparing two or more datasets on the same axes
- Takes less space than multiple histograms
- Shows the overall distribution shape (symmetry or skewness)
8. When should you use a frequency polygon instead of a histogram?
Use a frequency polygon when you want to compare multiple grouped datasets or show overall trends clearly. It is preferred when:
- Comparing two or more frequency distributions
- Highlighting the shape of data (normal, skewed)
- A cleaner, less crowded graph is needed
9. What does the shape of a frequency polygon tell you?
The shape of a frequency polygon shows the distribution pattern of the data. It can indicate:
- Symmetrical distribution (balanced shape)
- Positively skewed distribution (tail on the right)
- Negatively skewed distribution (tail on the left)
- Presence of peaks (modes)
10. What are common mistakes when drawing a frequency polygon?
Common mistakes in drawing a frequency polygon include plotting class limits instead of midpoints and not closing the graph properly. Avoid these errors:
- Using lower or upper class limits instead of midpoints
- Incorrect calculation of midpoints
- Not maintaining equal class widths
- Forgetting to add zero-frequency classes at the ends
