A frequency polygon is a type of line graph used in statistics to show how often data values occur. It is drawn by plotting the frequencies of class intervals against their midpoints and then joining these points with straight lines.
It helps in understanding the shape and trend of the data and is useful for comparing different data sets. A frequency polygon can be drawn with or without a histogram and is an effective way to represent quantitative data clearly.
Midpoint (Class Mark) Formula
If you want to plot a frequency polygon graph, you must figure out the midpoint or class mark for each of the class intervals.
Class Mark (Midpoint) = (Upper Limit + Lower Limit) / 2
Steps to Draw a Frequency Polygon
A frequency polygon is drawn on a set of x- and y-axes, just like a regular graph. The x-axis represents the values of the data (class intervals or class marks), while the y-axis shows the frequency (number of observations). The key element in drawing a frequency polygon is the midpoint of each class interval, known as the class mark. A frequency polygon can be drawn either with the help of a histogram or without it.
Below are the clear and easy steps to draw a frequency polygon without using a histogram:
Step 1: Draw the x-axis and y-axis on graph paper. Mark the class intervals along the x-axis and the frequencies along the y-axis.
Step 2: Find the midpoint (class mark) of each class interval using the formula:
Class mark = (Upper limit + Lower limit) ÷ 2
Step 3: Mark these class marks on the x-axis.
Step 4: Plot points on the graph by taking class marks on the x-axis and their corresponding frequencies on the y-axis. Each frequency must be plotted directly above its class mark.
Step 5: Join all the plotted points with straight line segments in order.
Step 6: The line formed by joining these points is called the frequency polygon.
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 |
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 (Ogive)
A cumulative frequency polygon, also called an Ogive, is a line graph used in statistics to represent the cumulative frequencies of a data set. In this graph, the class boundaries are plotted on the x-axis and the cumulative frequencies are plotted on the y-axis. The points are joined with straight lines to show how the data values accumulate.
Ogives help students understand the shape and pattern of frequency distributions and make it easy to compare two different frequency distributions.
There are two types of ogives:
• Less than ogive – It shows the cumulative frequency of observations less than or equal to a given value.
• More than (greater than) ogive – It shows the cumulative frequency of observations greater than or equal to a given value.
In the above diagram, the red line shows the less than ogive and the green line shows the more than ogive.
Read More: Bar Graphs and Histograms
Histogram vs 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. |
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Frequency Polygons Solved 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 |
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