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A graphical representation of data is an easy way to understand the various features of a data set. Frequency polygons are a type of graphical representation of data distribution that helps in understanding its shape. Frequency polygons are very similar to histograms but are helpful and useful while comparing two or more data.

## What is a Frequency Polygon Graph?

A frequency polygon is a visual representation of a distribution. It is used to understand the shape of a distribution. The frequency polygon can serve as an alternative to a histogram. Both visual representations perfectly reflect the shape of a distribution. However, unlike the histogram, the frequency polygon can be easily utilized to compare multiple distributions on the same graph. In some cases, a histogram and a frequency polygon can be used simultaneously to get a more accurate picture of the distribution shape.

A frequency polygon 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.

## Difference Between Frequency Polygon Graph and Histogram

Following are the differences between a frequency polygon graph and a histogram.

Frequency Polygon | Histogram |

A frequency polygon is a line graph | A histogram is a collection of adjacent rectangles |

The frequency polygon is multi-dimensional | A histogram is a two-dimensional figure |

Several frequency distributions can be plotted on the same axis as the frequency polygon | In the case of a histogram, we must have a separate graph for each distribution |

The frequency polygon is a continuous curve and it is easy to determine the scope and rate of change estimates | This is not possible in the case of a histogram |

## Advantages of Frequency Polygon Graph

Following are the advantages of a frequency polygon graph over a histogram

- The frequency polygons of several distributions may be plotted on the same graph, thereby making certain comparisons possible, whereas histograms cannot be usually employed in the same way. To compare histograms we must have a separate graph for each distribution. Because of this limitation for purposes of making a graphic comparison of frequency distributions, frequency polygons are preferred.
- The frequency polygon is simpler than its histogram counterpart.
- It sketches an outline of the data pattern more clearly.
- The polygon becomes increasingly smooth and curve-like as we increase the number of classes and the number of observations.

## How to Construct Frequency Polygon Graph

The graph of a frequency polygon is drawn on a Cartesian system. Like any other graph, the $x$-axis represents the value in a dataset and the $y$-axis shows the number of occurrences(frequency) of each category. While plotting a frequency polygon graph, the most important aspect is the mid-point which is called the class mark.

The frequency polygon graph can be drawn with or without a histogram.

### Constructing Frequency Polygon Graph Using Histogram

Following are the steps to drawing a frequency polygon graph using a histogram:

**Step 1:** Draw a histogram for the given data set. For drawing with a histogram, draw rectangular bars against the class intervals with the height(or length) equal to its frequency.

**Step 2:** Join the midpoints of the bars to get the frequency polygons.

### Examples

Let’s consider an example to understand how a frequency polygon is constructed using a histogram.

The weekly earnings of 100 workers in a factory are given below. Draw a frequency polygon of the given frequency distribution.

For the given frequency distribution, construct a histogram. To understand the steps in the construction of a histogram, see here.

Locate the classmarks of the class intervals. The classmarks are the midpoints of the class intervals.

$\text{Classmark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$

Now, starting from the origin(0), draw straight lines joining the midpoints of the top of consecutive bars of the histogram.

Remove(Erase) the histogram.

The straight line curve so obtained is the frequency polygon graph.

**Note:** Do not join the points by a free-hand curve. Join the points by straight lines.

### Constructing Frequency Polygon Graph Without Using Histogram

Following are the steps to drawing a frequency polygon graph without a histogram:

**Step 1:** On the horizontal axis ($x$-axis) mark the class intervals for each class while on the vertical axis ($y$-axis) mark the points of a scale based on the minimum and maximum $y$-values with appropriate intervals.

**Step 2:** Calculate the midpoint of each of the class intervals which is commonly called classmarks using the formula $\text{Classmark} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$.

**Step 3:** Mark the classmarks so obtained on the $x$-axis.

**Step 4:** Plot the frequency of each class interval against its class mark.

**Step 5:** Join the points with a line segment similar to a line graph.

**Step 6:** The curve that is obtained by this line segment is the frequency polygon.

### Examples

Let’s consider an example to understand how a frequency polygon is constructed without using a histogram.

The weekly earnings of 100 workers in a factory are given below. Draw a frequency polygon of the given frequency distribution.

Calculate the classmarks of the class intervals.

For $1000 – 3000$, classmark = $\frac{1000 + 3000}{2} = 2000$

For $3000 – 5000$, classmark = $\frac{3000 + 5000}{2} = 4000$

For $5000 – 7000$, classmark = $\frac{5000 + 7000}{2} = 6000$

For $7000 – 9000$, classmark = $\frac{7000 + 9000}{2} = 8000$

Locate the corresponding $y$-coordinates for each of the classmarks. The $y$-coordinates are the respective frequencies of the class intervals.

The points are $\left(2000, 30 \right)$, $\left(4000, 40 \right)$, $\left(6000, 20 \right)$, and $\left(8000, 10 \right)$.

Extend the line on the left by joining the points $\left(2000, 30 \right)$ and $\left(0, 0 \right)$. Similarly, extend the line on the right by joining the points $\left(8000, 10 \right)$ and $\left(10000, 0 \right)$.

## Points to Remember While Drawing a Frequency Polygon Graph

Following are the points that one should keep in mind while drawing a frequency polygon graph

## Practice Problems

**1.** The frequency polygon of a frequency distribution is shown below.

Answer the following about the distribution from the histogram.

- What is the frequency of the class interval whose class mark is 15?
- What is the class interval whose class mark is 45?
- Construct a frequency table for the distribution.

**2.** The following frequency polygon displays the weekly incomes of laborers of a factory.

Answer the following.

- Find the class interval whose frequency is 25.
- How many labourers have a weekly income of at least ₹500 but not more than ₹700?
- What is the range of weekly income of the largest number of labourers?
- Prepare the frequency distribution table.

**3.** In a batch of 400 students, the height of students is given in the following table. Represent it through a frequency polygon.

**4.** In a city, the weekly observations made in a study on the cost of living index are given in the following table: Draw a frequency polygon for the data above (without constructing a histogram).

## FAQs

### What is a frequency polygon graph?

A frequency polygon is a line graph of class frequency plotted against the class midpoint. It can be obtained by joining the midpoints of the tops of the rectangles in the histogram.

### What is a frequency polygon used for?

Frequency polygons are a graphical device for understanding the shapes of distributions. They serve the same purpose as histograms but are especially helpful for comparing sets of data.

### How do you read a frequency polygon?

The area under the frequency polygon is the same as the area under the histogram and is, therefore, equal to the frequency values that would be displayed in a distribution table. The frequency polygon also shows the shape of the distribution of the data, and in this case, it resembles a bell curve.

### Where does a frequency polygon start?

You can construct a frequency polygon by joining the midpoints of the tops of the bars. Frequency polygons are particularly useful for comparing different sets of data on the same diagram.

## Conclusion

A frequency polygon is a visual representation of a distribution. It is used to understand the shape of a distribution. It is drawn by joining the points with $x$-coordinate as the midpoints of the class interval and $y$-coordinate as their corresponding frequencies by straight lines.

## Recommended Reading

- Data Collection & Organization(Methods, Tools, Types & Techniques)
- Discrete and Continuous Data(Meaning, Differences & Examples)
- How to Make a Pictograph – Definition, Advantages & Examples
- How to Make a Bar Graph – Definition, Advantages & Examples
- How to Make Double Bar Graph – Definition, Advantages & Examples
- What is Pie Chart? (Definition, Formula & Examples)
- What is a Histogram – Definition, Advantages & Examples
- How to Make Stacked Bar Graph – Definition, Advantages & Examples