Frequency Density

Here we will learn about frequency density, including how to calculate it from a grouped frequency table.

There is also a histogram worksheet based on Edexcel, AQA and OCR exam questions, and further guidance on where to go next if you’re still stuck.

What is frequency density?

Frequency density is the frequency per unit for the data in each class. It is calculated by dividing the frequency by the class width ie the difference between the upper limit of the class interval and the lower limit of the class interval.

We use frequency density to plot histograms.

Step-by-step guide: Frequency density formula (coming soon)

To work out frequency density we first need to identify the class width of the interval by subtracting the lower bound from the upper bound of the class interval.

Once we have identified the class width the frequency density can be calculated:

In a histogram:

• the frequency density is the height of the bar
• the frequency of a class interval is equal to the area of the bar
• the total area of the bars is equal to the total frequency

What is frequency density?

How to calculate frequency density

In order to calculate frequency density:

1. Identify the upper and lower bounds of the class interval.
2. Find the class width of the class interval by finding the difference of the upper and lower bounds.
3. Divide the frequency of the class interval by its class width.

How to calculate frequency density

Related lessons on histograms

Frequency density is part of our series of lessons to support revision on histograms. You may find it helpful to start with the main histograms lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Histogram
• Frequency density formula 

Frequency density examples

Example 1: calculating frequency density

The table shows information about the ages of people at a cinema.

Calculate the frequency densities.

1. Identify the upper and lower bounds of the class interval.

2Find the class width of the class interval by finding the difference of the upper and lower bounds.

3Divide the frequency of the class interval by its class width.

Example 2: calculating frequency density

The table shows information about the heights of pupils in a mathematics class.

Calculate the frequency densities.

Identify the upper and lower bounds of the class interval.

Find the class width of the class interval by finding the difference of the upper and lower bounds.

Divide the frequency of the class interval by its class width.

Example 3: calculating frequency density

The table shows information about the mass of fish in a lake.

Calculate the frequency densities.

Identify the upper and lower bounds of the class interval.

Find the class width of the class interval by finding the difference of the upper and lower bounds.

Divide the frequency of the class interval by its class width.

Example 4: finding frequencies from the frequency density

The table shows information about the heights of plants in a garden.

Calculate the missing frequencies and frequency densities.

Identify the upper and lower bounds of the class interval.

Find the class width of the class interval by finding the difference of the upper and lower bounds.

Divide the frequency of the class interval by its class width.

Here we have some missing frequencies as well as missing frequency densities.

We can calculate the frequency densities for the final two class intervals in the usual way.

To calculate the missing frequencies we need to work backwards.

\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}

therefore

\text{Frequency} = \text{Frequency density} \times \text{Class width}

Multiply the frequency density by the class width to find the frequency.

Example 5: finding the frequencies from the frequency density

This table shows information about the ages of people playing bingo.

Calculate the missing frequencies and frequency densities.

Identify the upper and lower bounds of the class interval.

Find the class width of the class interval by finding the difference of the upper and lower bounds.

Divide the frequency of the class interval by its class width.

\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}

therefore

\text{Frequency} = \text{Frequency density} \times \text{Class width}

Example 6: finding the frequencies from the frequency density

This table shows how far some children travel to school.

Work out the missing frequencies and frequency densities.

Identify the upper and lower bounds of the class interval.

Find the class width of the class interval by finding the difference of the upper and lower bounds.

Divide the frequency of the class interval by its class width.

\text{Frequency density} = \frac{\text{Frequency}}{\text{Class width}}

therefore

\text{Frequency} = \text{Frequency density} \times \text{Class width}

Common misconceptions

• Midpoint of the group

A common error is to use the midpoint when calculating frequency densities rather than the class width. This is because the midpoint is used for estimating the mean from a frequency table and frequency polygons.

Frequency density is the frequency per unit for the data in each class so it is important to use the width of the class in the calculation.

The next lessons are

• Types of sampling methods
• Cumulative frequency
• Frequency table

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