GCSE Maths Statistics Histograms

Frequency Density

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, along with 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.

We use frequency density to plot histograms.

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

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.

Frequency Density Image 1

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

Frequency Density Image 2

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?

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

How to calculate frequency density

Histograms worksheet

Histograms worksheet

Histograms worksheet

Get your free histograms worksheet of 20+ questions and answers including frequency density and frequency density formula questions. With reasoning and applied questions.

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Histograms worksheet

Histograms worksheet

Histograms worksheet

Get your free histograms worksheet of 20+ questions and answers including frequency density and frequency density formula questions. With reasoning and applied questions.

DOWNLOAD FREE

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:

Frequency density examples

Example 1: calculating frequency density

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

Frequency Density Example 1

Calculate the frequency densities.

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

Frequency Density Example 1 Step 1

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

Frequency Density Example 1 Step 2

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

Frequency Density Example 1 Step 3

Example 2: calculating frequency density

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

Frequency Density Example 2

Calculate the frequency densities.

Frequency Density Example 2 Step 1

Frequency Density Example 2 Step 2

Frequency Density Example 2 Step 3

Example 3: calculating frequency density

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

Frequency Density Example 3

Calculate the frequency densities.

Frequency Density Example 3 Step 1

Frequency Density Example 3 Step 2

Frequency Density Example 3 Step 3

Example 4: finding frequencies from the frequency density

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

Frequency Density Example 4

Calculate the missing frequencies and frequency densities.

Frequency Density Example 4 Step 1

Frequency Density Example 4 Step 2

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.


Frequency Density Example 4 Step 3

Example 5: finding the frequencies from the frequency density

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

Frequency Density Example 5

Calculate the missing frequencies and frequency densities.

Frequency Density Example 5 Step 1

Frequency Density Example 5 Step 2

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


therefore


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


Frequency Density Example 5 Step 3

Example 6: finding the frequencies from the frequency density

This table shows how far some children travel to school.

Frequency Density Example 6

Work out the missing frequencies and frequency densities.

Frequency Density Example 6 Step 1

Frequency Density Example 6 Step 2

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


therefore


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


Frequency Density Example 6 Step 3

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.

Practice frequency density questions

1. To calculate the frequency density we use the formula

\text{Frequency density }=\frac{\text{Class width}}{\text{Frequency}}
GCSE Quiz False

\text{Frequency density }=\frac{\text{Frequency}}{\text{Class width}}
GCSE Quiz True

\text{Frequency density }=\frac{\text{Frequency}}{\text{Midpoint}}
GCSE Quiz False

\text{Frequency density }=\frac{\text{Cumulative frequency}}{\text{Class width}}
GCSE Quiz False

Frequency density is the frequency per unit for the data in each class.

2. A class interval was given as 25 \leq x < 30. How would you calculate the class width?

25-30
GCSE Quiz False

30-25
GCSE Quiz True

(25+30) \div 2
GCSE Quiz False

(30-25) \div 2
GCSE Quiz False

Class width = upper bound – lower bound

3. A class interval given as 15 \leq x < 20. has a frequency density of 1.6. What is its frequency?

16
GCSE Quiz False

3.125
GCSE Quiz False

17.5
GCSE Quiz False

8
GCSE Quiz True

We need to multiply the frequency density by the class width.

 

1.6 \times 5=8

4. The table shows information about the heights of a group of children.

 

Frequency Density Practice Question 4 Image 1

 

Which table shows the correct frequency densities?

Frequency Density Practice Question 4 Image 2

GCSE Quiz False

Frequency Density Practice Question 4 Image 3

GCSE Quiz False

Frequency Density Practice Question 4 Correct Answer Image

GCSE Quiz True

Frequency Density Practice Question 4 Correct Answer Image-1

GCSE Quiz False
\text{Frequency density }=\frac{\text{Frequency}}{\text{Class width}}

5. The table shows the frequency densities for the mass of some stones in a garden.

 

Frequency Density Practice Question 5

 

Which table shows the correct frequencies?

Frequency Density Practice Question 5-1

GCSE Quiz True

Frequency Density Practice Question 5 Image 1

GCSE Quiz False

Frequency Density Practice Question 5 Image 2

GCSE Quiz False

Frequency Density Practice Question 5 Image 4

GCSE Quiz False

We need to multiply the frequency density by the class width.

6. This table shows information about the heights of trees in a wood.

 

Frequency Density Practice Question 6

 

Which table shows the correct missing values?

Frequency Density Practice Question 6 Image 1

GCSE Quiz False

Frequency Density Practice Question 6 Image 2

GCSE Quiz False

Frequency Density Practice Question 6 Correct Answer

GCSE Quiz True

To find the missing frequencies, we need to multiply frequency density by class width.

 

To find the missing frequency densities we need to divide the frequency by the class width.

Frequency density GCSE questions

1. The table shows information about the heights of sunflowers in an allotment.

 

Frequency Density GCSE Question 1 Image 1

 

Complete the frequency density column.

 

(3 marks)

Show answer

Frequency Density GCSE Question 1 Image 2

Attempt to divide frequency by class width

(1)

Three correct

(1)

All correct

(1)

2. The table shows the frequency densities for the mass of sheep in a field.

 

Frequency Density GCSE Question 2 Image 1

 

Complete the frequency column.

 

(3 marks)

Show answer

Frequency Density GCSE Question 2 Image 2

 

Attempt to multiply frequency density by class width

(1)

Three correct

(1)

All correct

(1)

3. The table shows information about the ages of guests at a hotel.

 

Frequency Density GCSE Question 3 Image 1

 

Complete the missing values.

 

(4 marks)

Show answer

Frequency Density GCSE Question 3 Image 2

Attempt to multiply frequency density by class width

(1)

Attempt to divide frequency by class width

(1)

Three correct

(1)

All correct

(1)

Learning checklist

You have now learned how to:

  • Construct and interpret diagrams for grouped discrete data and continuous data, i.e. histograms with equal and unequal class intervals and cumulative frequency graphs, and know their appropriate use 

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