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.

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

## 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.

## 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.

### Example 3: calculating frequency density

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

Calculate the frequency densities.

### 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.

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 density} = \text{Frequency} \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.

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

therefore

\text{Frequency density} = \text{Frequency} \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.

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

therefore

\text{Frequency density} = \text{Frequency} \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.

### Practice frequency density questions

1. To calculate the frequency density we use the formula

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

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

\text{Frequency density }=\frac{\text{Frequency}}{\text{Midpoint}}

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

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

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

25-30

30-25

(25+30) \div 2

(30-25) \div 2

Class width = upper bound – lower bound

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

16

3.125

17.5

8

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.

Which table shows the correct frequency densities?

\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.

Which table shows the correct frequencies?

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

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

Which table shows the correct missing values?

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.

Complete the frequency density column.

(3 marks)

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.

Complete the frequency column.

(3 marks)

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.

Complete the missing values.

(4 marks)

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

## The next lessons are

• Frequency density formula
• Histograms

## Still stuck?

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#### FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.