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Types of data

Grouped frequency tables

Averages from grouped frequency tables

This topic is relevant for:

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.

**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**

In order to calculate frequency density:

**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.**

Get your free frequency density worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free frequency density worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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

Calculate the frequency densities.

**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.**

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.**

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.**

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

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

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

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

**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.

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.

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

Complete the frequency density column.

**(3 marks)**

Show answer

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)**

Show answer

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)**

Show answer

Attempt to multiply frequency density by class width

**(1)**

Attempt to divide frequency by class width

**(1)**

Three correct

**(1)**

All correct

**(1)**

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

- Frequency density formula
- Histograms

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

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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.