Syllabus Edition
First teaching 2023
First exams 2025
|
Histograms (CIE IGCSE Maths: Extended)
Revision Note
Author
PaulExpertise
Maths
Frequency Density
What is frequency density?
- Frequency density is given by the formula
- Frequency density is used with grouped data (class intervals)
- it is particularly useful when the class intervals are of unequal width
- it provides a measure of how spread out data within its class interval is, relative to its size
- For example,
- 10 data values spread over a class interval of 20 would have a frequency density of
- 20 data values spread over a class interval of 100 would have a frequency density of
- As the data in the first interval is more densely spread (closer together) than in the second interval, despite the second interval having twice as many data values
How do I calculate frequency density?
- In questions it is usual to be presented with grouped data in a table
- So add two extra columns to the table
- one to work out and write down the class width of each interval
- the second to then work out the frequency density for each group (row)
Worked example
The table below shows information regarding the average speeds travelled by trains in a region of the UK.
The data is to be plotted on a histogram.
Work out the frequency density for each class interval.
Average speed s m/s |
Frequency |
5 | |
15 | |
28 | |
38 | |
14 |
Add two columns to the table - one for class width, one for frequency density.
Writing the calculation in each box helps to keep accuracy.
Average speed s m/s |
Frequency | Class width | Frequency density |
5 | 40 - 20 = 20 | 5 ÷ 20 = 0.25 | |
15 | 50 - 40 = 10 | 15 ÷ 10 = 1.5 | |
28 | 55 - 50 = 5 | 28 ÷ 5 = 5.6 | |
38 | 60 - 55 = 5 | 38 ÷ 5 = 7.6 | |
14 | 70 - 60 = 10 | 14 ÷ 10 = 1.4 |
Drawing Histograms
What is a histogram?
Isn't a histogram just a really hard bar chart?!
- No!
- The main difference is that bar charts are used for discrete (and non-numerical) data whilst histograms are used with continuous data, usually grouped in unequal class intervals
- In a bar chart, the height (or length) determines the frequency
- In a histogram, it is the area of a bar that determines the frequency
- the frequency of a class interval is proportional to the area of the bar for that interval
- This means, unlike any other chart you have come across, it is very difficult to tell anything from simply looking at a histogram
- some basic calculations will need to be made for conclusions and comparisons to be made
How do I draw a histogram?
- Drawing a histogram first requires the calculation of the frequency densities for each class interval (group)
- Most questions will get you to finish an incomplete histogram, rather than start with a blank graph
- As frequency is proportional to frequency density
- In the majority of questions, , so the proportionality element can be ignored
- Once the frequency densities are known
- bars (rectangles) are drawn with widths being measured on the horizontal (x) axis
- the height of each bar is that class' frequency density and is measured on the vertical (y) axis
- as the data is continuous, bars will be touching
Exam Tip
- Always work out and write down the frequency densities
- It is easy to make errors and lose marks by going straight to the graph
- Method marks are available for showing you know to use frequency density rather than frequency
Worked example
A histogram is shown below representing the distances achieved by some athletes throwing a javelin.
There are two classes missing from the histogram. These are:
Distance, m | Frequency |
8 | |
2 |
Add these to the histogram.
Before completing the histogram, remember to show clearly you've worked out the missing frequency densities.
Distance, m | Frequency | Class width | Frequency density |
8 | 70 - 60 = 10 | 8 ÷ 10 = 0.8 | |
2 | 100 - 80 = 20 | 2 ÷ 20 = 0.1 |
Interpreting Histograms
How do I interpret a histogram?
- It is important to remember that the frequency density (y-) axis does not tell us frequency
- The area of the bar is proportional to the frequency
- The frequency will be the area of the bar directly and is found by using
- You may be asked to estimate the frequency of part of a bar/class interval within a histogram
- Find the area of the bar for the part of the interval required
- Once area is known, frequency can be found as above
Exam Tip
- The frequency density axis will not always be labelled
- look carefully at the scale, it is unlikely to be 1 unit to 1 square
- look carefully at the scale, it is unlikely to be 1 unit to 1 square
Worked example
The table below and its corresponding histogram show the mass, in kg, of some new born bottlenose dolphins.
Mass m kg |
Frequency |
4 ≤ m < 8 | 4 |
8 ≤ m < 10 | 15 |
10 ≤ m < 12 | 19 |
12 ≤ m < 15 | |
15 ≤ m < 30 | 6 |
(a)
Find the missing frequency in the table for the group 12 ≤ m < 15.
The frequency is the area
Frequency is 9
(b)
Complete the histogram.
Mass m kg |
Frequency | Class width | Frequency density |
10 ≤ m < 12 | 19 | 2 | 9.5 |
15 ≤ m < 30 | 6 | 15 | 0.4 |
(c)
Estimate the number of dolphins whose weight is greater than 13 kg.
We can see from the table that there are 6 dolphins in the interval 15 ≤ m < 30.
So we need to estimate the number of dolphins that are in the interval 13 ≤ m < 15.
Find the area of that rectangle.
3 × (15 - 13) = 6
6 + 6 = 12
There are approximately 12 dolphins with a weight greater than 13 kg
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