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Histograms (Cambridge O Level Maths)

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Frequency Density

What is frequency density?

  • Frequency density is given by the formula

frequency space density equals fraction numerator frequency over denominator class space width end fraction

  • 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 10 over 20 equals 1 half
      • 20 data values spread over a class interval of 100 would have a frequency density of 20 over 100 equals 1 fifth
      • As 1 half greater than 1 fifth 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
20 less or equal than s less than 40 5
40 less or equal than s less than 50 15
50 less or equal than s less than 55 28
55 less or equal than s less than 60 38
60 less or equal than s less than 70 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
20 less or equal than s less than 40 5 40 - 20 = 20 5 ÷ 20 = 0.25
40 less or equal than s less than 50 15 50 - 40 = 10 15 ÷ 10 = 1.5
50 less or equal than s less than 55 28 55 - 50 = 5 28 ÷ 5 = 5.6
55 less or equal than s less than 60 38 60 - 55 = 5 38 ÷ 5 = 7.6
60 less or equal than s less than 70 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

table row frequency proportional to cell frequency space density end cell row cell frequency space density end cell equals cell k cross times fraction numerator frequency over denominator class space width end fraction end cell row blank blank blank end table

  • In the majority of questions, k equals 1, 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.Histogram Question Bars 1, IGCSE & GCSE Maths revision notes

There are two classes missing from the histogram.  These are:

Distance, x m Frequency
60 less or equal than x less than 70 8
80 less or equal than x less than 100 2

Add these to the histogram.

Before completing the histogram, remember to show clearly you've worked out the missing frequency densities.

Distance, x m Frequency Class width Frequency density
60 less or equal than x less than 70 8 70 - 60 = 10 8 ÷ 10 = 0.8
80 less or equal than x less than 100 2 100 - 80 = 20 2 ÷ 20 = 0.1

Histogram Question Bars 1, IGCSE & GCSE Maths revision notes

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

frequency equals area

  • 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

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

1-2-4---histogram-we-diagram

 

(a)

Find the missing frequency in the table for the group 12 ≤ m < 15.

The frequency is the area

Area equals 3 cross times open parentheses 15 minus 12 close parentheses

Frequency is 9

(b)

Complete the histogram.

Frequency space density equals fraction numerator Frequency over denominator Class space width end fraction

Mass
m kg		 Frequency Class width Frequency density
10 ≤ m < 12 19 2 9.5
15 ≤ m < 30 6 15 0.4

histograms-cie-2025-1

 

(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.

histograms-cie-2025-2

3 × (15 - 13) = 6

6 + 6 = 12

There are approximately 12 dolphins with a weight greater than 13 kg

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    Paul

    Author: Paul

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.