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Histograms & Frequency Density (CIE IGCSE Maths: Extended)

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Paul

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29 November 2022

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Frequency Density

What is frequency density?

Frequency density is given by the formula

$frequency density = \frac{frequency}{class width}$

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 $\frac{10}{20} = \frac{1}{2}$
  - 20 data values spread over a class interval of 100 would have a frequency density of $\frac{20}{100} = \frac{1}{5}$
  - As $\frac{1}{2} > \frac{1}{5}$ 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
$20 \leq s < 40$	5
$40 \leq s < 50$	15
$50 \leq s < 55$	28
$55 \leq s < 60$	38
$60 \leq s < 70$	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
$20 \leq s < 40$	5	40 - 20 = 20	5 ÷ 20 = 0.25
$40 \leq s < 50$	15	50 - 40 = 10	15 ÷ 10 = 1.5
$50 \leq s < 55$	28	55 - 50 = 5	28 ÷ 5 = 5.6
$55 \leq s < 60$	38	60 - 55 = 5	38 ÷ 5 = 7.6
$60 \leq s < 70$	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

$\begin{array}{rcl} frequency & \propto & frequency density \\ frequency density & = & k \times \frac{frequency}{class width} \end{array}$

In the majority of questions, $k = 1$ , 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

Examiner 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
$60 \leq x < 70$	8
$80 \leq x < 100$	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
$60 \leq x < 70$	8	70 - 60 = 10	8 ÷ 10 = 0.8
$80 \leq x < 100$	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
Most of the time, the frequency will be the area of the bar directly and is found by using

$frequency = area$

Occasionally the frequency will be proportional to the area of the bar so use

$frequency = k \times area$

- You will need to work out the value of k from other information given in the question
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

Examiner 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

2-2-3-histograms-we-diagram

(a)

Use the table and histogram to find the value of $k$ in the formula

$frequency density = k \times \frac{frequency}{class width}$

Start by finding the frequency density in terms of $k$ - add two columns to the table, one for class width, one for frequency density.

Mass m kg	Frequency	Class width	Frequency density
4 ≤ m < 8	4	8 - 4 = 4	$k (\frac{4}{4}) = k$
8 ≤ m < 10	15	10 - 8 = 2	$k (\frac{15}{2}) = 7.5 k$
10 ≤ m < 12	19	2	$k (\frac{19}{2}) = 9.5 k$
12 ≤ m < 15		3
15 ≤ m < 30	6	15	$k (\frac{6}{15}) = 0.4 k$

We can use either of the two intervals that feature both in the table and on the histogram to find the value of $k$ .
Using the first bar.

$frequency density = k = 0.5$

$k = 0.5$

Check using the other (2^nd) bar to check; $7.5 k = 3.75, k = 0.5$

(b)

Estimate the number of dolphins whose weight is greater than 13 kg.

We can see from the table that their 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.
For 13 ≤ m < 15, the histogram shows the frequency density is 1.5 and we found the value of $k$ in part (a).
Using the formula given in the question,

$\begin{array}{rcl} ∴ 1.5 & = & 0.5 \times \frac{frequency}{15 - 13} \\ frequency & = & 6 \end{array}$

So the total number of dolphins can be estimated by

$6 + 6 = 12$

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

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Histograms & Frequency Density (CIE IGCSE Maths: Extended)

Revision Note

What is frequency density?

How do I calculate frequency density?

What is a histogram?Isn't a histogram just a really hard bar chart?!

How do I draw a histogram?

How do I interpret a histogram?

You've read 0 of your 10 free revision notes

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Paul

What is a histogram?
Isn't a histogram just a really hard bar chart?!