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i
ESSENTIAL
E
Further
Mathematics
PL
Third edition
SA
M
PETER JONES
MICHAEL EVANS
KAY LIPSON
TI-Nspire and Casio ClassPad material
prepared in collaboration with
Russell Brown
Kevin McMenamin
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CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
www.cambridge.edu.au
Information on this title: www.cambridge.edu.au/0521613280
Peter Jones, Michael Evans & Kay Lipson 2005
First published 1998
Reprinted 1998
Second edition 1999
Reprinted 2000, 2001, 2002, 2003, 2005
Third edition 2005
Reprinted 2006
E
C
PL
Cover designed by Modern Art Production Group
Text designed by Sylvia Witte
Typeset in India by Techbooks
Printed in China through Everbest Printing Company Pty Ltd
National Library of Australia Cataloguing in Publication data
Jones, Peter, 1943-.
Essential further mathematics.
3rd ed.
ISBN-13 978-0-521-74051-7 paperback
ISBN-10 0-521-61328-0 paperback
1. Mathematics – Problems, exercises, etc. I. Evans,
Michael (Michael Wyndham). II. Lipson, Kay. III. Title
510.76
SA
M
ISBN-13 978-0-521-74051-7 paperback
ISBN-10 0-521-61328-0 paperback
Reproduction and communication for educational purposes
The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this
publication, whichever is the greater, to be reproduced and/or communicated by any educational institution for
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remuneration notice to Copyright Agency Limited (CAL) under the Act.
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Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or
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websites is, or will remain, accurate or appropriate.
All Victorian Curriculum and Assessment Authority material copyright VCAA. Reproduced by permission of
the Victorian Curriculum and Assessment Authority Victoria, Australia.
Disclaimer: This publication is independently produced for use by teachers and students. Although references
•¥ 2008
Jones, Evans,
Lipson
Cambridge
University
Presswith
¥• Uncorrected
pages ¥•the
978-0-521-61328-6
have been
reproduced
permissionSample
of the VCAA
publication is in no
way ©connected
with or
endorsed
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E
Contents
Acknowledgements
xiv
CORE
CHAPTER 1 — Organising and displaying data
Classifying data
1
Organising and displaying categorical data
Organising and displaying numerical data
What to look for in a histogram
20
Stem-and-leaf plots and dot plots
26
Key ideas and chapter summary
34
Skills check
35
Multiple-choice questions
35
Extended-response questions
37
SA
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1.1
1.2
1.3
1.4
1.5
1
3
8
CHAPTER 2 — Summarising numerical data: the median,
range, IQR and box plots
2.1
2.2
2.3
2.4
2.5
40
Will less than the whole picture do?
40
The median, range and interquartile range
(IQR)
41
The five-number summary and the box plot
45
Relating a box plot to distribution shape
52
Interpreting box plots: describing and
comparing distributions
54
Key ideas and chapter summary
57
Skills check
58
Multiple-choice questions
59
Extended-response questions
60
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Contents
CHAPTER 3 — Summarising numerical data: the mean and
the standard deviation
3.3
The mean
63
Measuring the spread around the mean:
the standard deviation
67
The normal distribution and the 68–95–99.7% rule:
giving meaning to the standard deviation
73
Standard scores
79
Populations and samples
83
Key ideas and chapter summary
88
Skills check
90
Multiple-choice questions
91
Extended-response questions
92
PL
3.4
3.5
63
E
3.1
3.2
CHAPTER 4 — Displaying and describing relationships
between two variables
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
95
Investigating the relationship between two
categorical variables
95
Using a segmented bar chart to identify
relationships in tabulated data
99
Investigating the relationship between a numerical
and a categorical variable
102
Investigating the relationship between two
numerical variables
104
How to interpret a scatterplot
107
Calculating Pearson’s correlation
coefficient r
112
The coefficient of determination
118
Correlation and causality
121
Which graph?
122
Key ideas and chapter summary
123
Skills check
124
Multiple-choice questions
125
Extended-response questions
128
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iv
CHAPTER 5 — Regression: fitting lines to data
5.1
5.2
5.3
5.4
5.5
131
Least squares regression line: the theory
131
Calculating the equation of the least squares
regression line
133
Performing a regression analysis
140
A graphical approach to regression: the three
median line
153
Extrapolation and interpolation
157
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Contents
v
Key ideas and chapter summary
159
Skills check
160
Multiple-choice questions
160
Extended-response questions
162
CHAPTER 6 — Data transformation
E
Data transformation
166
Transforming the x axis
169
183
Transforming the y axis
Choosing and applying the appropriate
transformation
189
Key ideas and chapter summary
197
Skills check
197
Multiple-choice questions
197
Extended-response questions
200
PL
6.1
6.2
6.3
6.4
CHAPTER 7 — Time series
7.1
7.2
7.3
204
Time series data
204
Smoothing a time series plot (moving
means)
210
Smoothing a time series plot (moving
medians)
215
Seasonal indices
220
Fitting a trend line and forecasting
207
Key ideas and chapter summary
233
Skills check
234
Multiple-choice questions
235
Extended-response questions
237
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7.4
7.5
CHAPTER 8 — Revision of the core
8.1
8.2
8.3
8.4
8.5
166
239
Displaying, summarising and describing
univariate data
239
Displaying, summarising and describing
relationships in bivariate data
243
Regression and data transformation
245
Time series
249
Extended-response questions
253
MODULE 1 — Number patterns and
applications
CHAPTER 9 — Arithmetic and geometric
sequences
259
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Contents
9.4
9.5
9.6
9.7
9.8
297
PL
9.9
9.10
Sequences
259
Arithmetic sequences
260
The nth term of an arithmetic sequence
and its applications
264
The sum of an arithmetic sequence
and its applications
274
Geometric sequences
281
The nth term of a geometric
sequence
285
Applications modelled by geometric
sequences
289
The sum of the terms in a geometric
sequence
294
The sum of an infinite geometric sequence
Rates of growth of arithmetic and geometric
sequences
302
Key ideas and chapter summary
307
Skills check
308
Multiple-choice questions
309
Extended-response questions
310
E
9.1
9.2
9.3
CHAPTER 10 — Difference equations
10.1
10.2
312
Introduction
312
The relationship between arithmetic and geometric
sequences and difference equations
320
First-order difference equations
322
Solving first-order difference equations that
generate arithmetic sequences
324
Solving difference equations that generate
geometric sequences
325
Solution of general first-order difference equations
(optional)
327
Summary of first-order difference equations
328
Applications of first-order difference
equations
329
The Fibonacci sequence
338
Key ideas and chapter summary
345
Skills check
346
Multiple-choice questions
346
Extended-response questions
348
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vi
10.3
10.4
10.5
10.6
10.7
10.8
10.9
CHAPTER 11 — Revision: Number patterns and
applications
11.1
11.2
350
Multiple-choice questions
350
Extended-response questions
355
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Contents
vii
MODULE 2 — Geometry and trigonometry
CHAPTER 12 — Geometry
E
Properties of parallel lines – a review
360
Properties of triangles – a review
362
Properties of regular polygons – a review
364
Pythagoras’ theorem
367
Similar figures
371
Volumes and surface areas
375
Areas, volumes and similarity
382
387
Key ideas and chapter summary
Skills check
389
Multiple-choice questions
390
PL
12.1
12.2
12.3
12.4
12.5
12.6
12.7
CHAPTER 13 — Trigonometry
392
Defining sine, cosine and tangent
The sine rule
396
The cosine rule
401
Area of a triangle
404
Key ideas and chapter summary
Skills check
407
Multiple-choice questions
408
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13.1
13.2
13.3
13.4
360
392
406
CHAPTER 14 — Applications of geometry and
trigonometry
14.1
14.2
14.3
410
Angles of elevation and depression, bearings, and
triangulation
410
Problems in three dimensions
417
Contour maps
421
Key ideas and chapter summary
424
Skills check
424
Multiple-choice questions
424
Extended-response questions
426
CHAPTER 15 — Revision: Geometry and
trigonometry
15.1
15.2
431
Multiple-choice questions
431
Extended-response questions
438
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Contents
MODULE 3 — Graphs and relations
CHAPTER 16 — Constructing and interpreting linear
graphs
The gradient of a straight line
441
The general equation of a straight line
443
Finding the equation of a straight line
445
Equation of a straight line in intercept form
449
Linear models
450
Simultaneous equations
452
Problems involving simultaneous linear
equations
456
Break-even analysis
458
Key ideas and chapter summary
460
Skills check
461
Multiple-choice questions
461
PL
16.8
441
E
16.1
16.2
16.3
16.4
16.5
16.6
16.7
CHAPTER 17 — Graphs
17.1
17.2
17.3
17.4
17.5
465
Line segment graphs
465
Step graphs
468
Non-linear graphs
470
Relations of the form y = kxn for
n = 1, 2, 3, −1,
−1, −
−2
2
472
Linear representation of non-linear relations
Key ideas and chapter summary
482
Skills check
483
Multiple-choice questions
483
Extended-response questions
486
SA
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viii
CHAPTER 18 — Linear programming
18.1
18.2
18.3
18.4
488
Regions defined by an inequality
488
Regions defined by two inequalities
490
Feasible regions
492
Objective functions
493
Key ideas and chapter summary
503
Skills check
504
Multiple-choice questions
505
Extended-response questions
507
CHAPTER 19 — Revision: Graphs and relations
19.1
19.2
475
510
Multiple-choice questions
510
Extended-response questions
514
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Contents
ix
MODULE 4 — Business related mathematics
CHAPTER 20 — Principles of financial mathematics
Percentage change
521
Simple interest
526
Compound interest
534
Reducing balance loans
546
548
Key ideas and chapter summary
Skills check
549
Multiple-choice questions
549
Extended-response questions
551
E
20.1
20.2
20.3
20.4
521
CHAPTER 21 — Applications of financial
Percentage changes and charges
553
Bank account balances
558
Hire purchase
561
Inflation
567
Depreciation
571
Applications of Finance Solvers
581
Key ideas and chapter summary
597
Skills check
599
Multiple-choice questions
600
Extended-response questions
602
SA
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21.1
21.2
21.3
21.4
21.5
21.6
553
PL
mathematics
CHAPTER 22 — Revision: Business-related
mathematics
22.1
22.2
606
Multiple-choice questions
606
Extended-response questions
610
MODULE 5 — Networks and decision
mathematics
CHAPTER 23 — Undirected graphs
23.1
23.2
23.3
23.4
23.5
614
Introduction and definitions
614
Planar graphs and Euler’s formula
619
Complete graphs
622
Euler and Hamilton paths
623
Weighted graphs
626
Key ideas and chapter summary
630
Skills check
632
Multiple-choice questions
632
Extended-response questions
636
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Contents
CHAPTER 24 — Directed graphs
639
Introduction, reachability and dominance
Network flows
645
The critical path problem
649
Allocation problems
656
Key ideas and chapter summary
662
Skills check
663
Multiple-choice questions
663
Extended-response questions
667
639
E
24.1
24.2
24.3
24.4
CHAPTER 25 — Revision: Networks and decision
mathematics
671
Multiple-choice questions
671
Extended-response questions
677
PL
25.1
25.2
MODULE 6 — Matrices and applications
CHAPTER 26 — Matrices and applications 1
26.1
26.2
26.3
26.4
CHAPTER 27 — Matrices and applications II
27.1
27.2
27.3
27.4
690
What is a matrix?
690
Using matrices to represent information
696
Matrix arithmetic: addition, subtraction and scalar
multiplication
699
Matrix arithmetic: the product of two
matrices
706
Key ideas and chapter summary
715
Skills check
717
Multiple-choice questions
717
Extended-response questions
720
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x
722
The inverse matrix
722
Applications of the inverse matrix:
solving simultaneous linear equations
729
Matrix powers
737
Transition matrices and their applications
739
Key ideas and chapter summary
749
Skills check
751
Multiple-choice questions
751
Extended-response questions
754
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Contents
xi
CHAPTER 28 — Revision: Matrices and
applications
756
Multiple-choice questions
756
Extended-response questions
760
Appendix TI-nspire
763
Appendix ClassPad
768
771
SA
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PL
E
Answers
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Ess Maths IN-BOOK BROCHUR.qxd 11/3/05 7:37 PM Page xii Quark08 Quark08:Books:CUAT006_From Pooja:CUAT013:
The new Essential
series for the 2006
study design
1
CORE
Organising and
displaying data
What is the difference between categorical and numerical data?
What is a frequency table, how is it constructed and when is it used?
What is the mode and how do we determine its value?
What are bar charts, histograms, stem plots and dot plots? How are they
constructed and when are they used?
How do you describe the features of bar charts, histograms and stem plots when
In each chapter you will find …
writing a statistical report?
1.1
Classifying data
Statistics is a science concerned with understanding the world through data. The first step in
this process is to put the data into a form that makes it easier to see patterns or trends.
Some data
The data contained in Table 1.1 is part of a larger set of data collected from a group of
university students.
Chapter 9 – Arithmetic and geometric sequences
237 data
Table 1.1 Student
Height
(cm)
How to use a graphics calculator to generate the terms of an arithmetic sequence
on the
Home screen
Weight
(kg)
Age
(years)
Sex
M male
F female
female
57
58
62
84
64
74
60
50
18
19
18
18
18
22
19
34
M
M
M
F
M
F
F
M
Plays sport
1 regularly
2 sometimes
3 rarely
rarely
2
2
1
1
3
3
3
3
Pulse rate
(beats/min)
86
82
96
71
90
78
88
70
E
173
Generate the first five terms of the arithmetic sequence: 2, 7, 12, 17, 22, . . . 179
167
Steps
195
173
1 Start on the Home screen. Clear. Enter the value of the
184
first term 2. Press Í.
175
140
Source: www.statsci.org/data/oz/ms212.html. Used with permission.
2 The common difference for this sequence is 5. So, type
in + 5. Press Í.. The second term in the sequence,
7, is generated.
3 Pressing Í again generates the next term, 12.
4 Pressing Í again generates the next term, 17.
Keep pressing Í until the required number
of terms is generated.
Being able to recognise an arithmetic sequence is another skill that you need to develop. The
key idea here is that the successive terms in an arithmetic sequence differ by a constant amount
(the common difference).
PL
a vibrant full colour text with a
clear layout that makes maths
more accessible for students
1
‘Using a graphics calculator’
boxes within chapters explain
how to do problems using the
TI-83/Plus and TI-84 graphics
calculators, and include screen
shots to further assist students
Example 1
Testing for an arithmetic sequence
20, 17
17,, 14,
14, 11
11,, 8,
8, . . . arithmetic?
a Is the sequence 20,
Chapterr 4 — Displaying and describing relationships between two variables
Chapte
107
Solution
Clearly, traffic volume is a very good predictor of carbon monoxide levels in the air. Knowing
Strategy:: Subtract successive terms in the sequence to see whether they differ by a constant
Strategy
the traffic volume will enable us to predict carbon monoxide levels with a high degree of
amount. If they do, the sequence is arithmetic.
accuracy. This contrasts with the next example, which concerns the ability to predict
20,, 17,
20
17, 14,
14, 11,
11, 8,
8, . . .
1 Write down the terms of the sequence.
mathematical ability from verbal ability.
17 − 20 = −3
−3
2 Subtract successive terms.
14 − 17 = −3
−3
Example 3
Calculating and interpreting the coefficient of determination
11 − 14 = −3
−3 and so on
Sequence
Sequ
ence is arithmetic as terms differ by a
Scores on tests of verbal and mathematical ability are linearly related with:
constant amount.
3 Write down your conclusion
rmathematical, verbal = +0.275
Determine the value of the coefficient of determination, write it in percentage terms, and
interpret. In this relationship, mathematical ability is the DV.
Solution
SA
M
a wealth of worked examples
that support theory explanations
within chapters
carefully graduated exercises
that include a number of easier
lead-in questions to provide
students with a greater
opportunity for immediate
success
chapter summaries at the end of
each chapter provide students
with a coherent overview
chapter reviews that include key
ideas and chapter summary and
skills check lists, and multiplechoice and extended-response
questions
Appendices that include a TIAppendices
83/84 Plus help guide and stepby-step worked examples using
TI-89 Graphics Calculators
revision chapters to help
consolidate student knowledge
The coefficient of determination is:
r 2 = (0.275)2 = 0.0756 . . . or 0.076 × 100 = 7.6%
Therefore, only 7.6% of the variation observed in scores on the test of mathematical ability can
be explained by the variation in scores obtained on the test of verbal ability.
Clearly, scores on the verbal ability test are not good predictors of the scores on the
mathematical ability test; 92.4% of the variation in mathematical ability is explained by other
factors.
Exercise 4G
1 For each of the following values of r, calculate the value of the coefficient of determination
and convert to a percentage (correct to one decimal place).
a r = 0.675
b r = 0.345
c r = −0.567
d r = −0.673
e r = 0.124
2 a For the relationship described by the scatterplot
173
shown opposite, the coefficient of determination = 0.8215.
Determine the value of the correlation coefficient r
(correct to three decimal places).
Chapter 6 — Data transformation
Review
New Essential Mathematics Series
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C H A P T E R
Key ideas and chapter summary
Data transformation
This means changing the scale on either the x or y axis. It is
performed when a residual plot shows that the underlying
b For the relationship described by the scatterplot shown
relationship in a set of bivariate data is clearly non-linear.
opposite, the coefficient of determination = 0.1243.
x2 or y2 transformation
the value of the correlation coefficient r
The square transformation stretches out the upper end Determine
of
the scale on an axis.
(correct to three decimal places).
The log transformation compresses the upper end of the
scale on an axis.
log x or log y transformation
1
1
or transformation
x
y
The reciprocal transformation compresses the upper end
of the scale on an axis to a greater extent than the log
transformation.
Residual plots
Residual plots are used to assess the effectiveness of each
data transformation.
Coefficient of determination (r2 )
The transformation which results in a linear relationship
and which has the highest value of the coefficient of
determination is considered to be the best transformation.
The circle of transformations
The circle of transformations provides guidance in
choosing the transformations that can be used to linearise
various types of scatterplots. See page 166.
Skills check
Having completed this chapter you should be able to:
1
1
recognise which of the x 2 , log x, , y 2 , log y or transformations might be used to
x
y
linearise a bivariate relationship
apply each of these transformations to a data set
use residual plots and the coefficient of determination r 2 to decide which
transformation gives the best model for the relationship
use the transformed variable as part of a regression analysis to give a model for the
relationship
Multiple-choice questions
1 The missing data values, a and b, in the table are:
value
(value)2
log(value)
1
a
0
A a = 0, b = 0.5
D a = 1, b = 0.602
2
4
b
3
4
9
16
0.477 0.602
B a = 1, b = 0.5
E a = 1, b = 0.693
Glossary
C a = 1, b = 0.301
Assignment problem: [p. 602] See allocation
problem.
A
B
Activity (CPA): [p. 595] A task to be completed as
part of a project. Activities are represented by the
edges in the project diagram.
Acute angle: An angle less than 90◦ .
Bar chart: [p. 4] A statistical graph used to display
the frequency distribution of categorical data.
Bearing: [p. 371] See true bearing.
Adjacency matrix: [pp. 561, 586] A square matrix
showing the number of edges joining each pair of
vertices in a graph.
Algorithm: A step-by-step procedure for solving a
particular problem that involves applying the same
process repeatedly. Examples include Prim’s
algorithm and the Hungarian algorithm.
Allocation problem: [p. 602] A problem that
involves finding the best way to match a given
number of objects (people, machines, etc.) to a given
number of activities.
Alternate angles: [p. 320]
Angle of depression: [p. 371] The angle between the
horizontal and a direction below the horizontal.
Angle of elevation: [p. 371] The angle between the
horizontal and a direction above the horizontal.
Angle sum of a triangle: [p. 322] In triangle
ABC, <A + <B + <C = 180◦ .
Annuity equation: [pp. 531, 534] The equation:
A = PRn −
Q(R n − 1)
r
, where R = 1 +
.
R−1
100
The annuity equation can be used to determine
either the amount owed on a reducing balance loan
with regular repayments, or the value of an
investment (annuity) with regular payments or
withdrawals.
Area of a triangle: [p. 364] See also Heron’s
formula.
Arithmetic sequence: [p. 236] A sequence whose
successive terms differ by a constant amount (d)
called the common difference. Given the value of the
first term in an arithmetic sequence (a), there are
rules for finding the nth term and the sum of the first
n terms.
Bipartite graph (bigraph): [p. 562] A graph whose
set of vertices can be split into two subsets, X and Y,
in such a way that each edge of the graph joins a
vertex in X and a vertex in Y.
Bivariate data: [p. 85] Data associated with two
related variables.
Book value: [p. 526] The value of an item after
depreciation.
Box plot (standard): [p. 41] A graphical
representation of a five number summary.
Box plot (with outliers): [p. 42] A modified form of
the standard box plot in which possible outliers are
shown. Possible outliers are defined as data values
greater than Q 3 + 1.5 × IQR and less than
Q 1 − 1.5 × IQR.
Break-even analysis: [p. 418] Finding the point
where the revenue of a business first equals the costs
of running the business. Past this point, the business
is running at a profit: profit = revenue − costs.
C
Capacity of a cut: [p. 592] The sum of the
capacities (weights) of the edges directed from X to Y
that the cut passes through.
Categorical data: [p. 2] Data obtained when
classifying or naming some quality or attribute, for
example: place of birth, hair colour.
Causal relationship: [p. 108] When the change in
one variable in a relationship can be said to be the
direct result of a change in a second variable, the
relationship is said to be causal. A high correlation
between two variables does not necessarily mean that
the variables are causally related.
a comprehensive glossary of
mathematical terms with page
.
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Explaining icons in the book ...
Links to Teacher CD-ROM
Gives an extra hint in extended-response
questions that a numerical approach is
required
Indicates that there is an explanation in
Appendix B as to how this example may be
done using TI-89 Graphics Calculators
Indicates that a skillsheet is available to
provide further practice and examples in this
area. If students are having difficulty they can
approach
pproach their teacher who can access this
material on the Teacher CD-ROM.
Links to Student CD-ROM
E
Calculator icons
PL
Live links to interactive files on the Student CD-ROM.
What teachers and students will find on the CD-ROMs ...
Teacher CD-ROM
The Essential Further Mathematics Teacher CD-ROM
contains a wealth of time-saving assessment and
classroom resources including:
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modifiable chapter tests and answers containing
multiple-choice and short-answer questions
chapter review assignments with extended
problems that can be given to students in class or
can be completed at home
printable versions of the multiple-choice questions
from the Student CD-ROM
print-ready skillsheets to revise the prerequisite
knowledge and skills required for the chapter
editable Exam Question Sets from which teachers
can create their own exams.
Student CD-ROM
The textbook includes a Student CD-ROM that contains
a PDF of the book, interactive multiple-choice questions
and unique drag-and-drop activities. Technology
aapplets
pplets such as PowerPoint and Excel activites are also
included.
Additional resources ...
Solutions Supplements
Websites
The Essential Further Mathematics Third Edition
Solutions Supplement book provides solutions to the
extended-response questions, highlighting the process
as well as the answer.
Teacher Website
This dynamic website enables teachers to interact with
each other through teacher forums, to send questions
to the authors and to obtain updates.
www.essentialmaths.com.au
Student Website
This free student website contains a student forum
allowing keen mathematics students to interact with
each other, as well as interactive tests and links to
other useful sites.
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E
Acknowledgements
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PL
Cambridge University Press and the authors would like to acknowledge all the reviewers who
provided invaluable feedback throughout the development of this text. In particular we would
like to thank Cathy Ashworth (Sandringham Secondary College), Anthony Gale (Catholic
Regional College, Sydenham), Tim Grant (St Bernard’s College, Essendon), David Greenwood
(Trinity Grammar School, Kew), Fran Petrie (Melbourne High School), Paul Rice
(St Bernard’s College, Essendon), Inna Smith (St Michael’s Grammar School, St Kilda), Kyle
Staggard (Bendigo Senior Secondary College), Leah Whiffin (Bendigo Senior Secondary
College), and Joe Wilson (Mill Park Secondary College). We also acknowledge the work of
James Wan and James Hillis who checked all of the answers in this book.
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C H A P T E R
1
CORE
PL
E
Organising and
displaying data
What is the difference between categorical and numerical data?
What is a frequency table, how is it constructed and when is it used?
What is the mode and how do we determine its value?
What are bar charts, histograms, stem plots and dot plots? How are they
constructed and when are they used?
How do you describe the features of bar charts, histograms and stem plots when
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writing a statistical report?
1.1
Classifying data
Statistics is a science concerned with understanding the world through data. The first step in
this process is to put the data into a form that makes it easier to see patterns or trends.
Some data
The data contained in Table 1.1 is part of a larger set of data collected from a group of
university students.
Table 1.1 Student data
Height
(cm)
Weight
(kg)
Age
(years)
Sex
M male
F female
173
179
167
195
173
184
175
140
57
58
62
84
64
74
60
50
18
19
18
18
18
22
19
34
M
M
M
F
M
F
F
M
Plays sport
1 regularly
2 sometimes
3 rarely
2
2
1
1
3
3
3
3
Pulse rate
(beats/min)
86
82
96
71
90
78
88
70
Source: www.statsci.org/data/oz/ms212.html. Used with permission.
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Essential Further Mathematics – Core
E
Variables
In a data set, we call the things about which we record information variables. An important
first step in analysing any set of data is to identify the variables involved, their units of
measurement (where appropriate) and the values they take. In this particular data set there are
six variables:
height (in centimetres)
sex (M = male, F = female)
weight (in kilograms)
plays sport (1 = regularly, 2 = sometimes, 3 = rarely)
age (in years)
pulse rate (beats/minute)
Types of variables: categorical and numerical
PL
Having identified the variables we are working with, the next step is to decide the variable type.
Some variables represent qualities or attributes. For example, ‘F’ in the Sex column
indicates that the person is a female, while a ‘2’ in the Plays sport column indicates that the
person is someone who plays sport sometimes.
Variables that represent qualities are called categorical variables.
Other variables represent quantities. For example, a ‘179’ in the Height column indicates
that the person is 179 cm tall, while an ‘82’ in the Pulse rate column indicates that they have a
pulse rate of 82 beats/minute.
Variables that represent quantities are called numerical variables.
Numerical variables come in two types, discrete and continuous.
Discrete numerical variables represent quantities that are counted. The number of mobile
phones in a house is an example. Counting leads to discrete data values because it results in
values like 0, 1, 2, 3 etc. There can be nothing in between. As a guide, discrete numerical
variables arise when we ask the question ‘How many?’
Continuous numerical variables represent quantities that are measured rather than counted.
Thus, even though we might record a person’s height as 179 cm, in reality that could be any
value between 178.5 and 179.4 cm. We have just rounded off the height to 179 cm for
convenience, or to match the accuracy of the measuring device.
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2
Warning!!
It is not the variable name itself that determines whether the data is numerical or categorical, it is
the way the data for the variable is recorded.
For example:
‘weight’ recorded in kilograms, is a numerical variable
‘weight’ recorded as 1 = underweight, 2 = normal weight, 3 = overweight, is a categorical
variable
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Chapter 1 — Organising and displaying data
3
Exercise 1A
1 What is:
a a numerical variable? Give an example.
b a categorical variable? Give an example.
2 There are two types of numerical variables. Name them.
length of bananas (in centimetres)
h
number of cars in a supermarket car park i
daily temperature in ◦ C
j
eye colour (brown, blue, . . . )
k
shoe size (6, 8, 10, . . . )
l
the number of children in a family
m
city of residence (NY, London, . . . )
n
number of people who live in your city/area
time spent watching TV (hours)
the TV channel most watched by students
salary (high, medium, low)
salary (in dollars)
whether a person smokes (yes, no)
the number of cigarettes smoked per day
PL
a
b
c
d
e
f
g
E
3 Classify each of the following variables as numerical or categorical. If the variable is
numerical, further classify the variable as discrete or continuous.
Recording information on:
4 Classify the data for each of the variables in Table 1.1 as numerical or categorical.
Organising and displaying categorical data
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1.2
The frequency table
With a large number of data values, it is difficult to identify any patterns or trends in the raw
data. We first need to organise the data into a more manageable form. A statistical tool we use
for this purpose is the frequency table.
The frequency table
A frequency table is a listing of the values a variable takes in a data set, along with how
often (frequently) each value occurs.
Frequency can be recorded as a
count: the number of times a value occurs, or
percent: the percentage of times a value occurs
count
× 100%
total count
A listing of the values a variable takes, along with how frequently each of these values
occurs in a data set, is called a frequency distribution.
percent =
Example 1
Frequency table for a categorical variable
The sex of 11 preschool children is as shown (F = female, M = male):
F M M F F M F F F M M
Construct a frequency table to display the data.
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Essential Further Mathematics – Core
Solution
Frequency
Sex
Count
Percent
Female
Male
6
5
54.5
45.5
Total
11
100.0
E
1 Set up a table as shown. The variable Sex has two
categories, ‘Male’ and ‘Female’.
2 Count up the number of females (6) and males (5).
Record in the ‘Count’ column.
3 Add the counts to find the total count, 11 (6+5).
Record in the ‘Count’ column opposite ‘Total’.
4 Convert the counts into percentages.
Record in the ‘Percent’ column. For example:
6
× 100% = 54.5%
11
5 Finally, total the percentages and record.
PL
percentage of females =
There are two things to note in constructing the frequency table in Example 1.
1 In setting up this frequency table, the order in which we have listed the categories ‘Female’
and ‘Male’ is quite arbitrary; there is no natural order. However, if the categories had been,
for example, ‘First’, ‘Second’ and ‘Third’, then it would make sense to list the categories in
that order.
2 The Total count should always equal the total number of observations, in this case 11.
The percentages should add to 100%. However, if percentages are rounded to one decimal
place a total of 99.9 or 100.1 is sometimes obtained. This is due to rounding error. Totalling
the count and percentages helps check on your counting and percentaging.
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4
How has forming a frequency table helped?
The process of forming a frequency table for a categorical variable:
displays the data in a compact form
tells us something about the way the data values are distributed (the pattern of the data)
The bar chart
Once categorical data has been organised into a frequency table, it is common practice to
display the information graphically to help identify any features that stand out in the data. A
statistical tool we use for this purpose is the bar chart.
The bar chart pictures the key information in a frequency table. It is designed for categorical
data. In a bar chart:
frequency (or percentage frequency) is shown on the vertical axis
the variable being displayed is plotted on the horizontal axis
the height of the bar (column) gives the frequency (count or percentage)
the bars are drawn with gaps between them to indicate that each value is a separate
category
there is one bar for each category
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Chapter 1 — Organising and displaying data
5
Example 2
Constructing a bar chart from a frequency table
Construct a bar chart for this frequency table.
3
14
6
23
13.0
60.9
26.1
100.0
15
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PL
1 Label the horizontal axis with the variable
name, ‘Climate type’. Mark the scale off
into three equal intervals and label, ‘Cold’,
Moderate’ and ‘Hot’.
2 Label the vertical axis ‘Frequency’. Scale
allowing for the maximum frequency, 14.
Fifteen would be appropriate. Mark the
scale off in fives.
3 For each interval, draw in a bar. There are
gaps between the bars to show that the
categories are separate. The height of the
bar is made equal to the frequency.
Cold
Moderate
Hot
Total
E
Solution
Climate type
Frequency
Count Percent
Frequency
10
5
0
Cold
Moderate
Climate type
Hot
The mode
One of the features of a data set that is quickly revealed with a bar chart is the mode or modal
category. This is the most frequently occurring value or category. This is given by the
category with the tallest bar. For the bar chart above, the modal category is clearly ‘Moderate’.
That is, for the countries considered, the most frequently occurring climate type is ‘Moderate’.
However, the mode is only of interest when a single value or category in the frequency table
occurs much more often than the others. Modes are of particular importance in ‘popularity’
polls. For example, in answering questions like ‘Which is the most frequently watched TV
station between the hours of 6.00 and 8.00 p.m.?’ or ‘What are the times when a supermarket
is in peak demand, ‘morning’, ‘afternoon’ or ‘night’?’
What to look for in a frequency distribution of a categorical
variable: writing a report
A bar chart, in combination with a frequency table, is useful for gaining an overall view of a
frequency distribution of a categorical variable, the so-called ‘big picture’.
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Essential Further Mathematics – Core
Describing a bar chart
In describing a bar chart, we focus on two things:
the presence of a dominant category (or group of categories) in the distribution. This
is given by the mode. If there is no dominant category, then this should be stated.
the order of occurrence of each category and its relative importance
E
In commenting on these features, it is usual to support your conclusions with percentages.
When quoting percentages, it is also advisable to indicate at the beginning the total number of
cases involved. Using the information in Example 2 to describe the distribution of climate
type, you might write as follows:
Report
PL
The climate types of 23 countries were classified as being, `cold', `moderate' or `hot'. The
majority of the countries, 60.9%, were found to have a moderate climate. Of the remaining
countries, 26.1% were found to have a hot climate while 13.0% were found to have a cold
climate.
Stacked or segmented bar charts
Percentage
Frequency
25
A variation on the standard bar chart is the
Climate
Hot
segmented or stacked bar chart. In a
Moderate
20
segmented bar chart, the bars are stacked
Cold
on one another to give a single bar with
15
several components. The lengths of the
10
segments are determined by the frequencies.
When this is done, the height of the bar gives
5
the total frequency. Segmented bar charts
should only be used when there are
0
a relatively small number of components, usually no more than four or five, because it becomes
difficult to distinguish the components. The segmented bar chart above was formed from the
climate data used in Example 2. Note that a legend has been included to identify the segments.
In a percentage segmented bar chart,
100
Climate
the lengths of each of the segments in the
90
Hot
bar are determined by the percentages.
80
Moderate
70
When this is done, the height of the bar is
Cold
60
100. The percentage segmented bar chart
50
opposite was formed from the climate data
40
used in Example 2.
30
Percentage segmented bar charts are most
20
10
useful when we come to analyse the
0
relationship between two categorical
variables, as we will see in Chapter 4.
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6
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Chapter 1 — Organising and displaying data
7
Exercise 1B
1 a In a frequency table, what is the mode?
b Identify the mode in the following data sets:
i Grades:
A A C B A B B
B
ii Shoe size: 8 9 9 10 8 8 7 9 8
B
10
D C
12 8
10
E
2 The following data;
2 1 1 1 3 1 3 1 1 3 3
identifies the state of residence of a group of people, where 1 = Victoria, 2 = SA
and 3 = WA.
PL
a Form a frequency table (with both counts and percentages) to show the distribution of
state of residence for this group of people. Use the table in Example 1 as a model.
b Construct a bar chart using Example 2 as a model.
3 The size (S = small, M = medium, L = large) of 20 cars was recorded as follows:
S
M
S
S
L
L
M
S
M
M
M
M
L
M
S
S
S
S
M
M
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a Form a frequency table (with both counts and percentages) to show the distribution of
size for these cars. Use the table in Example 1 as a model.
b Construct a bar chart using Example 2 as a model.
4 The table shows the frequency distribution of School type for a number of schools. The table
is incomplete.
a Write down the information missing from the table.
b How many schools are categorised
as ‘Independent’?
c How many schools are there in total?
d What percentage of schools are
categorised as ‘Government’?
e Use the information in the frequency table
to complete the following report.
School type
Catholic
Government
Independent
Total
Frequency
Count Percent
4
11
5
20
25
100
Report
schools were classified according to school type. The majority of these schools,
%,
schools. Of the remaining schools,
were
while
were found to be
schools.
20% were
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8
Essential Further Mathematics – Core
5 The table shows the frequency distribution of the place of birth for 500 Australians.
a Is Place of birth a categorical or a numerical variable?
b Display the data in the form of a percentage
segmented bar chart.
c Use the information in the frequency table to
write a brief report.
Place of birth
Percent
Australia
Overseas
Total
78.3
21.8
100.1
Type of vehicle
Private
Commercial
Total
Frequency
Count
Percent
132 736
49 109
PL
a Copy and complete the table giving the
percentages correct to the nearest
whole number.
b Display the data in the form of a
percentage segmented bar chart.
E
6 The table records the number of new cars sold in Australia during the first quarter of 2005
categorised by type (private vehicle or commercial vehicle).
7 The table shows the frequency distribution of eye colour of 11 preschool children.
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a Use the information in the table to construct a
bar chart. Place the columns in order of
decreasing frequency.
b Use the information in the table to construct a
percentage segmented bar chart.
c Use the information in the table to write a brief
report.
8 Twenty-two students were asked the question, ‘How
often do you play sport?’ with the possible response:
‘Regularly’, ‘Sometimes’ or ‘Rarely’. The
distribution of responses is summarised in the
frequency table.
a Write down the information missing from the table.
b Use the information in the frequency
table to complete the following report.
Frequency
Eye colour Count Percentage
Brown
Hazel
Blue
Total
6
2
3
11
Plays sport
Frequency
Count Percent
Regularly
Sometimes
Rarely
Total
54.5
18.2
27.3
100.0
5
10
22.7
31.8
22
Report
When
students were asked the question, `How often do you play sport', the dominant
% of the students. Of the remaining students,
response was `Sometimes', given by
% of the students responded that they played sport
while
% said they that
.
they played sport
1.3
Organising and displaying numerical data
Frequency tables can also be used to organise numerical data. For discrete numerical data, the
process exactly mirrors that for categorical data. For continuous data, some modifications need
to be made because groups of data values, rather than individual values are listed.
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Chapter 1 — Organising and displaying data
9
Example 3
Frequency table for discrete numerical data
The family size of 11 preschool children (including the child itself) is as follows:
3 3 4 4 5 3 2 4 3 5 3
Display the data in the form of a frequency table.
Solution
E
Frequency
Family size
Count
Percent
2
3
4
5
1
5
3
2
9.1
45.5
27.3
18.2
Total
11
100.1
PL
1 Set up a table as shown. In the data set, the
variable family size takes the values 2, 3, 4
and 5. List these values under ‘Family size’
in some order, here increasing.
2 Count up the number of 2s, 3s, 4s and 5s in
the dataset. For example, there are five 3s.
Record these values in the ‘Count’ column.
3 Add the counts to find the total count, 11. Record this value in the ‘Count’ column opposite
‘Total’.
4 Convert the counts into percents. Record in the ‘Percent’ column. For example,
5
× 100% = 45.5%
11
5 Finally, total the percentages and record.
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percentage of 3s =
Grouping data
Some variables can only take on a limited range of values, for example, the number of children
in a family. Here, it makes sense to list each of these values individually when forming a
frequency distribution.
In other cases, the variable can take a large range of values, for example age (0–100). Listing
all possible ages would be tedious and would produce a large and unwieldy display. To solve
this problem, we group the data into a small number of convenient intervals. There are no hard
and fast rules for the number of intervals but, normally, between five and fifteen intervals are
used. Usually, the smaller the number of data values, the smaller the number of intervals.
Note that the intervals are defined so that it is quite clear into which interval each data value
falls. For example, you cannot define intervals as, 1–5, 5–10, 10–15, 15–20, . . . etc, as you
would not know into which interval to put the values, 5, 10, 15, etc.
Guideline for choosing the number of intervals
There are no hard and fast rules for the number of intervals to use but, normally, between
five and fifteen intervals are used.
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Essential Further Mathematics – Core
Example 4
Grouping data
The ages of a sample of 200 people aged from 16 to 72 years are to be recorded. Group the
ages into six equal-sized categories that will cover all of these ages.
Solution
E
Number of intervals: 6
57
Interval width =
= 9.5: use 10
6
PL
1 Write down the required number of intervals.
2 Determine interval width.
Ages range from 16 to 72, which covers
57 years. Six intervals will give intervals
57
= 9.5.
of width
6
Set the interval width to 10, the nearest
whole number above 9.5.
3 Choose a starting point that ensures that
the intervals cover the full range of values.
15 would be a suitable starting point.
4 Write down the intervals.
Starting point: 15
Intervals: 15−24, 25−34, . . . , 65−74
Once we know how to group data, we can form a frequency distribution for grouped data.
Example 5
A grouped frequency distribution for a continuous numerical variable
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10
The data below gives the average hours worked per week in 23 countries.
35.0, 48.0, 45.0, 43.0, 38.2, 50.0, 39.8, 40.7, 40.0, 50.0, 35.4, 38.8,
40.2, 45.0, 45.0, 40.0, 43.0, 48.8, 43.3, 53.1, 35.6, 44.1, 34.8
Form a grouped frequency table with five intervals.
Solution
1 Set up a table as shown. For five intervals and
data values ranging between 34.8 and 53.1,
use the intervals: 30.0–34.9, 35.0–39.9, . . . ,
50.0–54.9.
2 List these intervals, in ascending order, under
‘Average hours worked’.
3 Count the number of countries whose
average working hours fall into each of
the intervals. For example, six countries have
average working hours between 35.0 and 39.9.
Record these values in the ‘Count’ column.
4 Add the counts to find the total count, 23.
Record this value in the ‘Count’ column
opposite ‘Total’.
Average hours
worked
30.0−34.9
35.0−39.9
40.0−44.9
45.0−49.9
50.0−54.9
Total
Frequency
Count
Percent
1
6
8
5
3
4.3
26.1
34.8
21.7
13.0
23
99.9
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Chapter 1 — Organising and displaying data
11
5 Convert the counts into percentages. Record in the ‘Percent’ column.
For example, for 35.0–39.9 hours,
6
× 100% = 26.1%
percentage =
23
6 Finally, total the percentages and record.
E
There are two things to note in the frequency table in Example 5.
1 The intervals in this example are of width five. For example, the interval 35.0–39.9, is an
interval of width 5.0 because it contains all values from 34.9500 . . . to 39.9499.
2 The modal interval is 40.0–44.9 hours; eight (34.8%) of the countries have working hours
that fall into this interval.
The histogram
PL
How has forming a frequency table helped?
The process of forming a frequency table for a numerical variable:
orders the data
displays the data in a compact form
tells us something about the way the data values are distributed (the pattern of the data)
helps us identify the mode (the most frequently occurring value or interval of values)
SA
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The frequency histogram, or histogram for short, is a graphical way of presenting the
information in a frequency table for numerical data. Later in the chapter, you will learn about
two other graphical displays for numerical data, the stem plot and the dot plot.
Constructing a histogram from a frequency table
In a frequency histogram:
frequency (count or percent) is shown on the vertical axis
the values of the variable being displayed are plotted on the horizontal axis
for continuous data, each bar in a histogram corresponds to a data interval. For discrete
data, where there are gaps between values, the intervals start and end halfway between
values. Empty classes or missing discrete values have bars of zero height.
the height of the bar gives the frequency (usually the count, but it can equally well be the
percentage)
Example 6
Constructing a histogram from a frequency table:
continuous numerical variable
Construct a histogram for this frequency table.
Average hours worked
30.0–34.9
35.0–39.9
40.0–44.9
45.0–49.9
50.0–54.9
Total
Frequency (count)
1
6
8
5
3
23
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Essential Further Mathematics – Core
Solution
Example 7
9
8
Frequency
7
6
5
4
E
3
2
1
0
25
30
35 40 45 50 55
Average hours worked
60
PL
1 Label the horizontal axis with the variable
name, ‘Average hours worked’. Mark in the
scale using the beginning of each interval
as the scale points: that is 30, 35, . . .
2 Label the vertical axis ‘Frequency’. Scale
allowing for the maximum frequency, 8.
Ten would be appropriate. Mark in the scale
in units.
3 Finally, for each interval, 30.0–34.9,
35.0–39.9, . . . , draw in a bar with the base
starting at the beginning of each interval
and finishing at the beginning of the next.
The height of the bar is made equal to the
frequency.
Constructing a histogram from a frequency table:
discrete numerical variable
Construct a histogram for this frequency table.
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Family size
2
3
4
5
Total
Frequency (count)
1
5
3
2
11
Solution
1 Label the horizontal axis with the variable name,
‘Family size’. Mark the scale in units, so that it
includes all possible values.
2 Label the vertical axis ‘Frequency’. Scale to
allow for the maximum frequency, 5. Five
would be appropriate. Mark the scale in units.
3 Draw in a bar for each data value. The width of
each bar is 1, starting and ending halfway between
data values. For example, the base of the bar
representing a family size of 2 starts at 1.5 and
ends at 2.5. The height of the bar is made equal to
the frequency.
5
4
Frequency
12
3
2
1
0
1
2
3
4
Family size
5
6
Constructing a histogram from raw data
It is relatively quick to construct a histogram from a preprepared frequency table. However, if
you only have raw data (as you mostly do), it is a very slow process because you have to
construct
the frequency
table first.
Fortunately,
a graphics calculator
do this
forLipson
us.
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Chapter 1 — Organising and displaying data
13
How to construct a histogram using the TI-Nspire CAS
Display the following set of 27 marks in the form of a histogram.
11
12
4
18
25
22
15
17
7
18
14
23
13
15
12
17
15
18
13
22
16
23
14
PL
Steps
1 Start a new document: Press c and
select 6:New Document (or press / +
N ). If prompted to save an existing
document move cursor to No and
press enter .
14
13
E
16
15
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2 Select 3:Add Lists & Spreadsheet.
Enter the data into a list named marks.
a Move the cursor to the name space of
column A (or any other column) and
type in marks as the list name. Press
enter .
b Move the cursor down to row 1, type
in the first data value and press enter .
Continue until all the data has been
entered. Press enter after each entry.
3 Statistical graphing is done through the
Data & Statistics application.
Press c and select 5:Data & Statistics.
A random display of dots will
appear – this is to indicate list data are
available for plotting. It is not a
statistical plot.
a Move cursor to the text box area
below the horizontal axis. Press x
(click) when prompted and select the
variable marks. Press enter to paste
the variable marks to that axis.
Note:
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Essential Further Mathematics – Core
Note:
PL
Your screen should now look like that
shown opposite. This histogram has a
column (or bin) width of 2 and a
starting point of 3.
E
b A dot plot is displayed as the default
plot. To change the plot to a
histogram press b /1:Plot
Type/3:Histogram.
Keystrokes: b 1 3
Count is the same as frequency.
To change the count axis to a
percentage axis, press / +
b /4:Scale/1:Percent
Hint: Pressing / + b gives you
a contextual menu that enables you
to do things that relate only to histograms.
1
2
4 Data analysis
a Move the cursor onto any column and
a will show and the column will
become highlighted. Holding down
the centre mouse button x until a
appears displays the end points of the
selected column (here 11 to < 13)
and its frequency or count (i.e. 3).
b Press x to change the hand back to
. To view other column data values
move the cursor to another column
and repeat the steps in 4a above.
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14
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Chapter 1 — Organising and displaying data
15
PL
E
Hint: To unshade previously selected columns move the cursor to the open area and
press x .
to undo the
Hint: If you accidentally move a column or data point, press / +
move.
5 Change the histogram column (bin) width to 4 and the starting point to 2.
a Press / + b to get the contextual menu as shown (below left).
b Select 5:Bin Settings.
c In the settings menu (below right) change the Width to 4 and the Starting Point
(Alignment) to 2 as shown. Press enter .
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d A new histogram is displayed with column width of 4 and a starting point of 2 but it
no longer fits the viewing window (below left). To solve this problem press / +
b /6:Zoom/2:Zoom-Data to obtain the histogram shown below right.
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Essential Further Mathematics – Core
How to construct a histogram using the ClassPad
Display the following set of 27 marks in the form of a histogram.
11
12
4
18
25
22
15
17
7
18
14
23
13
15
Steps
1 From the application menu screen,
locate the built-in Statistics application.
12
17
15
18
13
22
16
23
14
PL
to open.
Tap
from the icon panel (just
Tapping
below the touch screen) will display the
application menu if it is not already
visible.
14
13
E
16
15
2 Enter the data into a list named marks.
To name the list:
a Highlight the heading of the first list
by tapping it.
b Press k on the front of the
tab.
calculator and tap the
c Type the word marks and press E.
To enter the data:
d Type in each data value and press E
or
(which is found on the cursor
button on the front of the calculator)
to move down to the next cell.
The screen should look like the one
shown below (left).
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16
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PL
3 Set up the calculator
to plot a statistical graph.
from the toolbar at
a Tap
the top of the screen. This
opens the Set StatGraphs
dialog box.
b Complete the dialog
box as given below. For
• Draw: select On
• Type: select Histogram ( )
• XList: select main \
marks ( )
• Freq: leave as 1
c Tap h to confirm your
selections.
Note: To make sure only this
graph is drawn, select SetGraph
from the menu bar at the top and
confirm there is a tick only beside
StatGraph1 and no others.
E
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Chapter 1 — Organising and displaying data
17
SA
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4 To plot the graph:
in the toolbar at the
a Tap
top of the screen.
b Complete the Set Interval
dialog box for the histogram
as given below. For
• HStart: type in 2 (i.e. the
starting point of the first
interval)
• HStep: type in 4 (i.e. the
interval width)
Tap OK.
The screen is split into two halves,
with the graph displayed in the
bottom half, as shown opposite.
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PL
Tapping r from the icon panel
will allow the graph to fill the entire
screen.
The graph is drawn in an
automatically scaled window. Tap
r again to return to half-screen
size.
from the toolbar places
Tapping
a marker at the top of the first
column of the histogram (see
opposite) and tells us that
a the first interval begins
at 2 (xc = 2)
b for this interval, the frequency
is 1(Fc = 1)
To find the frequencies and
starting points of the other intervals,
use the arrow ( ) to move from
interval to interval.
E
Essential Further Mathematics – Core
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18
Exercise 1C
1 The number of occupants in nine cars stopped at a traffic light was:
1 1 2 1 3 1 2 1 3
What is the mode of this data set? What does this tell us?
2 The number of surviving grandparents for 11 preschool children is listed below.
0 4 4 3 2 3 4 4 4 3 3
Form a frequency table to show the distribution of the number of surviving grandparents.
3 a Write down the missing information in the
frequency table.
b How many families had only one child?
c How many families had more than one
child?
d What percentage of families had no
children?
e What percentage of families had fewer
than three children?
No. of children
in family
0
1
2
3
4
Total
Frequency
Count
%
3
10
6
2
21
47.6
28.6
9.5
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Chapter 1 — Organising and displaying data
19
4 a Salaries of women teaching in a school range from $20 106 to $63 579. Group the salaries
into five equal-sized categories that cover all teaching salaries.
b The number of students in VCE Further Mathematics classes ranges from 6 to 33. Group
class size into six equal-sized categories that cover all Further Mathematics class sizes.
c The amount of money carried by a sample of 23 students ranges from nothing to $8.75.
Group the amount of money carried by the students into five equal-sized categories that
cover all amounts of money carried by the students.
5 The histogram opposite was formed by recording the
number of words in 30 randomly selected sentences.
E
Frequency (%)
30
25
20
15
10
PL
a What percentage of these sentences contained:
ii 25–29 words?
i 5–9 words?
iv
fewer than 15 words?
iii 10–19 words?
Give answers correct to the nearest per cent.
b How many of these sentences contained:
ii more than 25 words?
i 20–24 words?
c What is the mode (modal interval)?
35
5
0
5 10 15 20 25 30
Number of words in sentence
Population density
0–199
200–399
400–599
600–799
800–999
Total
Frequency (count)
11
4
4
2
1
22
7 Use the information in the table opposite to
help you construct a histogram to display the
distribution of the number of rooms in the
houses of 11 preschool children. Use the
histogram in Example 7 as a model. Label
axes and mark in scales.
Number of rooms
4
5
6
7
8
Total
Frequency (count)
3
0
1
3
4
11
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6 Use the information in the table opposite to
help you construct a histogram to display
population density. Use the histogram in
Example 6 as a model. Label axes and
mark in scales.
8 The pulse rates of 23 students are given below.
86
70
82
78
96
69
71
77
90
64
78
80
68
83
71
78
68
88
88
70
76
86
74
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20
Essential Further Mathematics – Core
E
a Use a graphics calculator to construct a histogram. So that the first column starts at 63
and the column width is two.
b For this histogram:
i what is the starting point of the third column?
ii what is the ‘count’ for the third interval? What actual data column width is five values
does this include?
c Use the window menu to redraw the histogram so that the first starts at 60.
d For this histogram, what is the count in the interval ‘65 to <70’?
9 The following data values are the number of children in the families of 25 VCE students:
1 6 2 5 5 3 4 1 2 7 3 4 5 3 1 3 2 1 4 4 3 9 4 3 3
What to look for in a histogram
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1.4
PL
a Use a graphics calculator to construct a histogram. So that the first column width is one
and starts at 0.5.
b For this histogram; what is the starting point for the fourth column and what is the count?
c Use the window menu to redraw the histogram so that the first interval width is two and
starts at 0.
d For this histogram:
i what is the count in the interval from 6 to less than 8?
ii what actual data value(s) does this interval include?
Histograms are useful for gaining an overall view of a frequency distribution of a numerical
variable, the so-called ‘big picture’.
Describing a histogram
In describing a histogram, we focus on three things:
shape and outliers (values in the data set that appear to stand out from the rest)
centre
spread
Shape
How is the data distributed? Is the histogram peaked, that is, do some data values tend to occur
much more frequently than others, or is it relatively flat, showing that all values in the
distribution occur with approximately the same frequency?
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Chapter 1 — Organising and displaying data
21
lower tail
peak
upper tail
10
8
6
4
2
0
Histogram 1
peak
peak
E
10
8
6
4
2
0
Frequency
Frequency
Symmetric distributions
If a histogram is single-peaked, does the histogram region tail off evenly on either side of the
peak? If so, the distribution is said to be symmetric (see Histogram 1).
Histogram 2
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PL
A single-peaked symmetric distribution is characteristic of the data that derives from
measuring variables such as people’s heights, intelligence test scores, weights of oranges in a
storage bin, or any other data for which the values vary evenly around some central value. The
histogram for average hours worked (see Example 6) would be classified as approximately
symmetric.
The double-peaked distribution (Histogram 2) is symmetric about the dip between the two
peaks. A histogram which has two distinct peaks indicates a bimodal (two modes) distribution.
A bimodal distribution often indicates that the data has come from two different
populations. For example, if we were studying the distance the discus is thrown by Olympic
level discus throwers, we would expect a bimodal distribution if both male and female throwers
were included in the study.
Skewed distributions
Sometimes a histogram tails off primarily in one direction. Such distributions are said to be
skewed.
If a histogram tails off to the right we say that it is positively skewed (Histogram 3). The
distribution of salaries of workers in a large organisation tends to be positively skewed. Most
workers earn a similar salary with some variation above or below this amount, but a few earn
more and even fewer, like the senior manager, earn even more. The distribution of house prices
also tends to be positively skewed.
long upper tail
+ve skew
Histogram 3
10
Frequency
Frequency
peak
10
8
6
4
2
0
8
long lower tail
peak
–ve skew
6
4
2
0
Histogram 4
If a histogram tails off to the left we say that it is negatively skewed (Histogram 4). The
distribution of age at death tends to be negatively skewed. Most people die in old age, a few in
middle age and even fewer in childhood.
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Essential Further Mathematics – Core
Outliers
Centre
E
PL
Frequency
Outliers are any data values that stand out from the main body of data. These are data values
that are atypically high or low. See for example, Histogram 5, which shows an outlier. In this
case it is a data value which is atypically low compared to the rest of the data values.
outlier
main body of data
Outliers can indicate errors made collecting
10
or processing data, for example a person’s
8
age recorded as 365. Alternatively, they may
6
indicate data values that are very different
4
2
from the rest of the values. For example,
0
compared to her students’ ages, a teacher’s
Histogram 5
age is an outlier.
Frequency
Histograms 6 to 8 display the distribution
8
of test scores for three different classes
7
6
taking the same subject. They are identical
5
in shape, but differ in where they are
4
located along the axis. In statistical terms
3
we say that the distributions are ‘centred’
2
at different points along the axis.
1
But what do we mean by the centre of a
0
50 60 70 80 90 100 110 120 130 140 150
distribution? This is an issue we will return
Histograms 6 to 8
to in more detail later. For the present we
will take centre to be the middle of the
distribution.
The middle of a symmetric distribution is reasonably easy to locate by eye. Looking at
Histograms 6 to 8, it would be reasonable to say that the centre or middle of each distribution
lies roughly halfway between the extremes; half the observations would lie above this point
and half below. Thus we might estimate that Histogram 6 (yellow) is centred at about 60,
Histogram 7 (light blue) at about 100, and Histogram 8 (dark blue) at about 140.
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Chapter 1 — Organising and displaying data
23
E
Frequency
For skewed distributions, it is more difficult to estimate the middle of a distribution by eye.
The middle is not halfway between the extremes because, in a skewed distribution, the scores
tend to bunch up at one end. However, if we
5
line that divides
imagine a cardboard cut-out of the histogram,
the area of the
4
the midpoint lies on the line that divides the
histogram in half
histogram into two equal areas (Histogram 9).
3
Using this method, we would estimate the
2
centre of the distribution to lie somewhere
between 35 and 40, but closer to 35, so we
1
might opt for 37. However, remember this is
0
15 20 25 30 35 40 45 50
only an estimate.
PL
Histogram 9
Spread
If the histogram is single peaked, is it narrow? This would indicate that most of the data values
in the distribution are tightly clustered in a small region. Or is the peak broad? This would
indicate that the data values are more widely spread out. Histograms 10 and 11 are both single
peaked. Histogram 10 has a broad peak indicating that the data values are not very tightly
clustered about the centre of the distribution. In contrast, Histogram 11 has a narrow peak,
indicating that the data values are tightly clustered around the centre of the distribution.
2
4
6
8 10 12 14 16 18 20 22
Histogram 10
Frequency
Frequency
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wide central region
10
8
6
4
2
0
narrow central region
20
16
12
8
4
0
2
4 6
8 10 12 14 16 18 20 22
Histogram 11
But what do we mean by the spread of a distribution? We will return to this in more detail
later. For a histogram we will take it to be the maximum range of the distribution.
Range
Range = largest value − smallest value
For example, Histogram 10, has a spread (maximum range) of 22 (22 − 0) units, which is
considerably greater than the spread of Histogram 11, which has a spread of 12 (18 − 6) units.
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Essential Further Mathematics – Core
Example 8
Describing a histogram in terms of shape, centre and spread
35
30
25
20
15
10
E
Frequency (count)
The histogram opposite shows the distribution of the
number of phones per 1000 people in 85 countries.
a Describe its shape and note outliers (if any).
b Locate the centre of the distribution.
c Estimate the spread of the distribution.
5
0
170 340 510 680 850 1020
Number of phones (per 1000 people)
Solution
The distribution is positively skewed.
There are no outliers.
The distribution is centred in the interval
170−340 phones/1000 people.
Spread = 1020 − 0
= 1020 phones/1000 people
PL
a Shape and outliers
b Centre Count up the frequencies from
either end to find the middle interval.
c Spread Use the maximum range to
estimate the spread.
It should be noted that with grouped data, it is difficult to precisely determine the location of
the centre of a distribution from a histogram. So, when working with grouped data, it is
acceptable to state that the centre of a distribution lies in the interval 170–340. We will learn
how to solve this problem later in the chapter.
If you were using the histogram above to describe the distribution in a form suitable for a
statistical report, you might write as follows.
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Report
For the 85 countries, the distribution of the number of phones per 1000 people is positively
skewed. The centre of the distribution lies somewhere in the interval 170−340 phones/1000
people. The spread of the distribution is 1020 phones/1000 people. There are no outliers.
Exercise 1D
1 Label each of the following histograms as approximately symmetric, positively skewed or
negatively skewed, and identify the following:
i the mode
a
Frequency
20
15
10
ii any potential outliers
iii the approximate location of the centre
b
80
Frequency
24
60
40
5
20
0
0
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Histogram
B Evans, Lipson
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Chapter 1 — Organising and displaying data
25
c
d
15
20
Frequency
Frequency
20
10
5
15
10
5
0
0
Histogram D
Histogram C
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E
PL
Frequency
10
2 These three histograms show
the marks obtained by a group
9
8
of students in three subjects.
7
a Are each of the distributions
6
approximately symmetric or
5
skewed?
4
b Are there any clear outliers?
3
c Determine the interval
2
containing the central mark
1
for each of the three subjects.
0
d In which subject was the
2 6 10 14 18 22 26 30 34 38 42 46 50
spread of marks the least? Use
Subject A
Subject B
Subject C
Marks
the range to estimate the spread.
e In which subject did the marks vary most? Use the range to estimate the spread.
3 Label each of the following histograms as approximately symmetric, positively skewed or
negatively skewed, and identify the following:
ii any potential outliers
iii the approximate location of the centre
b
20
Frequency
Frequency
i the mode(s)
a
15
10
5
0
80
60
40
20
0
Histogram B
Histogram A
20
15
10
5
0
d
Frequency
Frequency
c
20
15
10
5
0
Histogram D
Histogram C
20
15
f
Frequency
Frequency
e
10
5
0
80
60
40
20
0
Histogram E
Histogram F
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26
Essential Further Mathematics – Core
Use the histogram to complete the report
below describing the distribution of
pulse rate in terms of shape, centre,
spread and outliers (if any).
6
5
Frequency (count)
4 This histogram shows the distribution
of pulse rate (in beats per minute) for
28 students.
4
3
2
1
E
0
60 65 70 75 80 85 90 95 100 105 110 115
Pulse rate (beats per minute)
Report
1.5
PL
For the
students, the distribution of pulse rates is
with an outlier. The
beats per minute and the spread of the
centre of the distribution lies in the interval
beats per minute. The outlier lies in the interval
beats per minute.
distribution is
Stem-and-leaf plots and dot plots
Stem plots
SA
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A stem-and-leaf plot, or stem plot for short, is an alternative to the histogram. It is
particularly useful for displaying small to medium sized sets of data (up to about 50 data
values) and has the advantage of retaining all the original data values. This makes it useful for
further computations. A stem plot is also a very quick and easy way to order and display a set
of data by hand. Like a histogram, the stem plot gives information about the shape, outliers,
centre and spread of the distribution.
One of the stem plot’s advantages over a histogram in describing distributions is being able
to see all the actual data values. This enables the centre and the range of the distribution to be
located more precisely. It also enables the clear identification of outliers.
Constructing a stem plot
In a stem-and-leaf plot, each data value is separated into two parts: the leading digit(s) form
the stem, and the trailing digit becomes the leaf. For example, in a stem-and-leaf plot, the
data values 25 and 132 are represented as follows:
25 is represented by
132 is represented by
Stem Leaf
2 5
13 2
and so on.
To construct a stem plot, enter the stems to the left of a vertical dividing line, and the leaves
for each data point to the right. Usually we first construct an unordered stem plot by
systematically plotting each data point as listed in the data set. From the unordered
stem-and-leaf plot an ordered stem plot is then easily obtained. In an ordered stem plot the
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Chapter 1 — Organising and displaying data
27
leaves increase in value as they move away from the stem. It is usually the ordered stem plot
that we want, because an ordered stem plot makes it easy to find the key values.
Example 9
Constructing an ordered stem plot
Solution
0
1
2
3
4
5
PL
1 The data set has values in the units, tens,
twenties, thirties, forties and fifties. Thus,
appropriate stems are 0, 1, 2, 3, 4, and 5.
Write these down in ascending order,
followed by a vertical line.
E
University participation rates (%) in 23 countries are given below.
26 3 12 20 36 1 25 26 13 9 26 27 30 1 15 21 7 8 22 3 37 17 55
Display the data in the form of an ordered stem plot.
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2 Now attach the leaves.
The first data value is ‘26’. The stem is ‘2’
and the leaf is ‘6’. Opposite the 2 in the stem
write down the number 6, as shown.
The second data value is ‘3’ or ‘03’. The stem
is ‘0’ and the leaf is ‘3’. Opposite the 0 in the
stem, now write down the number 3, as shown.
Continue systematically working through
the data following the same procedure until
all points have been plotted. You will then
have the unordered stemplot as shown.
3 Ordering the leaves in increasing value as
they move away from the stem gives the
ordered stem plot as shown.
0
1
2
3
4
5
0
1
2
3
4
5
6
3
6
0 3 1 9 1 7
1 2 3 5 7
2 6 0 5 6 6
3 6 0 7
4
5 5
unordered stemplot
0 1 1 3 3 7
1 2 3 5 7
2 0 1 2 5 6
3 0 6 7
4
5 5
ordered stemplot
8 3
7 1 2
8 9
6 6 7
Using a stem plot to describe a distribution
Stem plots are just like histograms, except that you can see all the data values. This enables
more precise
estimates
to be madeSample
of thepages
centre
and spread.
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Essential Further Mathematics – Core
Spread (range)
E
Methods for determining the centre, spread and outliers from a stem plot
Centre (middle) Count up from either end of the distribution until you find the middle
value; the value which has an equal number of data values either side.
n+1
th
For an odd number of data values, n, the middle value is the
2
value. Thus, the median will be an actual data value.
n+1
th
For an even number of data values, n, the middle value is the
2
value. Thus, the median will lie between two data values.
Subtract the smallest data value from the largest data value.
Range = largest value − smallest value
Example 10
Data values that stand out from the main body of data are called outliers.
Their values can be read directly from the stem plot.
PL
Outliers
Describing a stem plot in terms of shape, centre and spread
Test marks
The ordered stem plot opposite shows the
0
distribution of test marks of 23 students:
1 5 9 9 9
a Name its shape and note outliers (if any).
2 0 4 5 7 8 8 8
b Locate the centre of the distribution.
3 0 3 5 5 6 8
c Estimate the spread of the distribution.
4 1 2 3 3 5
d Write down the values of any outliers.
5
6 0
Solution
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28
a Shape
b Centre There are 23 data values, the middle
value is the 12th value. Check by counting.
c Spread Use the range to estimate the spread.
d Outlier Read off the value of the outlier
The distribution is approximately
symmetric with one outlier.
The distribution is centred at 30 marks.
Spread = 60 − 15 = 45 marks
Outlier = 60 marks
If you were using the stem plot to describe the distribution in a form suitable for a statistical
report, you might write as follows.
Report
For the 23 students, the distribution of marks is approximately symmetric with an outlier.
The centre of the distribution is at 30 marks and the distribution has a spread of 45
marks. The outlier is a mark of 60.
Split stems
In some instances, using the simple process outlined above produces a stem plot that is too
bunched up to give us a good overall picture of the variation in the data. This is often the case
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Chapter 1 — Organising and displaying data
29
when the data values all have the same first digit or the same one or two first digits. For
example, a group of 17 VCE students recently sat for a statistics test marked out of 20. The
results are as shown below.
2
12
13
9
18
17
7
16
12
10
16
14
11
15
16
15
17
0
1
2
0
7
1
9
2
2
3
4
5
5
6
6
6
7
E
Using the above process to form a stem plot we end up with a bunched-up plot like the one
below:
7
8
1 (10−19)
Single stem
PL
When this happens, the stem plot scale can be stretched out by ‘splitting’ the stems. Generally
the stem is split into halves or fifths. For example, for the interval 10–19, the split stem system
works as follows.
1 (10−14)
1 (15−19)
1 (10−11)
1 (12−13)
1 (14−15)
1 (16−17)
1 (18−19)
Stem split into halves
Stem split into fifths
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In a stem plot with a single stem, the ‘1’ represents the interval 10–19.
In a stem plot with its stem split into halves, the top ‘1’ represents the interval 10–14,
while the bottom ‘1’ represents the interval 15–19.
In a stem plot with its stem split into fifths, the top ‘1’ represents the interval 10–11, the
second ‘1’ represents the interval 12–13, the third ‘1’ represents the interval 14–15, the
fourth ‘1’ represents the interval 16–17, while the bottom ‘1’ represents the interval 18–19.
Comparison of stem plots with different split stems
Using a split stem plot to display the test marks can show features not revealed by a standard
plot. This can be seen in the next plot with the stem split into fifths, indicating that a mark of 2
is an outlier.
0 2 7 9
1 0 1 2 2 3 4 5 5 6 6 6 7 7 8
Single stem
0
0
1
1
2
7 9
0 1 2 2 3 4
5 5 6 6 6 7 7 8
Stem split into halves
0
0
0
0
0
1
1
1
1
1
2
7
9
0
2
4
6
8
1
2 3
5 5
6 6 7 7
Stem split into fifths
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Essential Further Mathematics – Core
Back-to-back stem plots
A back-to-back plot has a single stem with
8 6 4 2 1 1 1 5 6 8 9 8
two sets of leaves as shown.
1 0 2 1 1 1 4
The real value of back-to-back plots is
3 2 2 3 1 1 5
that they are a useful tool for comparing
the distribution of two sets of data values for the same variable.
Using a back-to-back stem plot to compare two distributions
Solution
Report
Test 1
Test 2
9
8 6 6 5
9 8 7 6 5 5
7 5 3 3 2 2
9
0
0
0
0
1
2
3
4
5
8
9
0
0
1
0
4
3
2
5 7 8 8
5 5 6 8 9
3 3 4 5
PL
Use the back-to-back stem
plot to write a report
comparing the distribution
of the two sets of test
marks in terms of shape,
centre, spread and outliers.
E
Example 11
8
The distribution of the Test 1 marks is negatively skewed while the distribution of the
Test 2 marks is approximately symmetric. The two distributions have similar centres,
36.5 and 35. The spread of the Test 1 marks is less than the Test 2 marks, 29
compared to 42. There are no outliers.
SA
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30
Dot plots
The simplest way to display numerical data is to form a dot plot. A dot plot consists of a
number line with each data point marked by a dot. When several data points have the same
value, the points are stacked on top of each other. Like stem plots, dot plots are a great way for
displaying small data sets and have the advantage of being very quick to construct by hand.
They are best when the data values are relatively close together.
Example 12
Constructing a dot plot
The ages (in years) of the 13 members of a sporting team are:
22 19 18 19 23 25 22 29 18 22 23 24 22
Construct a dot plot.
Solution
1 Draw in a number line, scaled to include all
data values. Label the line with the variable
being displayed.
17
18
19
20
21
22
23 24
25 26 27
28
29 30
Age (years)
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Chapter 1 — Organising and displaying data
31
2 Plot each data value by marking in a dot
above the corresponding value on the
number line.
17
18
19
20 21
22 23 24 25 26 27
28 29 30
Age (years)
E
Interpreting a dot plot
Dot plots are interpreted in much the same way as stem plots. However, usually there is little
we can say about the shape of the distribution from the dot plot because there are not sufficient
data points for any pattern to be revealed.
From the dot plot in Example 12, we see that the distribution of ages is centred at 22 years
(the middle value) with a spread of 11 years (29 − 18 = 11).
Which graph?
PL
One of the issues that you will face is choosing a suitable graph to display a distribution. The
following guidelines might help you in your decision making. They are guidelines only,
because in some instances, there may be more than one suitable graph.
Type of data
Graph
Qualifications on use
Categorical
Bar chart
Segmented bar chart
Not too many categories (4 or 5 maximum)
Histogram
Stem plot
Dot plot
Best for medium to large data sets (n ≥ 40)
Best for small to medium sized data sets (n ≤ 50)
Only suitable for small data sets (n ≤ 20)
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Numerical
Exercise 1E
1 The data below gives the urbanisation rates (%) in 23 countries.
54 99 22 20 31
3 22 9 25 3 56 12
16
9 29
6 28 100 17 9 35 27 12
a Construct an ordered stem plot.
b What advantage does a stem plot have over a histogram?
2 For each of the following stem plots (A, B and C):
a name its shape and note outliers (if any)
b locate the centre of the distribution
c determine the spread of the distribution
d write down the values of outliers (if any)
Stem plot A
Stem plot B
Stem plot C
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
1
2
3
4
5
6
0
2
0
2
0
2
0 1 1 2 6 7 7 9
2 3 5 5 5 5 6
1 4 7
2
0
1
0
2
1
2
2
3
0
2
2
3
6
1
2
4
9
5 6 8 8
4 5 9 9 9
4 6
1 3
2
0 2 4
1 1 3 5 8 8
0 0 4 4 4 7 7 8 9
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Essential Further Mathematics – Core
3 The data below gives the wrist circumference (in cm) of 15 men.
16.9 17.3 19.3 18.5 18.2 18.4 19.9 16.7 17.1 17.6 17.7 16.5 17.0 17.2 17.6
a Construct a stem plot for wrist circumference using:
i stems 16, 17, 18, 19
ii these stems split into halves
b Which stem plot appears to be more appropriate for this data?
c Use the stem plot with split stems to help you complete the report below
E
Report
For the
men, the distribution of their wrist circumference is
. The centre of
cm and it has a spread of
cm. There are no outliers.
the distribution is at
PL
4 The data below gives the weight (in kg) of 22 students.
57 58 62
84 64 74 57 55 56 60 75
68 59 72 110 56 69 56 50 60 75 58
a Construct a stem plot for weight using:
i stems 5, 6, 7, 8, 9, 10 and 11
ii these stems split into halves
b Use the stem plot with a split stem to write a brief report on the distribution of the
weights of the students in terms of shape (and outliers), centre and spread. Use the report
from Question 3 as a model.
5 The number of possessions (kicks, mark, handballs, knockouts, etc.) recorded for players in
a football game between Carlton and Essendon is shown below.
SA
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32
Carlton
Essendon
10 44 32 44 19 35 11 5 24 28 21 32 21 59 21 12 19 26 23 22 29 34
22 34 36 20 14 25 16 19 32 32
14 29 8 22 21 26 44 19 21 22
a Display the data in the form of an ordered back-to-back stem plot.
b Complete the following report comparing the two distributions in terms of shape (and
outliers), centre and spread.
Report
The distribution of the number of possessions is
for both teams. The two
and
possessions respectively. The spread of
distributions have similar centres, at
possessions, compared to
possessions for
the distribution is less for Carlton,
Essendon.
6 The following data gives the number of children in the families of 14 VCE students:
1 6 2 5 5 3 4 4 2 7 3 4 3 4
a Construct a dot plot.
b What is the mode?
c What is:
i the centre?
ii the spread?
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Chapter 1 — Organising and displaying data
33
7 The following data gives the life expectancies in years of 13 countries:
76 75 74 74 73 73 75 71 72 75 75 78 72
a Construct a dot plot.
b What is the mode?
c What is:
i the centre?
ii the spread?
PL
number of passengers in a bus — 1000 buses in sample
amount of petrol purchased (in litres) — 30 petrol purchases
type of petrol purchased (super, unleaded, premium)
prices of houses sold in Melbourne over a weekend
the number of medals won by countries winning medals at the Olympics
state of residence of a sample of 200 Australians
number of cigarettes smoked in a day (a sample of 120 people)
resting pulse rates of 7 students
SA
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a
b
c
d
e
f
g
h
E
8 Data has been collected for each of the following variables. The data is to be displayed
graphically. In each case, decide which is the most appropriate graph. Select from bar chart,
histogram, stem plot or dot plot. Sometimes more than one sort of graph is suitable.
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Essential Further Mathematics – Core
Key ideas and chapter summary
Data can be classified as numerical or categorical.
Frequency table
A frequency table is a listing of the values a variable takes in a data set,
along with how often (frequently) each value occurs.
Frequency can be recorded as a:
r count: the number of times a value occurs; for example, the number
of females in the data set is 32
r percent: the percentage of times a value occurs; for example, the
percentage of females in the data set is 45.5%
Categorical data
Categorical data arises when classifying or naming some quality or
attribute; for example, place of birth, hair colour.
Bar chart
Bar charts are used to display the frequency distribution of categorical
data.
PL
E
Types of data
For a small number of categories, the distribution of a categorical
Describing
variable is described in terms of the dominant category (if any), the
distributions of
categorical variables order of occurrence of each category and its relative importance.
Mode
The mode is the value or group of values that occurs most often
(frequently) in a data set. For example, for the data 2 1 1 3 3 2 5 1 6 1 1
2 1 1, the mode is 1, because it is the data value that occurs most often.
SA
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Review
34
Numerical data
Numerical data arises from measuring or counting some quantity; for
example, height, number of people, etc.
Numerical data can be discrete or continuous. Discrete data arises when
you count. Continuous data arises when you measure.
Histogram
A histogram is used to display the frequency distribution of a numerical
variable: suitable for medium to large sized data sets.
Stem plot
A stem plot is an alternative graphical display to the histogram: suitable
for small to medium sized data sets.
The advantage of the stem plot over the histogram is that it shows the
value of each data point.
Dot plot
A dot plot consists of a number line with each data point marked by a
dot: suitable for small sets of data only.
Describing the
distribution of a
numerical variable
The distribution of a numerical variable can described in terms of:
r shape:
symmetric or skewed (positive or negative)?
r outliers: values that appear to stand out
r centre:
the midpoint of the distribution (median)
r spread: one measure is the range of values covered
(Range = largest value − smallest value)
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Chapter 1 — Organising and displaying data
35
SA
M
PL
E
Having completed this chapter you should be able to:
differentiate between numerical and categorical data
interpret the information contained in a frequency table
identify and interpret the mode
construct a bar chart or histogram from a frequency table
decide when it is appropriate to use a histogram rather than a bar chart and vice
versa
construct a histogram from raw data, using a graphics calculator
construct a dot plot and a stem plot from raw data, using split stems if required
locate the mode of a distribution from a histogram, stem plot, dot plot or bar chart
recognise a symmetric, positively skewed and negatively skewed histogram or stem
plot
identify potential outliers in a distribution from its histogram or stem plot
write a brief report to describe the distribution of a numerical variable in terms of
shape, centre, spread and outliers (if any)
write a brief report to describe the distribution of a categorical variable in terms of
the dominant category (if any), the order of occurrence of each category and their
relative importance
Multiple-choice questions
The following information relates to Questions 1 to 3
A survey collected information about the number of cars owned by a family and the car
size (small, medium, large).
1 The variables Number of cars owned and car Size are:
A both categorical variables
B both numerical variables
C a categorical and a numerical variable respectively
D a numerical and a categorical variable respectively
E neither numerical nor categorical variables
2 To graphically display the information about car size you could use a:
A dot plot
B stem plot
C histogram
D segmented bar chart
E back-to-back stemplot
3 The Number of cars owned is:
A a continuous numerical variable
C a continuous categorical variable
E none of the above
B a discrete numerical variable
D a discrete categorical variable
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Skills check
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Essential Further Mathematics – Core
The following information relates to Questions 4 to 6
A group of teenagers were asked to
nominate their favourite leisure
Leisure activity
activity. Their responses have been
Sport
organised into a frequency table as
Listening to music
shown. Some information is missing.
Watching TV
Other
Total
Frequency
Count Percentage
73
70
29.2
19.2
23.6
E
59
250
PL
4 The percentage of students who said that listening to music was their favourite
leisure activity is:
A 17.5
B 28.0
C 29.2
D 50.0
E 70.0
5 The number of students who said watching TV was their favourite leisure activity
is:
A 19
B 48
C 62
D 125
E 70.0
6 For these students, the most popular leisure activity is:
A sport
B listening to music
C watching TV
D other
E can’t tell
7 The total number of students in the class is:
A 6
B 18
C 20
D 21
E 22
8 The number of students in the class who
obtained a test score less then 14 is:
A 4
B 10
C 14
Frequency
Questions 7 to 11 relate to the histogram shown below
This histogram displays the test scores of a class
6
of Further Mathematics students.
5
SA
M
Review
36
4
3
2
1
0
6 8 10 12 14 16 18 20 22 24 26 28
Test score
D 17
E 28
9 The histogram is best described as:
A negatively skewed
B negatively skewed with an outlier
C approximately symmetric
D approximately symmetric with outliers
E positively skewed
10 The centre of the distribution lies in the interval:
A 8–10
B 10–12
C 12–14
D 14–16
11 The spread of the students’ marks is:
A 8
B 10
C 12
D 20
E 22
12 For the stem plot shown opposite, the modal interval is:
A 20–24
B 25–29
C 20–29
D 25
E 29
E 18–20
1
1
2
2
3
0
5
3
5
0
2
5
3
7
1
6 9
4
9 9 9
2 4
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Chapter 1 — Organising and displaying data
37
60
50
40
30
20
10
0
Red
Black
Brown
Blonde
E
14 For these students, the most common
hair colour is:
A black
B blonde
C brown
Percentage
13 The number of students with
brown hair is closest to:
A 4
B 34
C 57
D 68
E 114
Other
D red
E other
PL
15 The ages of 11 primary school children were collected. The best graph to display
the distribution of ages of these children would be a:
A bar chart
B dot plot
C histogram
D segment bar chart
E stem plot
Extended-response questions
1 One hundred and twenty-one students were
asked to identify their preferred leisure activity.
The results of the survey are displayed in a
bar chart.
30
Percentage
SA
M
25
a What percentage of students nominated
watching TV as their preferred leisure
activity?
b What percentage of students in total
nominated either going to the movies or
reading as their preferred leisure activity?
c What is the most popular leisure activity for
these students? How many students rated this
activity as their preferred leisure activity?
20
15
10
5
TV
M
us
i
M c
ov
i
Re es
ad
in
g
O
th
er
Sp
or
t
0
Preferred leisue activity
2 The number of people killed in natural and non-natural disasters in 1997 by world
region is shown in the table below.
a Construct a bar chart.
Region
Number killed
b In which region was the:
Europe
874
i greatest number of people killed?
Africa
8 327
ii least number of people killed?
Asia
10 551
∗
Oceania
457
The Americas
1 581
∗
includes Australia (41)
Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson
TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin
Review
The following information relates to Questions 13 and 14
This percentage segmented bar chart
100
90
shows the distribution of hair
80
70
colour for 200 students.
P1: FXS/ABE
P2: FXS
9780521740517c01.xml
CUAT013-EVANS
September 1, 2008
17:55
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Essential Further Mathematics – Core
3 A group of 52 teenagers was asked
‘Do you agree that the use of marijuana
should be legalised?’ Their responses
are summarised in the table opposite.
Legalise
Agree
Disagree
Don’t know
Total
Frequency
Count Percent
18
26
8
52
E
a Construct a properly labelled and scaled
frequency bar chart for this data.
b Complete the table by calculating the
appropriate percentages correct to one decimal place.
c Use the percentages to construct a percentage segmented bar chart for this data.
d Use the frequency table to help you complete the following report.
PL
Report: In response to the question, `Do you agree that the use of marijuana
. Of the remaining
should be legalised?', 50% of the 52 students
% agreed while
% said that they
.
students,
4 The table below gives the distribution of the number of children in 50 families.
a Is the number of children in a
family a numerical or categorical
variable?
b Write down the missing information.
c What is the mode?
d Determine the number of
families with:
i three children
ii two or three children
iii less than three children
e Determine the percentage of
families with:
i six children
ii more than six children
iii less than six children
SA
M
Number of children
in family
0
1
2
3
4
5
6
7
8
Total
10
5 Students were asked how much they
spent on entertainment each month. The
8
results are displayed in the histogram.
Use this information to answer the
6
following questions.
4
a How many students:
i were surveyed?
2
ii spent $100–105 per month?
0
90
b What is the mode?
c How many students spent $110 or more per month?
d What percentage spent less than $100 per month?
Frequency
Count Percent
5
6
19
7
10
2
3
0
1
50
4
6
0
2
100
38
14
Frequency
Review
38
100
110 120
Amount ($)
130
Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson
TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin
140
P1: FXS/ABE
P2: FXS
9780521740517c01.xml
CUAT013-EVANS
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Chapter 1 — Organising and displaying data
39
i Name the shape of the distribution displayed by the histogram.
ii Locate the interval containing the centre of the distribution.
iii Determine the spread of the distribution using the range.
6 This stem plot displays the ages (in years) of a group of women.
a What was the age of the youngest woman?
2 = 17.2 years
3 4
6 6 8 8 9 9
1 3 3 3 4
5 5 5 5 5 6 7 8 8 8 9
2 2 3 3
PL
E
b In terms of age, one of the women is a
Note: 17
possible outlier. What is her age?
17 2
c How many women were aged between
17 5
18 0
17.0 and 17.4 years, inclusive?
18 5
d How many women were 19 years old
19 1
or older?
19 8
e What is the modal age category?
f What percentage of women were younger 20
20 6
than 20 years old?
g i Name the shape of the distribution
of ages, noting outliers.
ii Locate the centre of the distribution.
iii Determine the spread of the distribution.
10
8
Frequency
SA
M
7 The distribution of times spent by 37 cars
while stopped at traffic lights is as shown in
the histogram opposite. Use the histogram to
write a report on the distribution of waiting
times in terms of shape, centre, spread
and outliers.
6
4
2
0
5 10 15 20 25 30 35 40 45 50 55
Waiting time (seconds)
8 Use a graphics calculator to construct histograms for the following sets of data:
a Use intervals of width 5 starting at 90.
Monthly expenditure on entertainment (in dollars)
110 115 105
98 118 114 125
112 107 135 121
95 114 104
97 130 122
94 108 118 106 121 125 107 109
93
b Use intervals of width 8 starting at 32.
Life span (in years)
58
65
68
74
73
73
75
71
72
61
67
66
50
66
64
72
74
48
41
44
44
49
48
48
37
Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson
TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin
Review
e