Data Handling

Grades 7, 8 and 9

your leading partner in quality statistics

CENSUS@SCHOOL

Foreword
Statistics South Africa (Stats SA) is committed towards building statistical capacity and
promoting statistical literacy within the organization, other government departments and
within schools and the public as a whole.

The Census@School (C@S) 2009 project was undertaken nationally in all provinces. Data
were collected from a sample of 2 500 schools selected from the Department of Basic
Education’s database of approximately 26 000 registered schools (EMIS database). The
main objective of the project is to raise awareness of the national census, how it gathers
data, and its benefits to society. In addition, data collected should be used to provide
contextual material for teachers and learners to use for teaching and learning of data
handling, and promoting statistical literacy relating to a variety of subjects.

The first series of Mathematics Study Guides on Data Handling and Probability for the
Senior Phase (Grades 7–9) using the 2009 C@S data has been developed. This milestone
has been achieved through the collaboration with and support from the national and
provincial Departments of Basic Education with regard to the C@S projects.

This material has been written for Statistics South Africa by

Meg Dickson
Mmatladi Khembo
Farhana Motala
Ingrid Sapire
Elsie Venter
Kerryn Vollmer

EDITOR
Jackie Scheiber

University of the Witwatersrand

CENSUS@SCHOOL

CONTENTS
Chapter 1 Collecting Data 1
Chapter 2 Organising and Summarising Data 21
Chapter 3 Mode, Mean, Median 33
Chapter 4 Representing Data 47
Chapter 5 Interpreting Data 71
Chapter 6 Relative Frequency and Probability 93
Chapter 7 Chapter 1 Answers 115
Chapter 2 Answers 118
Chapter 3 Answers 120
Chapter 4 Answers 123
Chapter 5 Answers 128
Chapter 6 Answers 131
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Chapter

1
Collecting data
In this chapter you will:
x Define data
x Learn about the Investigative Cycle
x Learn about methods used for collecting data
x Learn about the difference between samples and populations
x Select and justify appropriate methods for collecting data
x Learn about the design and use of questionnaires

I n statistics you often work with tables of data. Statistical results are also often
shown in graphs rather than just in words and numbers. These graphs make it
easier to interpret and analyse information. In this chapter you will learn about
collecting data.

What is data?
Data can be numbers, words, measurements, observations
NOTE
or even just a description of things.
Data is a collection Statistics is the name given to the study of data. It involves:
of facts such as 1. Collecting data
values or 2. Sorting data
measurements. 3. Displaying data in diagrams or charts
4. Analysing the results
5. Coming to conclusions.

9 If, for example, someone tells you that 55% of the learners at their school are
boys while only 5% of the teaching staff is male, they have given you some
statistical information.
9 What these percentages (which are statistics) tell you is that more than half of
the children at the school are boys while only a very small percentage (5%) of
the teachers are men.

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The investigative cycle

When we do a statistical investigation, we use the statistical enquiry cycle, often
called the PPDAC (Problem, Plan, Data, Analysis, Conclusion) cycle.

PPDAC reminds us that we have to consider:

x The Problem – what we are going to research
x A Plan – how we are going to collect the data
x The Data – how the data is managed and organised
x The Analysis – exploring and analysing the data
x The Conclusion – the conclusions that we come to at the end of the cycle.

The South African Census@School activities have been developed using the
investigative cycle. Statisticians use this cycle and so can you.
Data
9 The cycle starts with a problem, and
goes on to planning and so on, but it
does not have to end at the
Plan Analysis
conclusion.
9 The conclusion could just be the
beginning of a new problem that you
find to investigate. That’s why it is a
cycle. Problem
Conclusion

We are going to go through each of the stages in the PPDAC cycle so that you will
get an idea of the stages in the cycle and how they fit together.

Stage 1: Problem
The problem section is where we make decisions about
o what data to collect
o who to collect it from
o why it is important.

Some questions that can help you to think about the problem and to develop a
statistical question of your own are:
x How do I go about answering this question?
x What do I need to know?
x How will I find the information that I need?
x What will I do with the information that I collect?
x Who will find this information useful?
x Is this information relevant to the problem?

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9 Formulating and defining a statistical question is important as it tells you what

to investigate and how to investigate it. This is where you start.
9 Most investigations begin with a wondering about something.
For example: ‘I wonder how many different pets there are in the pet shop down
the road?’
9 From this general question a statistical question needs to be developed so that a
meaningful investigation can be carried out.

Stage 2: Plan
The planning section is about how you will gather the data.

Questions that can help you to think about planning your data
gathering project are:
x How will I gather this data?
x What data will I gather?
x How am I going to record this information?

For example: at a certain pet shop there are 205 goldfish, 6 puppies, 15 kittens, 37
budgies, 17 hamsters and 4 cockatiels. The different types of pets and the number
of each type is data.
o You need to make predictions and then test them. You might think to
yourself “There are mostly kittens and puppies at the pet shop”.
o After that you can reflect on the difference between your prediction and the
result.
o You need to plan things such as the sample size and method of data
collection. You think about whether you will count all of the pets (the whole
pet shop population), or if the shop is big, maybe only the pets in every
second cage (a sample of the pets). But to make this decision you need to
plan.

Stage 3: Data
The data section is how the data is managed and organised.

Questions you need to answer here are:

x How am I going to record my data?
x How will I present my data?

Raw Data is data which has been collected but not yet sorted out in any way.
For example: In order to organise the raw data about the animals in the pet shop, you
could make a table with the different types of pets in a list down one side and then
make a tick for every pet you count.

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You may record their data in any format as long as it is clear and easy to work with.

9 A table is usually the best format. Tables are the most common
organisational tool that statisticians use.
9 You could count the number of ticks to see how many of each pet type there
are and you can record this information in a table.
9 Then you’ll think about how to present the data – probably with a graph
show that you can give the “picture” of what the information you have
found looks like.

Stage 4: Analysis
The analysis section is about exploring the data and reasoning
with it.

Some questions that can help you think about how to analyse
your data are:
x What do I notice?
x Why do I think it looks like this?
x Are there relationships between some of the variables I
have investigated?

We can:
9 Read the data – take information directly off a graph
9 Read between the data – interpret the data
9 Read beyond the data – extend, predict or infer
9 Read behind the data – connect the data to the context.

For example:
o Look at the table of information and graphs that you may have drawn of the pet
shop statistics and see what it tells you.
o You might realise “Kittens and puppies may be bigger and easier to see from the
pet shop window, but there are far more fish for sale in this pet shop than
kittens and puppies put together”.
o You might wonder about which pets are most popular (reading beyond), which
pets are the easiest/hardest to look after (reading behind) or simply compare
the data you see in the graph (reading the data).

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Stage 5: Conclusion
The conclusion section is about answering the questions in
the problem section and providing reasons based on the
analysis of these questions.

Some questions that can help you to think about your

conclusions are:
x What do I notice?
x Why do I think it looks like this?
x Are there relationships between some of the variables I have investigated?

9 Your conclusions should relate back to your original question.

9 You should mention any features you noticed or wondered about and
investigated.
9 You should give reasons based on what you find out in your investigation.

For example: Based on the pet shop data, you could conclude that fish are the
most popular pet and easiest for the pet shop to keep since they have the most fish.

The five stages (Problem; Plan; Data; Analysis; Conclusion) take you through the
full “investigative research cycle”.

The example on the next page illustrates how the five stages of the research cycle
can be used:

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)
Example 1.1
The Census@School researchers wanted to investigate whether
South African learners had access to libraries, computers and the
internet at school. Explain how you can use the investigative cycle
to do this.

Solution
1) PROBLEM:
The general question that C@S wanted to investigate was something like – ‘I
wonder how accessible libraries, computers and the internet are to the learners
in South Africa?’
2) PLAN:
C@S had to plan how to collect the data. The project collected data for many
questions, including questions about the accessibility of libraries, computers and
the internet. They had to plan things such as:
a) How to gather this data – C@S decided to take questionnaires into schools.
Two questionnaires were developed, one for Grades 3 – 7 and one for
Grades 8 –12.
b) What data to gather – the questionnaire included questions about how
easy/difficult it is for learners to get to libraries.
c) How to record this information – the learners would write their answers on
the questionnaires which could be collected.
3) DATA:
C@S had to think about the way in which they would organise the data.
This would involve:
a) How to record the data – C@S entered all of the data onto a table.
b) How to present the data – a bar graph representing the accessibility of
libraries, computers and the internet was presented in the C@S report.
4) ANALYSE data:
What the bar graph showed was that more learners have greater access to
libraries than to computers and the internet.
Fewer learners have access to the internet than to libraries or computers.
5) CONCLUSION:
a) The analysis showed that, of the three, libraries are the most accessible and
the internet is the least accessible. Computers are somewhere in the middle.
b) Because learners do not have to pay money to take books out of a library,
more learners can access a library. Computers cost a lot and are not always
available. Access to the internet costs even more money and so it is even
less common.
c) There is a relationship between ownership of a computer and access to the
internet, although internet cafes do provide internet access to people who
do not own computers.

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Exercise 1.1

Make up your own example of a statistical investigation, like the pet shop example
used on the previous pages to illustrate the stages in the investigative cycle.

If you can’t think of an example of your own try one of the following:
x What is the range of ages of learners in your class or school?
x What is the range of learner heights in your class or school?

In the process, think of the following:

1) What is your PROBLEM? – State your research question.
2) What is your PLAN for data collection?
a) How are you going to gather this data?
b) What data will you gather?
c) How are you going to record this information?
3) What DATA will you record and how?
a) How are you going to record your data?
b) How will you present you data?
4) How do you think you will ANALYSE your data?
When you analyse the data, think about:
a) What you notice?
b) Why do you think it looks like this?
c) Are there relationships between some of the variables you have
investigated?
5) What kinds of questions do you hope to answer in your CONCLUSION?
When you reach your conclusions, think about:
a) What you notice?
b) Reasons for why you think it looks like this.
c) An explanation of the relationships between some of the variables that you
investigated.

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Methods used for collecting data

When you collect data, you have to think carefully how you are going to do it.

How you collect data depends on

9 the data you have to find
9 the time available for the research
9 the money available for the research, and so on.

You could use:

9 questionnaires
9 observation
9 experiments
9 telephonic research
9 group or individual interviews which you record on tape.

To be able to make conclusions from data, we need to know how the data were
collected; that is, we need to know the method(s) of data collection.

There are four main methods of data collection:

1) A census.
A census is a study that gets data from every member of a population. In most
research that you do yourself, a census is not practical, because of the high cost
involved and the time it takes.
2) A sample survey.
A sample survey is a study that gets data from a smaller group within the
population, and is used to estimate population characteristics.
3) An experiment.
An experiment is a controlled study in which the researcher tries to understand
cause-and-effect relationships.
Experiments are usually "controlled" in the sense that the researcher controls
(1) how subjects are assigned to groups and (2) which treatments each group
receives.
In the analysis phase of an experiment, the researcher compares the different
effects of the experimental treatment.
4) An observational study.
Like an experiment, an observational study tries to understand the cause-and-
effect of relationships.
However, in observational studies, the researcher does not have any control
over who goes into which group and so the findings may not be as clear to the
observer.

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)
Example 1.2
You have been asked to investigate what the favourite sport is of
the learners in your school. Think about how you are going to do
it, and then answer the following questions about the methods you
are going to use:

1) What data do you want to collect?

2) How do you plan to collect you data? Explain how you would go
about it.
3) If you collected data initially as raw data, how would you record
this data?
4) What form of data collection could you use?
a) Could you have used a census? Explain why or why not.
b) Could you have used a sample survey? Explain why or why
not.
c) Could you have used an experiment? Explain why or why not.
d) Could you have used an observational study? Explain why or
why not.
Solution
1) I want to collect information on favourite sports.
2) I will draw up a questionnaire with questions about different sports so that I can find
out about learners’ favourite sports. I need to have a list of sports so that learners
can say which they like best from the list.
3) If I let the learners each tick on a separate questionnaire I would have raw data.
I could make a table in which I could tally (count up) the number of learners who like
each sport.
4) Form of data collection:
a) NO, I could NOT have used a census because one person cannot carry out a
census. With a census, the whole population gets involved.
b) YES, I could have used a sample survey because I could have decided to ask a
sample (a smaller representative group) of learners from each of the grades,
instead of asking every learner in the whole school. This would have been much
easier and more efficient to do.
c) NO, I could NOT have used an experiment because I was not trying to
understand cause-and-effect relationships.
d) NO, I could NOT have used an observational study because, again, I was not
trying to understand cause and effect relationships.

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Exercise 1.2

You have been asked to investigate what the favourite tuck shop food is of the
learners in your class. Think about how you are going to do it, and then answer the
following questions about the methods you are going to use:
1) What data do you want to collect?
2) How do you plan to collect you data? Explain how you would go about it.
3) If you collected data initially as raw data, how would you record this data?
4) What form of data collection could you use?
a) Could you have used a census? Explain why or why not.
b) Could you have used a sample survey? Explain why or why not.
c) Could you have used an experiment? Explain why or why not.
d) Could you have used an observational study? Explain why or why not.

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The difference between samples and populations

Before you go any further in planning statistical research you need to learn about
populations and samples so that you’ll be able to plan effectively for your data
collection.

A Population
A population is the whole group of people (or things)
that you want to speak about or investigate.

Examples:
a) In the example about what the favourite foods from the tuck shop are,
o If you wanted to find out about the favourite foods of the learners in your class
then your class would be the population
o If you wanted to find out about the favourite foods of the learners in the whole
school then all of the learners in the school would be the population.
b) In the example about the pet shop, all of the pets in the shop would be the
population.

9 Normally a population is quite large, but the size of your population depends on
your statistical research.
9 The larger your population (for example you might want give information relating to
“male South Africans") the more impossible it becomes to ask each member of that
population the questions you want to ask. You then need to develop some way to
help you with selection of individuals from your population, and to give a generalised
result based on information obtained from your sample.

A Sample
x When you want to collect information from a certain
population, which can sometimes be large, you could
take a smaller group, which could represent the whole
population.
x Such a group, used to represent the population, is called
a sample.

Examples:
a) If your favourite food survey was only to be done in your class you would not have
to find a sample.
o If the survey needed to be extended to the whole school, you will not have time
to ask everyone in the school the questions that you want to ask.
o What you could do is ask a selection of learners whose answers you will use to
make conclusions about the whole school.

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o You would try to ask learners from each grade in the school, with a good spread
across the school, to make your sample representative of the whole school
population.
b) In the example of the pet shop, counting pets in every second cage would be a way
of sampling.

)
Example 1.3
Explain the difference between a population and sample by using
the investigation to find out about the favourite sport of the
learners
a) In your class
b) In your school.

Solution
a) If I wanted to find out about the favourite sports that the learners in my class
like to play, the class is the population for my research. I can ask every learner in
my class and then talk about the sports they like. I don’t have to take a sample
of the learners in the class, because the population is small.

b) If I want to find out about the favourite sports that the learners in my school like
to play, the population for my research is every learner in the whole school. This
is harder to do! That could be quite a big number. I would rather ask a sample of
the learners in the school what their favourite sport is. Then I could use this
information to decide about the favourite sport of the learners in the school.

This means that I would ask a smaller group of learners selected from the whole
school population. But to be sure that what I say about the school is realistic I
would have to be careful to choose a sample that represents the full spread of
learners in my school. This sample could consist of:
x Some (a percentage) from each grade
x Some (a percentage)from each class in each grade

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Exercise 1.3

x Thembi is part of a group of 7 learners who are friends.

It is break time and Thembi is looking forward to playing soccer with them.
He makes a quick survey of his friends to find out who wants to play soccer with
him.
o James, Hlengani and Thoko are keen to join in.
o Dan and Lebo say that they want to sit and play with their latest computer
game.
o Habib wants to stand around and talk to some other friends.
x Based on his quick survey, 4 out of the 7 friends (57% of them) wants to play
soccer. Thembi wonders whether 57% of South African learners play soccer at
break time.
x Habib thinks that Thembi’s quick survey is flawed (that it is not accurate) if he
wants information on all South African learners.
1) What is a sample?
2) What is a population?
3) What is the statement of Thembi’s research problem?
4) Describe Thembi’s quick survey. Is the sample he chose representative of the
whole population?

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Random samples
You need to be able to distinguish between samples and populations.
One way of selecting a sample is by using random
sampling.
9 If each member of the population has an equal
chance of being chosen for a sample, then you
have a random sample.
9 Taking a random sample makes sure that the
sample represents the population.

9 One way of selecting a random sample is to place all the names in a hat or box; to
mix the names up; and to then take the names out of the hat or box without looking.
The mixing up of the names is important or else the names which are placed last in
the hat will have a better chance of selection than those which were placed first into
the hat. This will lead to bias in your data.

For example
If you want to find out what the favourite tuck shop food is of all the learners in the
school, be sure not just to ask your friends (who might all have similar tastes) or people
only from one class.
o You need to think of a way of getting the information from as wide a selection of
learners as possible. That way, what you hear from your sample (your selection
carefully chosen to be representative) will reflect what is going on in the whole
school.

For example
Two ways you could select a random sample of learners from your school are:
1) You could cut up all of the school class lists and then draw 30 names out of the
hat. This might not give you a random sample if the hat is not well enough
shaken.
2) You could take each class list and select every 10th name on the lists. This could
be the easiest way to get a representative random sample.

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Exercise 1.4

Think again of Thembi’s quick soccer survey in Exercise 1.3.

1) What is the population represented by the sample that Thembi collected?
2) Did Thembi use random selection to find his data?
3) What aspects of Thembi’s survey does Habib think are flawed? Do you agree
with Habib? Explain your answer.
4) How can Thembi improve his survey if he wants information on all South African
learners? Explain.
5) Write down some other statistical research problems that you think of.
Try to think of problems which are different to one another, such as:
a) Write about one problem where you could use the whole population in your
research.
b) Write about one problem where you could not use the whole population in your
research.
c) Write about one problem that requires random selection to obtain the sample.
d) Write about one problem that does not require random selection to obtain the
sample.

Select and justify appropriate methods for

collecting data
9 To select the right method for your research you need to know about the pros
(good things) and cons (bad things) about the different methods of data
collection. Each method of data collection has advantages and disadvantages,
depending on the research you want to do.
9 One main factor that affects data collection is the available resources.
Resources are things like money, time and people.
9 When the population is large, a sample survey is cheaper to do than a census.
This is because a sample is smaller and cheaper and easier to work with.
x A well-designed sample survey can provide very good estimates of
population characteristics, but more quickly and cheaply than a census.
x When the population is small, you can design research that you can manage.
This will give you experience in small-scale statistical research.

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Exercise 1.5

Jane wants to find out what is the favourite sport of South African children. She spends a
week asking every learner she sees at her school what their favourite sport is. She tallies
up the information and decides that Athletics is the favourite sport.
My school
Favourite sport Boys Girls Total
Athletics 34 25 59
Soccer 39 5 44
Netball 2 30 32
Softball 15 12 27
Cricket 16 4 20
No Favourite sport 10 11 21
Other 8 6 14

She decides to check up on her finding. She goes to the South African Census@School
website and finds the following table:
Eastern Cape
Favourite sport Male Female Total
Soccer 6 931 775 7 706
Netball 135 5 626 5 761
Athletics 1 957 2 058 4 015
No Favourite sport 722 1 409 2 131
Rugby 1 276 44 1 319
Unspecified 595 476 1 071
Volleyball 293 435 728
Cricket 450 59 509
Dance Sport 80 387 467
Softball 185 281 465

1) How big is Jane’s sample?

2) Did Jane take a random sample?
3) Did Jane collect her data in an appropriate way? Explain.
4) What are the top five favourite sports according to Jane’s survey?
5) What are the top five favourite sports according to the Census@School table?
6) How does Jane’s finding that athletics is the favourite sport in her school compare
with the Census@School findings?
7) Is Jane’s table representative of the whole of South Africa? Explain.
8) Is the Census@School table representative of the whole of South Africa? Explain.

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The design and use of questionnaires

Statistical research is about asking and answering questions. There are different
kinds of questions involved. To do statistical research you need to develop your
skills of:
x Posing questions (the investigative questions) – thinking about and asking
questions that can lead to statistical research.
x Selecting appropriate sources for the collection of data (knowing where to find
your data) – planning research so that you can find the right data to help you to
answer the questions you have posed.
x Designing and using simple questionnaires to answer questions (making the
tools that you use to carry out the research)

Posing or asking questions

You need to know how to ask or pose questions relating to social, economic, and
environmental issues in your own environment.
9 To do this you need to develop your own awareness of these things by
reading newspapers, watching TV and listening to the radio.
9 Talking to your family, friends and teachers will also help!
9 Then you need to start thinking about the kinds of questions to ask.

You could ask about:

x Particular things
x Comparative things
x Relationships.

)
Example 1.4
Give an example of each of the different kinds of questions you
could ask when doing research at your school.

Solution
1) Particular things – What was the highest score in the Grade 7 maths final exam at
my school?
2) Comparative things – Are there more 14 year olds or 15 year olds in Grade 8 at my
school?
3) Relationships – Is there a relationship between the amount of money spent at the
tuck shop and the age of the learner at my school?

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Exercise 1.6

Give another example of each of the different kinds of questions you could ask
when doing research at your school.
1) A particular question
2) A comparative question
3) A relationship question

Selecting appropriate sources for the collection of data

You also need to know how to select appropriate sources for the collection of data.
9 You could ask people such as your friends and family.
9 You could look for data in a selection of newspapers, books, magazines.

When you decide to collect data, you need to decide whether you are working with
a sample or a population.

)
Example 1.5
Where would you find your data to answer the question you posed in
Example 1.4?

Solution
1) Particular question – What was the highest score in the Grade 7 maths final exam at
my school?
I could find out about this by speaking to the teachers and getting the mark lists for
all of the Grade 7 classes at my school.
2) Comparative question – Are there more 14 year olds or 15 year olds in Grade 8 at
my school?
I could find out about this by asking the Grade 8 learners at my school about their
age.
3) Relationship question – Is there a relationship between the amount of money spent
at the tuck shop and the age of the learner at my school?
I could ask a sample of learners who shop at the tuck shop about their age and the
amount that they spend. Then I would be able to see if there is a relationship. I
would have to be sure to ask a big enough representative sample.

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Exercise 1.7
Where would you find your data to answer the questions you posed in
Exercise 1.6?
Write down the question and then explain where you would find
your data.

Designing and using simple questionnaires to answer

questions

9 A questionnaire consists of a list of questions. These questions are used to

collect information and answers from people.
9 Always try out your questionnaire before you use it to check that it gets you the
data that you want. You may be surprised at how some questions are answered!
9 It is very common in questionnaires to use:
x Questions with yes/no type responses.
x Questions with multiple choice responses.
Both of these types of questions are popular because they are easy to analyse.
9 Here are some things to think about when you write your questions:
x Don’t use abbreviations (shortened forms of words) as some people may not
understand them.
x Keep the language as simple as possible.
x Don’t ask questions which have two parts in one (for example Do you like
hotdogs and coke? If they answer “yes” they could mean they like both or
either of the two).
x Be careful of asking open questions such as “what is your favourite food”
because you may get too many different answers.
x Closed questions (questions where respondents have to choose from given
answers) can limit the number answers you get, but this could also limit your
research. So you need to think carefully about the list of options you give in a
closed question.
9 A good questionnaire ends with a comments section that allows the respondent
to write down any other issues not covered by the questionnaire. This allows
them to express any thoughts, questions or concerns they might have.
9 Lastly, there should be a message at the end thanking the respondents for their
time and patience in completing the questionnaire.

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)
Example 1.6
1) Give an example of each of the following two types of questions
that could be asked in a questionnaire.
a) a yes/no response type question.
b) a multiple choice response type question.
2) Why are these types of questions easy to analyse?
Solution
1) a) Do you eat red meat? Circle yes/no
b) What is your favourite meat? (Circle the letter that represents your choice)
i) Beef
ii) Lamb
iii) Chicken
iv) I don’t eat meat
2) These types of questions easy to analyse because the responses are limited to the
options given on the questionnaire and so you don’t have to worry or think about
what to do with all sorts of different, uncommon answers that some people might
give.


Exercise 1.8

Pick one of your research questions that you posed in Exercise 1.6.
Design a 10 question questionnaire to collect data for your research.

9 Your questionnaire should have six yes/no questions and four multiple choice
questions.
9 If you want to design questionnaires for ALL of your questions, go right ahead.
The more you do, the better you’ll get at doing it.
9 If you had some difficulty making up questions, choose one of the following
questions:
1) What types of transport do the learners in my school use to go to school?
2) How far do the learners in my school live from the school?

Many of the questions that have been raised in this chapter were researched by the
C@S team. Results for South Africa in 2009 are available in the Census at School
Results 2009 report which can be found at the Stats SA website www.statssa.gov.za

You will find out lots more about working with data in the chapters that follow.

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Chapter

2
Organising and Summarising Data
In this chapter you will:
x Learn to recognise continuous and discrete data
x Organise and record data using tallies
x Draw an ungrouped and grouped frequency table
x Organise data using a stem and leaf diagram

T he word “data” is the plural of “datum”. So data are pieces of

information that are given or that one collects. Data can be words, numbers or
a mixture of both.

Types of data
Data can consist of two types of data: numerical data or
NOTE
non-numerical data. The data you collect in a survey or
x Data can be questionnaire may be varied – it may be about the colour of
numerical or non- learners’ eyes, their mode of transport to school, an opinion (e.g.
numerical. which chocolate do you prefer). It may also be numerical (e.g.
how many learners come to school by bus or taxi in the morning).

In this book we will be working with numerical data that is either discrete or
continuous as well as non-numerical or qualitative data.
x Qualitative data consist of descriptions using words.
Examples are: the colour of hair, the colour of eyes, shoe sizes.
x Discrete and continuous data both consist of numerical values.
o Discrete data is information that is collected by counting exact amounts.
Examples are: the number of children in a family; the number of children
with birthdays in October or the number of houses with electricity.
o Continuous data is collected by measurement and the values form part of a
continuous scale like a number line.

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Examples of continuous data are: measurements of physical quantities

such as mass/weight, height, length and time (e.g. the mass of learners in a
Grade 8 class measured in kilograms or the length of the right feet of the
Grade 7 learners measured in centimetres or the time it takes learners to
travel to school each morning measured in hours and minutes).

Organising data
When you first look at numerical data, all you may see is a jumble of information.
You need to sort or summarise the data and record it in a way that puts order into it
so that it makes more sense.
9 Data in this form are called raw data. Raw data haven’t been organised in
any way.
9 One of the most common ways of sorting data is by making a list. Data is
easy to sort into lists that are either numerical or alphabetical.


Exercise 2.1

Given below are the heights (in centimetres) of 90 Grade 8 boys in an Eastern Cape
school as recorded in the 2009 Census@School:
165 148 158 150 160 165 150 156 155 164 162 160 158 148 158
140 146 160 148 152 139 165 148 160 156 158 170 155 160 148
155 158 179 170 158 161 155 160 163 178 138 172 170 156 160
160 171 140 160 170 175 148 170 177 155 167 154 160 170 155
136 179 150 167 148 160 164 167 157 165 163 140 162 178 160
170 163 162 165 175 165 152 147 180 148 170 165 167 165 165

1) Write a list of the boys’ heights in ascending order.

2) Which height is the most common?
3) How many learners are shorter than 150 cm?
4) How many learners are taller than 170 cm?

Tally tables
9 You can also organise and summarise data using a tally table.
A tally is a mark which shows how often something happens.
9 For each score, a vertical stroke is entered in the appropriate row, with a
diagonal stroke being used to complete each group of five strokes.

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)
Example 2.1
Organise the list of heights (in centimetres) of 90 Grade 8 boys in
an Eastern Cape school given in Exercise 2.1 into a tally table.

Solution
Height (in cm) Tallies Frequency
136 1
137 0
138 1
139 1
140 3
141 1
142 0
143 0
144 0
145 0
146 1
147 1
148 8
149 0
150 3
151 0
152 2
153 0
154 1
155 6
156 3
157 1
158 6
159 0
160 12
161 1
162 3
163 3
164 2
165 9
166 0
167 4
168 0
169 0
170 8
171 1
172 1
173 0
174 0
175 2
176 0
177 1
178 2
179 2
180 1

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Frequency Tables
9 A frequency table is usually given without the tallies. The word “frequency”
may not appear in the table. Instead, the frequency column might be headed
“number of learners” or something like that. Always make sure that you
identify which numbers are the “values” and which ones are the “frequencies”.

Height (in cm) Frequency

136 1
137 0
138 1
139 1
140 3
141 0
142 0
143 0
144 0
145 0
146 1
147 1
148 8
149 0
150 3
151 0
152 2
153 0
154 1
155 6
156 3
157 1
158 6
159 0
160 12
161 1
162 3
163 3
164 2
165 9
166 0
167 4
168 0
169 0
170 8
171 1
172 1
173 0
174 0
175 2
176 0
177 1
178 2
179 2
180 1
TOTAL 90
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Exercise 2.2

Conduct a simple survey of the learners in your class to ask about the month of
their birthday.

1) Record the information in a frequency table like the one below:

Month Tally Frequency
January
February
March
April
May
June
July
August
September
October
November
December

2) In which month do the most birthdays occur?

3) In which month do the least birthdays occur?

Stem and Leaf Diagrams

Tally tables can become very long and difficult to work with if the range of possible
values is very large. In the 1960s John Tukey, an American mathematician and
statistician, devised a way of organising data called a stem and leaf diagram.

9 Stem-and-leaf diagrams can be used both with discrete data and with
continuous data (rounded off to the nearest whole number).
9 Stem-and-leaf diagrams retain the original data information, but present it in a
compact and more easily understandable way.
9 When entering a number like 56 onto a stem and leaf diagram, the tens digit (5)
forms the stem and the units digit (6) forms the leaf.

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)
Example 2.2
The members of your class got the following marks in a
mathematics test:
1) Organise the data using a stem and leaf diagram.
2) Write down 3 conclusions you can reach about the marks.

Marks scored in a mathematics test

32 56 45 78 77 59 65 54
54 39 45 44 52 47 50 52
40 69 72 36 57 55 47 33
39 66 61 48 45 53 57 56
55 71 63 62 65 58 55 51
Solution
1) Step 1: Decide on the values to use for the stem.

stem leaf
3
4
5
6
7

Look how to write 56 and 78 in the stem and leaf plot

The second number is

The first number
78: the stem is 7 and
is 56: the stem is
the leaf is 8
5 and the leaf is 6
stem leaf
3
4
5 6
6
7 8

Step 2 : Write down the leaves directly from the data without worrying about
the order:
stem leaf
3 2, 9, 6, 3, 9
4 5, 5, 4, 7, 0, 7, 8, 5
5 6, 9, 4, 4, 2, 0, 2, 7, 5, 3, 7, 6, 5, 8, 5, 1
6 5, 9, 6, 1, 3, 2, 5
7 8, 7, 2, 1

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Step 3 : Rewrite the leaves in ascending order. This makes the table easier to
read.

stem leaf
3 2, 3,6, 9, 9
4 0, 4, 5, 5, 5, 7, 7, 8
5 0, 1, 2, 2, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9
6 1, 2, 3, 5, 5, 6, 9,
7 1, 2, 7, 8
KEY: 3/2 = 32

2) We can read the following information from this stem and leaf diagram:
a) The lowest mark is 32
b) The highest mark is 78
c) More learners got marks in the 50s than in any of the other number ranges
d) More learners got marks in the 40s than in the 60s
e) The most common marks were 45 and 55 as three learners got 45 and three
got 55.
f) Four learners achieved marks of more than 69.

NOTE:
9 The leaf is the digit in the place furthest to the right in the number.
9 The stem is the digit or digits that remain when the leaf is dropped.
9 If the list of numbers included numbers like 120 ; 134 ; 127 then 12 and 13
would be the stems and 0, 4 and 7 would be the leaves.
9 If the list of numbers included a single digit number like 2, 3 or 9, then the stem
would be 0 and the leaves would be 2, 3 and 9.


Exercise 2.3

Given below are the heights (in centimetres) of 90 Grade 8 boys in an Eastern Cape
school as recorded in the 2009 Census@School:

165 148 158 150 160 165 150 156 155 164 162 160 158 148 158
140 146 160 148 152 139 165 148 160 156 158 170 155 160 148
155 158 179 170 158 161 155 160 163 178 138 172 170 156 160
160 171 140 160 170 175 148 170 177 155 167 154 160 170 155
136 179 150 167 148 160 164 167 157 165 163 140 162 178 160
170 163 162 165 175 165 152 147 180 148 170 165 167 165 165

1) Summarise the data using a stem and leaf diagram.

2) Write down three conclusions you can reach about the heights.

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Back-to-back stem-and-leaf diagram

9 When two sets of data are to be compared, the leaves can extend in opposite
directions from the same stem.

)
Example 2.3
The heights of 10 boys and 10 girls in Grade 7 were randomly
selected.
a) Draw a back-to-back stem-and-leaf diagram to illustrate the data
b) Write down at least 2 conclusions about the heights.
Heights of a random selection of 20 learners in Grade 7 in cm
Boys 171; 156; 154; 160; 145; 147; 149; 160; 150; 147
Girls 162; 155; 155; 155; 164; 170; 149; 156; 161; 164
Solution
a) STEP 1 : Decide on the values of the stems and draw up the table.
GIRLS’ HEIGHT BOYS HEIGHTS
in centimetres in centimetres
14
15
16
17

STEP 2 : Write down the leaves directly from the data.

GIRLS’ HEIGHT BOYS HEIGHTS
in centimetres in centimetres
9 14 5 7 9 7
6 5 5 5 15 6 4 0
4 1 4 2 16 0 0
0 17 1

STEP 3 : Rewrite the leaves in ascending order.

GIRLS’ HEIGHT BOYS HEIGHTS
in centimetres in centimetres
9 14 5 7 7 9
6 5 5 5 15 0 4 6
4 4 2 1 16 0 0
0 17 1
KEY: 9/14 = 149 cm and 14/5 = 145 cm

b) The heights of the girls range from 149 cm to 170 cm.

The heights of the boys range from 145 cm to 171 cm
Five of the girls have heights more than 159 cm whereas only 3 of the boys have
heights more than 159 cm.
Four of the boys have heights of 149 cm or less than 149 cm, whereas only 1 of
the girls has a height that is 149 cm or less.

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Grouped Frequency Tables

9 To obtain a clearer picture of a distribution, you can group data within class
intervals.

)
Example 2.4
Organise the heights (in centimetres) of 90 Grade 8 boys in an
Eastern Cape school as recorded in the 2009 Census@School in a
grouped frequency table:

Height Frequency Height Frequency Height Frequency

(in cm) (in cm) (in cm)
136 1 151 0 166 0
137 0 152 2 167 4
138 1 153 0 168 0
139 1 154 1 169 0
140 3 155 6 170 8
141 0 156 3 171 1
142 0 157 1 172 1
143 0 158 6 173 0
144 0 159 0 174 0
145 0 160 12 175 2
146 1 161 1 176 0
147 1 162 3 177 1
148 8 163 3 178 2
149 0 164 2 179 2
150 3 165 9 180 1

Solution
o The heights range from 139 cm to 180 cm.
o We can organise these heights
o 130 < h ч 140 means that the heights
into the following class intervals:
are more than 130 cm, but less than
130 < h ч 140
140 cm or equal to 140 cm.
140 < h ч 150
150 < h ч 160 o 140 < h ч 150 means that the heights
160 < h ч 170 are more than 140 cm, but less than
170 < h ч 180 150 cm or equal to 150 cm.
o The following grouped frequency table shows the heights of the learners:
Height in
Frequency
centimetres
130 < h ч 140 6
140 < h ч 150 13
150 < h ч 160 31
160 < h ч 170 30
170 < h ч 180 10

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NOTE:
9 The groups do not overlap at all.
Notice that the groups were written 130 < h ч 140 and then 140 < h ч 150.
In other words, you have to be careful that a particular height (e.g. 140 cm) does
not appear in two different intervals.
9 The data in this example – the heights of the Eastern Cape boys – is continuous
data rounded off to the nearest centimetre.


Exercise 2.4

1) The maths marks of a class of Grade 7 learners are given below.

a) Copy the following table and use it to organise the marks.

Mathematics Marks
32 ; 56 ; 45 ; 78 ; 77 ; 59 ; 65 ; 54 ; 54 ; 39 ; 45 ; 44 ; 52 ; 47 ; 50 ; 52 ;
51 ; 40 ; 69 ; 72 ; 36 ; 57 ; 55 ; 47 ; 33 ; 39 ; 66 ; 61 ; 48 ; 45 ; 53 ; 57

Mathematics marks Tally Frequency

30 < m ч 40
40 < m ч 50
50 < m ч 60
60 < m ч 70
70 < m ч 80

b) How many learners got 50 or less for the test?

2) Lerato wanted to do a survey about the height in metres of learners in her

Grade 9 class.

Heights of the Grade 12 learners in metres

1,82 1,64 1,71 1,86 1,64 1,67
1,73 1,76 1,84 1,52 1,63 1,65
1,80 1,67 1,71 1,64 1,58 1,81
1,67 1,74 1,69 1,56 1,68 1,74
1,79 1,83 1,69 1,58 1,57 1,73

a) Use intervals 1,50 < x d 1,55 ; 1,55 < x d 1,60 ; etc to draw a grouped
frequency table for Lerato’s data.
b) How many learners are taller than 1,75m?
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3) In a class in a school in Mpumalanga, Census@School recorded the time that 30

learners spent watching TV in a week. Their answers (to the nearest hour) are
given below:

12 20 13 15 22 3
6 24 20 15 9 12
5 6 8 30 7 12
14 25 2 6 12 20
18 3 18 8 9 20

a) Draw a stem and leaf diagram to organise this data.

b) Write down three conclusions about the data.

4) The metro bus company in Durban did a survey to find out how many learners
used a particular bus to come to school in town. They counted the number of
learners on the bus each time it arrived in town. The numbers are given below:

11 25 60 58 55 16 23 2 44 26
49 8 14 24 7 16 47 5 30 34
9 33 10 21 1 56 32 19 6 1
21 42 9 35 25 55 37 52 15 7
31 25 6

a) Draw a grouped frequency table to organise the data. You will need to think
of appropriate group intervals.
b) Why do you think the bus company might want to know the numbers of
learners on the bus?

5) A survey was conducted to find the colour of eyes of South African learners and
the following results were recorded and organised in a table:

Gender Brown Green Blue Other Unspecified Total

Male 60 531 1 615 2 235 5 095 704 70 180
Female 66 993 2 015 2 449 4 686 538 76 681
Unspecified 789 20 33 55 247 1 144

a) What fraction of all the learners has blue eyes?

b) What is the total number of learners with brown eyes?
c) Why do you think that there are such a high number of learners with brown
eyes?

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Chapter

3
Mode, Mean, Median
In this chapter you will:
x Learn about measures of central tendency: Mode, Mean, and Median
x Learn about measures of dispersion: Range and Extreme values

W hen you have a data set, it is possible to summarise the data with one single
number (also called a ‘statistic’). These single numbers are called either
Measures of Central Tendency or Measures of Dispersion or Spread.

Measures of central tendency

Measures of central tendency are numbers that are typical of a given set of data.
The three measures of central tendency that we calculate are the mode, the mean
and the median.

Mode
9 The mode is the number in your data that occurs most often. You can also say
the mode is the value that has the highest frequency.
9 Sometimes two scores (or numbers) occur equally often and then the data set
has more than one mode. If there are two modes we say that the data set is
bimodal. If there are more than two modes, we say that the data set is multi-
modal.
9 Other times there might not be any number that occurs more often than any
other number. This means that a data set may have no mode.

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C E N S U S @ S C H O O L

)
Example 3.1
For each of these sets of scores, find the mode.
a) 1, 7, 9, 4, 3, 5, 9, 3, 2, 9
b) 3, 7, 9, 4, 3, 5, 9, 3, 2, 9
c) 1, 2, 3, 4, 5, 6, 7, 8, 9

Solution
a) Order the scores from lowest to highest: 1, 2, 3, 3, 4, 5, 7, 9, 9, 9
It is now easy to see which score is the mode:
The score that occurs most, is 9, the mode of the data set.

b) Order the scores from lowest to highest: 2, 3, 3, 3, 4, 5, 7, 9, 9, 9

Both 3 and 9 occur three times in the data set, therefore the data has two
modes, 3 and 9.
You can also say this data are bi-modal.

c) The data is already ordered: 1, 2, 3, 4, 5, 6, 7, 8, 9

All the numbers occur only once in the data.
This means that this data has no mode.


Exercise 3.1
A random sample of ten learners was selected from the 2009
Census@School data base.

1) The following data set contains the ages of ten learners in years:
9, 15, 9, 15, 17, 15, 11, 18, 15, 19
a) Order the ages from the smallest to the biggest.
b) How many learners are 9 years old?
c) What is the highest age?
d) How many learners are older than 15?
e) How many learners are 15 years old?
f) Which age occurs most often?
g) What is the mode?

2) The following data set contains the heights of the ten learners in centimetres.
138, 161, 121, 170, 170, 165, 142, 160, 140, 182
a) Rank the heights from the lowest to the highest.
b) Which is the smallest height?
c) How many learners are taller than 160 centimetres?
d) How many learners are 170 centimetres tall?
e) What height is the mode?

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3) Learners were asked how many people lived in their home. The data set below
shows their answers: 10, 14, 6, 8, 3, 2, 6, 7, 7, 4
a) Rank the answers from smallest to largest.
b) How many learners are in the data set?
c) What is the fewest number of people in a home?
d) What is the most number of people in a home?
e) How many homes have 6 people?
f) How many homes have 7 people?
g) What is the mode of the people in the home?
h) If you only look at the first 7 learners in the data set, what will the mode be?
i) If you only look at the first 5 learners in the data set, what will the mode be?

4) Grade 9 learners had to say which sport is their favourite sport at school. The
frequency of the learners who selected a particular sport is represented in the
bar graph below. Study the graph and answer the questions.

a) How many learners played cricket?

b) How many learners played volleyball?
c) Which category was selected by most learners?
d) Which sport or sports were chosen by the fewest learners?
e) What is the mode category indicated by the learners?
f) How many learners took part in the survey?
g) If the category, “No favourite sport” is ignored, what sport will be the mode?
h) How many learners took part in the modal sport in question g)?

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5) A sample of 174 learners answered the question: “How do you travel to

school?” The percentages of learners using different ways of travelling to school
are shown in the graph below.

a) What method of travelling to school is the mode?

b) How many of the learners travelled to school by bus?
c) How many of the learners travelled to school by taxi?

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Mean
9 The mean is also known as the arithmetic mean or average.
9 To determine the mean you add all the scores together and then divide the sum
by the number (n) of scores.
sum of all the scores
9 We can use the formula: Mean = to calculate the mean.
number of scores
9 The “sum of all the numbers” is calculated by adding all the numbers together.

)
Example 3.2
Find the mean of the following set of scores:
1, 7, 9, 4, 3, 5, 9, 3, 2, 8

Solution
The sum of all the scores = 1  7  9  4  3  5  9  3  2  8 50

There are 10 scores in the data set, therefore to calculate the mean, you divide the
sum (50) by the number of scores or numbers (10).

sum of all the scores 50

Mean = =5
number of scores 10

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Exercise 3.2

1) The following data set consists of the ages (in years) of 5 learners:
9, 15, 11, 18, 19
a) What is the sum of the ages of the learners?
b) What is mean of the ages?

2) The data set below contains the ages of a few learners in years.
9, 15, 9, 15, 17, 15, 11, 18, 15, 19
a) The ages of how many learners are in this data set?
b) What is the sum of the ages of the learners?
c) What is mean of the ages?
d) What is the mean of the first 5 ages?

3) The following data set contains the heights of learners in centimetres.

138, 161, 121, 170, 170, 165, 142, 160, 140
a) What is the mean height of the learners in this data set? Give your answer
correct to 2 decimal places.
b) What is the mean height of the first 6 learners in the data set?

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Median
9 The median of a data set is the number right in the middle of an ordered data
set.
9 If there is an odd number of scores, the median is the middle number.
9 If there is an even number of scores, then two numbers will be the middle numbers.
In that case, you add the two numbers together and divide by 2.

)
Example 3.3
What is the median of each of the following sets of numbers?
a) 1, 7, 9, 4, 3, 5, 8, 3, 2
b) 2, 5, 7, 9, 10, 4, 12, 1, 15, 3

Solution
a) First order (or rank) the numbers in the data set from the smallest to the
greatest number: 1, 2, 3, 3, 4, 5, 7, 8, 9
There are 9 items in the data set.
The middle item is the number 4.
So the median = 4

b) First order (or rank) the scores from the smallest to the greatest number.
The data set looks as follows: 1, 2, 3, 4, 5, 7, 9, 10, 12, 15
There are 10 scores in this data set.
There is not only one score in the middle, but there are two, namely 5 and 7.
In this case, you add the two numbers in the middle and divide the sum by two
(2) to calculate the median:
57 12
Median = 6
2 2

NOTE: The median of this data set is 6, although the 6 did not actually appear in
the data set.

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Exercise 3.3

1) The following data set contains the ages, in years, of a few learners.
9, 15, 11, 18, 19
a) How many ages of learners are in this data set?
b) Is the number of ages an odd or an even number?
c) Rank the ages from the smallest to the biggest.
d) What is the median of the ages?

2) The data set below contains the ages of a few learners.

9, 15, 9, 15, 17, 15, 11, 18, 15, 19
a) Describe all the steps that you would take to calculate the median of the
ages.
b) What is median of the ages?

3) The following data set contains the heights of Grade 9 learners in metres.
1,81; 1,53; 1,6; 1,5; 1,28; 1,65; 1,75; 1,58; 1,13; 1,68; 1,77
a) Rank the heights of the learners from the smallest to tallest.
b) How many heights of learners are in this data set?
c) What is the median height in metres of the learners?

4) Grade 8 learners were asked how long it took them to travel to school.
The following data set contains these times (in hours).
0,50; 0,58; 0,67; 0,75; 0,50; 0,83; 1,00; 1,33; 0,58
a) Rank the travelling times from the shortest to the longest.
b) How many travelling times are in this data set?
c) What is the travelling time of the
(i) 3rd learner?
(ii) 7th learner?
(iii) 8th learner?
d) Which learner(s) travelled 30 minutes to get to school?
e) What is the position of the learner who travels 0,75 hours to get to school?
f) What is the position of the modal travelling time?
g) What is the modal travelling time (in hours)?

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5) a) A data set has 121 ordered data values. In which position will the
median be?
b) A data set has 502 ordered data values. Explain how you will determine
the median’s position and how you will calculate the median.

6) Thembi makes the following statement. “The median of a data set with 40
ordered data values is in the 20th position.”
Is this a valid statement? Explain your answer.

9 Do you know the song, ‘Row, row, row your boat’?

Use the tune to try to remember the difference between mode, mean and
median with the following song:

Mode, mode, mode is most,

Average is the mean,
Median, median, median, median, always in between!

9 If you do not know the song or the melody, write a song yourself to assist you in
remembering the difference between the three measures of central tendency.

9 If you have internet available, see also:

http://www.youtube.com/watch?v=oNdVynH6hcY&NR=1

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Measures of dispersion or spread

9 Sometimes we need to know how spread out data is.
9 For this we need to know two other statistics, the range and the extremes.
Both of these are called measures of dispersion (or measures of spread).

Range
The difference between the highest value and the lowest value in a data set is called
the range.
Range = highest value – lowest value

)
Example 3.4

Find the range of the following ranked data set:

1, 2, 3, 4, 5, 7, 9, 10, 12, 15

Solution
The lowest value is 1 and the highest value is 15.
So the Range = highest value – lowest value
= 15 – 1
= 14

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Exercise 3.4

1) The following data set contains the ages of a few learners: 9, 15, 11, 18, 19
a) Write the ages in order.
b) What is the highest age?
c) What is the lowest age?
d) What is the difference between the highest age and the lowest age?
e) What is the range of this data set?

2) The following data set contains the ages of a few learners.

9, 15, 9, 15, 17, 15, 11, 18, 15
a) What is the highest age?
b) What is the lowest age?
c) What is the range of this data set?

3) The data set below contains the heights of learners in centimetres.

138, 161, 121, 170, 170, 165, 142, 160, 140, 182
a) What is the height of the tallest learner in this data set?
b) What is the height the shortest learner in this data set?
c) What is the range of the heights in the data set?

4) Learners were asked how many minutes they travelled to school every day.
They answered as follows: 10, 45, 11, 5, 20, 10, 5, 24
a) What is the shortest travelling time?
b) What is the longest travelling time?
c) What is the range of travelling times?

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Extreme values
9 Extreme values are unexpected large or small values in a data set.

)
Example 3.5
The lengths (in centimetres) of the right feet of a group of 8-year
old learners are: 13, 18, 18, 19, 20, 20, 20, 21, 22, 31
What do you notice about this data set?

Solution
If you look at the foot lengths carefully, you can see that the lengths of the middle
eight of the foot lengths range from 18 cm to 22 cm.
13, 18, 18, 19, 20, 20, 20, 21, 22, 31
o The first learner has a very small right foot of only 13 cm.
o The largest right foot is 31 cm long.
o The very small value and the very big value are unexpected. They are called
extreme values.

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Exercise 3.5

1) A group of learners was asked to say how many minutes it took them to travel
to school every day.
They answered as follows: 45, 5, 20, 50, 13, 15, 18, 15, 90, 5, 15, 25, 15, 4
a) How many values are in the data set?
b) Rank the travelling times.
c) Identify the extreme value(s) and say why you think they are extreme.

2) In the Census@School survey, learners had to say how many children in their
home were still at school. The results are given in the following table:
Household Boys Girls
1 1 1
2 1 1
3 3 1
4 1 2
5 1 1
6 2 0
7 1 2
8 2 1
9 0 0
10 0 1
11 1 2
12 12 1
13 1 3
14 3 3
15 1 4

a) Which household had the highest number of school-going boys?

b) Which household had the highest number of school-going girls?
c) (i) Which household has the fewest school-going children?
(ii) How many children are school-going in this household?
(iii) How did you determine your answer?
(iv) Is this answer possible?
d) (i) Which household has the most school-going children?
(ii) How many children are school-going in this household?
(iv) Is this answer possible?
e) Are there any extreme values among the boys? Why do you say so?
f) Are there any extreme values among the girls? Why do you say so?

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3) The heights (in centimetres) of Grade 9 learners are given in the table below:
169 181 145 159 160
171 165 109 168 170
173 176 140 178 155
150 170 162 146 151
a) Rank the heights from shortest to tallest.
b) How many heights are in the data set?
c) What is the modal height?
d) What is the median of the heights?
e) What is the mean of the heights of the learners?
f) What is the range of the heights?
g) Identify an extreme value in the data.
h) Delete the extreme value from the data and recalculate the mean of the
data to 1 decimal place. What do you notice about the original mean and the
new mean?
i) Did the mode and the median change when you deleted the extreme value?

4) Explain the difference between a data set with no mode and a data set with a
mode of 0? Give your own example.

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Chapter

4
Representing Data
In this chapter you will:
x Draw graphs using pictograms
x Learn to draw 3 different types of bar graphs:
x Learn to draw Pie Charts, Histograms, Broken Line Graphs and Scatter plots
x Learn about scales on a graph
x Learn to choose an appropriate graph to represent given a set of data.

G raphs are often used to display data. Many people find graphs simpler to
understand than a table. They are also more attractive and interesting to look
at. A graph can show data clearly without lots of words or figures. This helps
you to see any patterns and to compare things easily.

PICTOGRAPHS
NOTE A pictograph gives you a quick impression of the
A pictograph uses given information.
simple pictures or
symbols to show data
Pictographs are often used in newspapers, magazines, books and on television because
comparing data in a pictograph is easy; just compare how many pictures each item has.

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Reading information off a pictograph

EXAMPLE 4.1

)
The 2009 Census@School was run at Anele’s school.
The results were analysed and it was found that the favourite sport
of the boys in her school was soccer, rugby, cricket and athletics.
Anele drew up a table to show the percentage of the boys that
liked each sport.
Draw a pictograph to represent this data

Favourite Sports – Males Percentage

Soccer 14
Rugby 6
Cricket 7
Athletics 3
No favourite Sport 3

SOLUTION:
In order to draw a pictograph, we need a title and a key. Title
x The title tells you what Favourite Sports – Males
the pictograph is about. Soccer -------
Rugby ---
x The key shows you what
each little picture “stands Cricket ----
for” or “represents”. Athletics -
x If each - = 2% of the No Favourite Sport -
males, then KEY: - = 2% of males
= half of 2%
= 1% of males Key

x So to represent 7% who
like cricket, we use three
full pictures and 1 half picture like this: Cricket - - -

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Drawing a pictograph
x Choose a simple picture or symbol that is easy to draw. Try to make it ‘look like’
the data in some way.
x Always give a key.
9 You must say what each picture stands for.
9 Each picture may represent more than 1 piece of data. Choose an
easy number to work with.
9 Part of a picture represents part (or a fraction) of that number
x Draw your pictograph on squared paper. This will help you to keep the pictures
in line.
9 Try to draw each picture the same size and space them out evenly.
9 Estimate fractions of a picture where necessary
x You could also use different pictures for different kinds of data.

EXAMPLE 4.2

)
The Census@School questionnaire also asks learners about the
type of dwelling or house in which they live. Anele drew a
pictograph to show the number of learners in her school living in
each type of dwelling.

Types of dwellings in which the learners live

Formal
Traditional
Informal or other
KEY: each dwelling = 10%
a) What percentage of the learners lives in Formal Dwellings?
b) What percentage lives in Traditional Dwellings?

SOLUTION
a) The pictograph uses 7 full houses plus half a house to represent the
percentage of the learners living in Formal Dwellings.
This means that 10% + 10% + 10% + 10% + 10% + 10% + 10% + 5%
= 75% of learners live in Formal Dwellings.

b) It also uses one full house plus half a house to represent the percentage of
the learners living in Traditional Dwellings.
This means that 10% + 5% = 15% of the learners live in Traditional Dwellings.

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EXERCISE 4.1

 1) Vonani is doing a Travel and Tourism project. He does a survey

of 100 people to find out where they like to go on holiday. The table shows the
result of his survey.
Place Number of people
The sea 56
The mountains 12
The game reserve 24
Other 8
Draw a pictograph to show the results.

2) The following results were obtained in David’s school after the Census@School
was run at his school. Draw a pictograph to show the results.

Favourite Subject Boys in Grades 3 to 7 Percentage

Maths 15
Literacy 6
Languages 7
Technology 3
Numeracy 3
Arts & Culture 5
Life Orientation 3

Pictographs have two main disadvantages

1) They can take a long time to draw well.
9 So, unless you are spending time on a special project, use simple
symbols that are quick and easy to draw.

2) They are not always a very accurate way of showing data because:
9 the data is often simplified before drawing;
9 fractions of pictures are estimated; etc.

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Bar Graphs
The length of the bar stands for the size of the data
NOTE
it represents. This makes the data easy to compare. Just
A bar graph uses bars
compare the lengths of the bars. The bars can be drawn
to display data. horizontally or vertically, and gaps are left between the bars.

Example 4.3

)
Draw a bar graph to illustrate the facilities and services at the
schools that took part in the 2009 Census@School
Facilities and services at school Percentage
Maths teacher 68,6
Electricity 65,9
Running water 60,5
Telephone 69,8
Library 24,6
Computer 53,0
Email 14,7
Internet 14,5
SOLUTION:
x First, you need a title for the graph.
For this graph the title is “Facilities and services at school”.
x Next you need two axes and a label for each axis.
In this example we want a vertical bar graph, so the bars will go up the page.
Since the percentage values will indicate the height of the graph, this will be on
the vertical axis.
o Vertical axis – Percentage
o Horizontal axis –Type of facility or service.
x Determine the scale on the vertical axis.
The highest percentage value is 68,6 %, so you can choose the vertical axis to go
up to 80. The interval of the scale will be 10 units.
x Finally, draw in each bar corresponding to the value of the category.
So, for the “maths teacher” category we draw a graph that is tall enough to
represent 68,6%.
Keep in mind that often we have to approximate the height of the bar since
some scales do not make drawing exact heights possible.

The final graph is on the next page.

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Select the starting point on the vertical axis carefully

When drawing a bar graph, the scale on the vertical axis has to be selected
carefully as it can affect how you interpret the graph.

o GRAPH 1 and GRAPH 2 contain the same information. They show the birthdays
per quarter of a random sample of 80 learners in Grade 8 in Kgomotso’s school.
o Look at how the scale of the vertical axis makes a difference in the way you
interpret the information displayed on the graphs.

Can you see that in Graph 1 ...

x The vertical scale begins at 10 and
ends at 30.
9 This is because the range of
values being graphed is from 16
to 24.
9 Since we want our smallest bar
to have some height, we start
the scale at 10.
x It looks like there 2-times as many
learners have a birthday in the 1st
term than have a birthday in the 4th
term.
9 This is not true because 24

learners have a birthday in the 1st quarter and 17 learners have a birthday in
the 4th term.

Can you see that in Graph 2 …

x The frequency scale begins at zero.
x It is clear that the bars for 1st quarter
and the 3rd quarter have nearly the
same height.
x The difference between the four
quarters does not seem as big as it
does in GRAPH 1.

NOTE: Direct comparison can only be made when the scale begins at zero.

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Select the intervals on the vertical axis carefully

9 When drawing the graph, the intervals (the distance between the marks) on
the vertical axis have also have to be selected asthey can affect how you
interpret a graph.

x On some bar graphs the scale is very easy. Each ‘small space’ stands for 1
unit.

0 1 2 3 0 1 2 3 4 5 6

0 5 10 15 0 10

x Some scales are not so easy. Each small space may stand for more than 1, or
for a fraction

0 100 0 20 0 1 2

10 spaces Æ 100 10 spaces Æ 20 5 spaces Æ 1

1 space Æ 100 y 10 = 10 1 space Æ 20 y 10 = 2 1 space Æ 1 y 5 = 0,2

o Changing the interval of the scale also affects the way a bar graph looks.

o Look again at GRAPH 2 and at GRAPH 3 showing different intervals on the

vertical axis ...

2 spaces Æ 5 units 5 spaces Æ 5 units

1 space Æ 5 y 2 = 2,5 1 space Æ 1 unit
So, each interval is 2,5 units So, each interval is 1 unit

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Exercise 4.2

 1) 200 people were asked which their favourite TV channel was.

The information is given in the following table. Draw a bar graph to illustrate
this information.

SABC1 78
SABC 2 44
SABC 3 75
MNET 33
e TV 23

2) 180 learners who took part in the 2009 Census@School were randomly selected
from the sample of schools. The following table was drawn showing the
provinces in which they were born. Draw a bar graph to illustrate this
information

NUMBER OF
PROVINCE
LEARNERS
Gauteng 39
Western Cape 25
Mpumalanga 25
Kwa-Zulu Natal 22
Northern Province 19
North West 14
Northern Cape 13
Free State 12
Eastern Cape 9
Outside of South Africa 2

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Compound Bar Graphs

NOTE
A compound bar graph When data is compared, either a compound bar
can be used to graph or sectional bar graph can be used.
compare different sets
of data.

o If you look at separate bar graphs, it is not very easy to compare them.

o We can combine two bar graphs and put the bars next to each other. This
makes it much easier to compare the two data sets.

EXAMPLE 4.4

)
The 2009 Census@School asked learners what their favourite sport
was. The following table gives the percentage girls and percentage
boys who reported that the following sports were their favourites.

Draw a compound bar graph to represent this information.

Percentage of Percentage of
FAVOURITE SPORTS
the boys the girls
Soccer 61,6 6,0
Rugby 8,4 1,5
Cricket 6,0 2,3
Athletics 7,8 10,2
Netball 1,8 50,5
No favourite Sport 14,4 29,5
Solution:
x Draw two axes at right angles to one another and chose a scale.
x For each sport draw a bar for the boys and a bar for the girls.
o So for soccer, first draw a bar with height 61,6 units for the boys.
o Then draw a second bar next to it without leaving a gap with a height of
6 units for the girls. Make this bar a different colour from the boys bar.
x Similarly, draw the sets of bars for each of the other sport categories.

The final graph is on the next page ...

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Here is another type of compound bar graph representing the data in Example 4.4:

9 Notice that the bars are not drawn alongside each other, but instead
are drawn on both sides of the vertical axis. This is useful because it helps to
make a quick visual comparison across each data category (sport).

9 You can see that 61,6% of boys like soccer, in comparison with just
6% of the girls. In contrast the netball seems to be the most popular sport for
girls; 50,5% of the girls chose this as their favourite, while only 0,8 % of the boys
chose netball.

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Exercise 4.3


Study the table showing the differences in the access to goods and services
between the learners in the 2001 Census@School and the 2009 Census@School.

Types of goods & Census@School Census @School

services at home 2001 2009
Telephone 47,6 30
Running water 55,3 55,3
Television 83,5 85,8
Radio 94,2 90,2

Draw a compound bar graph to show the percentage of learners’ access to goods
and services in the 2001 Census@School and the 2009 Census@School.

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Sectional or Stacked Bar Graphs

NOTE We use sectional bar graphs when we have two,
A sectional bar graph or more, different sets of information on the
consists of one bar on same topic. They are particularly useful when we are
top of another. interested in the total of the two of more bars.

EXAMPLE 4.5

)
A random sample of learners was taken from the results of the
2009 Census@School, and the following table was drawn.

Draw a sectional bar graph to represent the information.

Favourite Soccer / No
Netball Rugby Cricket Athletics Aerobics
Sport Football Favourite
Girls 4 14 18 1 1 4 2
Boys 19 7 2 5 5 1 1

Solution:
i. Use the data to draw the bars for the girls just as you would draw a single
bar graph.
o So, for soccer you would draw a bar with a length of 4 units.
ii. Draw all the rest of the bars to represent the girls.
iii. Now, for each category, draw the bar for the boys directly above the bars for
the girls, touching the bar for the girls. Shade this in a different colour from
the girls’ bar.
o This means that for soccer, you would draw a bar of length 19 units
above the girls’ bar you drew earlier. So the total height for bar for the
soccer category will be 4 + 19 = 23 units.

FAVOURITE SPORTS
25
Number of learners

20
15
10 B
5
G
0
Soccer No Netball Rugby Cricket Athletics Aerobics
favourite

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Exercise 4.4


In the 2009 Census@School, 600 learners from Grades 3 to 7 were asked which
subject was their favourite.

The results were as follows:

females males
Life Orientation 54 38
Languages 154 114
Technology 18 34
Natural Science 26 26
Maths 64 72

1) Draw a suitable sectional bar graph to illustrate the data.

2) Write down two conclusions about this data.

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Pie Charts
NOTE The whole circle stands for the whole amount of
A pie chart is a circular data being dealt with.
diagram used to Each slice stands for part of the data. Its size represents the size of
display data. that part of the data.

9 A pie chart is particularly suitable if you want to illustrate how the “whole” of some
data is divided up into different parts, and what portion of the whole each part
represents.
9 We can write this portion as a fraction, as a decimal fraction, or as a percentage of
the whole.

Drawing a Pie Chart

1) Draw circle centre O with a pair of compasses.
(Do not make your circle too small)
2) Draw radius OB as a starting line for measuring B O
angles.
o Remember that a radius is a straight line from
the centre of the circle to outside of the circle
C
(or circumference of the circle).
3) Use a protractor to draw the angle of each sector at
the centre of the circle.
o Measure the first angle from radius BO, by B O
placing the centre of the protractor at the
centre of the circle.

4) Measure the second angle from radius OC, then C

measure the next angle from the new line, and so
on.
o Measure the size of each angle very carefully.
Any mistakes will mean that the sectors will B O
not fit into the circle.
o Hint: Draw the smallest angle first, then the
next biggest, and so on.

5) Label each slice carefully. If it is difficult to fit the full name of each group on
each slice, label each with a letter and use a key to say what each stands for.
6) Give the pie graph a suitable title.
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How to work out the angles

1) Look at the given data, and decide what the “whole” or the total amount to be
shown on the pie graph (e.g. 24 hours) is going to be.
2) Find out how many different parts or sectors there are.
3) Calculate the angle at the centre of the circle for each of the sectors of the pie
graph. Do this by dividing 360º by the “whole”.
For example:
If we want to show how many hours in a day is spent on different activities, our
calculation is 360° y 24 = 15°
4) Multiply this answer by the number of items
For example: the angle for 3 hours is 3 x 15° = 45°
5) Always check that all the angles add up to 360q, but remember that when angles
are rounded off to the nearest degree, the total may not be exactly 360q.

EXAMPLE 4.6

)
The eye colours of a random sample of 120 learners who took part
in the 2009 Census@School were as follows. Draw a pie chart to
display this data.
Eye colour Frequency
Blue 2
Brown 104
Green 4
Other 10
TOTAL 120
Solution:
Calculate the angles of the pie chart as follows
1) The “whole” is 120 students
2) There are 4 different parts; that is, 4 eye colours.
3) Divide 360° by the “whole” : 360° y 120 = 3°
4) Multiply this answer by the number of items for each part:

Eye colour Frequency Angle

Blue 2 2 × 3q = 6q Notice that
all the
Brown 104 104 × 3q = 312q
angles add
Green 4 4 × 3q = 12q up to 360°
Other 10 10 × 3 = 30q
TOTAL 120 360q

The completed graph is shown on the next page ...

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EXERCISE 4.5

 1) Thirty learners were asked how they travel to school, and the
results were recorded in the given table.
a) Copy the table
b) Work out the angle at the centre of the circle for each learner by calculating
360° y 30
c) Fill in the rest of the table
d) Draw a pie chart to show this information

Method of travel Number of learners Angle

Walk 14 14 x 12° = …………….
Bus 7
Car 6
Bike 3
Helicopter 0
TOTAL: 30

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2. 90 people were asked which month they were born in. Here are the results:

Month No of people Angle Month No of people Angle

January 7 July 6
February 4 August 8
March 9 September 11
April 8 October 7
May 7 November 10
June 6 December 7

a) Copy the table

b) Work out the angle for one person (i.e. 360° y 90 = ……………)
c) Work out the angle for each month and write it on the table.
d) Draw a pie chart to show this information.

3. A survey was done of 120 learners from Grade 8, to find out what their favourite
subject was at school.
It was found that 30 prefer History, 40 prefer Geography and 50 prefer Maths.
a) Illustrate this information by drawing:
i) A table to illustrate this data.
ii) A bar chart
iii) A pie chart
b) Which one of the two graphs do you think represents this information
better? Give a reason for your answer.

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Histograms
NOTE Bars are drawn corresponding in height to the
A histogram is a graph frequency of each group.
of grouped data and The intervals or groups are shown along the horizontal axis
has no gaps between The frequency (or how often something happens) is shown along
the bars. the vertical axis.

EXAMPLE 4.7

)
The heights of the heights of 150 learners in Grade 7 at Makhosi’s
school are recorded in the following table.

Draw a histogram to display this data.

Frequency /
Height (cm)
number of learners
115 ч h < 120 6
120 ч h < 125 6
125 ч h < 130 29
130 ч h < 135 30
135 ч h < 140 35
140 ч h < 145 26
145 ч h < 150 12
150 ч h < 150 6

Solution:
x Draw the x-axis. Label this axis “Heights of learners”.
o In this example, the x-axis would have 8 intervals since the table has 8
intervals.
o Use the squares on the graph paper to make each interval the same size.
x Draw the y-axis. This will show the frequency.
o Label it “Number of learners”.
x Draw bars on the histogram with heights corresponding to the frequency of that
interval.

o So the first interval; 115 ч h < 120 will have a height of 6 units.

o Draw in the rest of the bars.

o The histogram bars are drawn next to and touch each other.
x Add a title to complete the histogram.

The required histogram is drawn on the next page…

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EXERCISE 4.6


1) In the 2009 Census@School, the learners Time in minutes frequency
from Grades 3 to 7 at Ruth’s school were 0 ч m < 10 45
asked how long in minutes, it takes for 10 ч m < 20 48
them to travel to school. 20 ч m < 30 35
30 ч m < 40 14
The table shows the results from a sample 40 ч m < 50 6
of 150 learners.
50 ч m < 60 1
60 ч m < 70 0
Draw a histogram to illustrate this data.
70 ч m < 80 1

2) The same group of 150 learners were Distance in km frequency

asked to fill in the distance they travelled 0чd<2 72
to school. The table shows the results. 2чd<3 48
3чd<4 18
Draw a histogram to illustrate the data.
4чd<5 12

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Scatter Plots
NOTE Each axis represents a different variable.
We plot values of one quantity against corresponding values of
A scatter plot is used to
display two sets of
another quantity.
data in order to find a
relationship between A scatter plot shows trends
them.

For example …
Joe has a job selling ice cream. When will he sell the most ice cream?
o Joe says he will sell the most ice cream on a hot day and less ice cream on a
cooler day.
o He thinks he will hardly sell any ice cream on a very cold day.
o Joe calls this common sense. In maths it is called a correlation.

A scatter graph showing the number of ice

creamsold

d
l 45
o
s 40
35
m
a 30
e
rc 25 Notice: The points on the
e 20
c
i 15 graph more or less form a
fo 10
re 5 straight line. We say there
b 0
m 0 10 20 30 40
is a positive linear
u
N Temperature correlation between the
number of ice creams sold
and the temperature.

The following scatter graph shows no correlation between the points:

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EXAMPLE 4.8

)
Use these results from a group of randomly selected learners who
participated in the 2009 Census@School to draw a scatter plot to
see if there is a correlation between age in years and grade
number.
Age in years Grade In school
13 8
16 10
16 11
11 6
14 7
13 6
18 12
10 4
9 3
10 5
Solution:
Draw the graph as follows:
x Draw two axes. Plot the age in years along the horizontal axis and the Grade
along the vertical axis.
x Choose an appropriate scale for each axis.
Here you can make both intervals 1 unit.
x Plot each set of points.
So the first point is Age = 13 and Grade = 8.
Similarly plot each set of point given. Here is the completed graph:

This point
corresponds to the
first set of data:
Age 13, Grade 8

CONCLUSION: There is a correlation between the age of a learner and the grade
they are in. As the age increases, the grade number increases.

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Exercise 4.7


1) 12 learners measure their heights and hand spans and tabulated their results.
Copy the given set of axes and draw a scatter plot to see if there is a relationship
between height and hand span.
Hand span in cm 26 25 20 19 25 18 22 27 23 24 23
Height in cm 177 175 165 164 172 160 180 164 175 171 169

2) A group of 14 learners measure their heights and foot sizes and tabulated their
results. Copy the given set of axes and draw a scatter plot to see whether foot
size increases with height.
Foot Height
(cm) (cm)
27 174
24 158
24 163
26 175
23 140
23 126
22 153
25 160
24 160
20 131
23 156
25 165
24 158
20 120

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Chapter

5
Interpreting Data
In this chapter you will:
x Look at bar graphs, pie charts and line graphs and interpret the information.
x Consider how information on bar graphs, pie charts and line graphs can be misleading
x Know more about errors when collecting and interpreting data

S o far in this workbook you have looked at different ways to collect, organise and
present information or data. Remember that the whole point of collecting data
is to help you understand more about the world you live in. In particular, by using
the Census@School data you find out about different schools and the learners who
attend them.

Interpreting bar graphs/bar charts

Bar graphs (which are also sometimes called bar charts) are very easy to read.
9 The lengths of the bars stand for the size of the data.
9 Gaps are usually left between each bar and the next.
9 The bars can be horizontal or vertical.
9 Remember that bar graphs are usually used to show discrete data (data that
can be counted) as opposed to continuous data (data that can be measured).
9 For every bar graph check the following points:
1. The title – what is the bar graph about?
2. The axes – check the labels of the axes. One axis gives the labels of the bars;
the other tells you how many items in each bar.
3. The scale – the scale on the number axis tells you how many of each thing
has been counted.

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)
Example 5.1
Study the bar graph below taken from the 2009 South African
Census@School results. It shows the modes of transport to school
from home of the learners who participated in the Census@School.

1) What is the graph about?

2) What information is shown on the axes?
3) Is the scale clear and easy to read?
4) What conclusions can the graph help you make?

Solution
1) In order to find out what the graph is about, we look at the title. The title tells
us that the graph shows how learners get to school each day.
2) The axes are clearly labelled.
The horizontal axis shows the different forms of transport: walk, car, train, bus,
bicycle, scooter, taxi and other.
The vertical axis shows percentage.
3) The scale is clear and easy to read.
You cannot read exact percentages for the number of learners walking to
school, but it is easy to estimate that nearly 80% (or over 70%) walk to school.
4) You can immediately see that most of the learners in the census walk to school
from home. The other learners mostly use car, bus or taxis to get to school.

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Misleading bar graphs

9 Bar graphs are often used to present misleading results.

9 An important thing to look at is the scale on the axes:
x Does the scale start at zero
x Is the scale distorted – too squashed up or too spread out. This is a trick that
is often used to make something look better than it is.

)
Example 5.2
The following graph (Graph B) illustrates the same data as the bar
graph on the previous page (Graph A).
Compare the two graphs. Why is it more difficult to read
information off Graph B than Graph A?

Solution
1) There is no heading to the graph so it is difficult to tell what the graph is about.
2) The vertical axis has no label. It might be showing actual numbers or
percentages. It is not clear.
3) The cones instead of bars do not give a clear ‘picture’ of what the information is
saying. Three dimensional (3D) shapes like these are usually more difficult to
read than two dimensional (2D) shapes like bars.

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Exercise 5.1

1) The following table shows the birth months of approximately 45 000 South
African learners who took part in the 2009 Census@School. The bar graph that
follows has been drawn to illustrate some of the information.

Month Frequency Month Percentage

January 3 568 January 7,9
February 3 592 February 8,0
March 3 922 March 8,7
April 3 490 April 7,8
May 3 888 May 8,6
June 3 981 June 8,9
July 3 597 July 8,0
August 3 876 August 8,6
September 4 476 September 10,0
October 3 502 October 7,8
November 3 253 November 7,2
December 3 809 December 8,5
TOTAL 44 954 TOTAL 100.0

a) In which month were the least learners born?

b) In which month were the most learners born?
c) Why was percentage rather than frequency used for the y-axis?
d) Is it easier to read the most and least values off the graph or the table?

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2) What is wrong or missing from this bar graph?

3) Graph A and Graph B show the number of cars sold each year by a particular car
dealer.

Explain why the graphs look different

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Comparative bar graphs

9 Comparative bar graphs use adjacent bars to compare two or more
subcategories.

)
Example 5.3
In the graph shown (taken from the South African Census@School
2009 results) the favourite subjects of male and female learners in
Grades 8 to 12 were compared.

1) What was the favourite subject of both the boys and the girls?
2) What was the next favourite subject of both the boys and the girls?
3) What percentage of the boy learners and the girl learners say that Technology is
their favourite subject?

Solution
1) Languages were the favourite subject of both boys and girls
2) The next favourite for both the boys and the girls is Mathematics
3) 3,6% of boys chose Technology as their favourite subject, while only 1,5% of the
girls chose it.

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Exercise 5.2

1) Look at the graph shown below.

a) What is the graph about?
b) What do the different coloured columns represent?
c) Write 2 sentences describing the information in the graph

2) The bar graph below shows the favourite subject of the learners in Grades 3 to
7. Compare this graph to the graph showing the favourite subjects of the
learners in Grades 8 to 12 on the previous page (page 74).
a) What subject is the favourite for Grades 3 to 7 and which subject is the
favourite for Grades 8 to 12?
b) List the favourite subjects of the Grade 3 to 7 girls from most favourite to
least favourite
c) What was the least favourite subject for all Grade 3 to 7 learners?

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3) The two graphs below show information about the favourite sport for boy and
girl learners in the 2001 and 2009 Census@School.
a) Write at least 3 sentences describing the differences in the favourite sport
for boy and girl learners.
b) Have the favourite sports changed at all from 2001 and 2009? How can you
tell?

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4) The graph below is taken from the Grade 8 to 12 Census@School done in

Ireland. Write at least three sentences describing what the graph shows about
learners’ preferences to be famous, happy, healthy or rich.

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Interpreting Pie Charts

9 The interpretation of pie charts is based on the fact that the largest slice of the
pie chart corresponds to the largest item of data and the smallest slice of the pie
chart corresponds to the smallest item. It is therefore easy to make
comparisons between the relative sizes of data items.
9 When interpreting a pie chart, the following parts of the pie chart are
important:
1. The title – what is the pie chart about?
2. The sectors or slices – check the labels of the sectors and any other
information given about the sectors.
3. The size of the sectors or slices – the size tells you the ‘how many’.
Sometimes the size of the slice is easy to estimate by eye, e.g. 90°at the
centre of a circle is a quarter of a circle.
9 One of the main advantages of pie charts is that you can see part of the data as
a fraction of the whole data.
9 Pie charts that are badly drawn can also display misleading information. As with
bar graphs if pie charts are missing headings or labels they can be difficult to
interpret.

)
Example 5.4
The following pie chart is taken from the 2009 South African
Census@School results. It shows the distances that learners
travelled from home to school each day.

1) What is the title of the bar chart?

2) What do the sectors (or slices) tell us?
3) Compare the sizes of the sectors and say what they tell us.

Solution
1) The title is “Travel distance to school from home” and tells us that the pie chart
shows the distance travelled by learners to get to school each day.

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2) The sectors/slices show different categories of distances: Less than 1 km; 1 to 5

km; 6 to 10 km and 11 km or more. They also show the percentage of learners
in each category.
3) There are only 4 sectors it is easy to compare the sizes.
o The 2 largest slices are almost the same size, so without knowing the
percentages it would be difficult to say with certainty which is the bigger.
o However it is easy to see that the smallest slice is those learners who travel
11 km or more from home to school.
o Most learners (42%) live less than 1 km from school.
o 42% + 36% = 78% of the learners live 5 km or less from school.


Exercise 5.3

1) Look at the two pie graphs below and then answer the following questions
about them:
a) What do the graphs represent?
b) Which food type is eaten most by males?
c) Which food type is eaten least by males?
d) Which food type is eaten most and eaten least by females?
e) What is the difference between the dairy consumption of males and the
dairy consumption of females?
f) Write down any other observations you can make about the data
represented in the pie charts.

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2) Look at the table and graphs shown below that give the gender by grade of
learners who participated in Census@School in 2009 and then answer the
questions on the next page.

GENDER
GRADE TOTAL
Boys Girls Unspecified
Grade 3 4 969 4 983 148 10 100
Grade 4 4 180 4 121 85 8 386
Grade 5 5 126 5 213 84 10 423
Grade 6 5 825 6 128 75 12 028
Grade 7 7 510 7 913 105 15 528
Grade 8 9 654 10 782 118 20 554
Grade 9 9 847 10 547 183 20 577
Grade 10 9 521 9 974 127 19 622
Grade 11 7 521 9 158 96 16 775
Grade 12 6 027 7 862 123 14 012
Total 70 180 76 681 1 144 148 005

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a) Is the table, the bar chart or the pie charts easier to read? Explain your answer.
b) In which grade were there most learners?
c) Did more boys or more girls participate in the survey? How do you know?
d) Can you tell how many learners participated in the census by looking at the bar
chart or the pie graph?
e) Look carefully at the graph below. In what way is it different to the bar graph on
the previous page? Does it show the same information or different
information?

f) By looking at any of the four graphs for this question, what conclusions can you
make about the gender by grade of the learners who participated in the census?

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Line Graphs
9 Line graphs are often used to show how a quantity varies.
9 Some important things to help you interpret line graphs are:
1. The title – what is the line graph about?
2. The axes – check the labels of the axes.
3. The scales on the axes – do the scales start at zero? What else do the scales
tell you?

)
Example 5.5
The following line graph is taken from the 2009 South African
Census@School results. It shows the average height of learners by
age and gender. Notice that the heights of the boys and the girls
are compared using two lines on one graph.

1) What is the title of the line graph?

2) What information can we read off the vertical and the horizontal axis?
3) How many units does each division on the vertical axis represent?
4) What conclusions can you make about the information shown on the graph?

Solution
1) The graph shows the average height by grade and gender
2) The vertical axis shows the age of learners and the horizontal axis shows the
grade.
3) Each division on the vertical axis represents 10 cm. The scales on the axes are
easy to read. From the information show you cannot tell actual heights but you
give a general comment about the ages.
4) You can easily see that the average height of boy and girl learners is similar
Grades 3 to 7. However, boys grow faster from Grade 8 to 12 and the average
height of boys in these grades is greater.

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Misleading line graphs

9 Some line graphs are purposely drawn to convey misleading information.

)
Example 5.6
The following two line graphs show the same information, but look
different.

What is the difference between the two graphs?

Solution
o The first graph clearly shows that most learners in Grade 3 are 7 years old, but
there are learners as old as 12 in Grade 3.
o In the second graph the information is much more squashed up and the graph
suggests that there are no learners older than 10 in Grade 3.

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Exercise 5.4

1) What do you think the person who drew these two graphs wants you to
believe? How would you correct the graphs so that they don’t mislead?

2) a) Why might this graph be misleading?

b) What might people believe from this graph?

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Histograms
A bar graph is a graph of ungrouped data and a histogram is a graph of grouped
data.
9 For a histogram, the bars must touch each other, with no spaces between the
bars.
9 Bars must begin and end at the boundaries of the intervals.
For example, if one interval starts just after 1 and ends at 5, the second interval
must start just after 5 and end at 10.
We write this as 1 < x ч 5 and 5 < x ч 10

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Exercise 5.5

1) The heights of one hundred Grade 12 boys were measured and recorded in the
following table.
Interval Percentage
80 < h ч 100 0
100 < h ч 120 2
120 < h ч 140 1
140 < h ч 160 5
160 < h ч 180 50
180 < h ч 200 40
200 < h ч 220 2
TOTAL 100

Two graphs were drawn to illustrate the data.

a) How wide (in centimetres) is the interval on the horizontal axis in Graph A
and how wide is the interval on the horizontal axis in Graph B?

b) Which of the two graphs best represents the spread of heights of the Grade
12 boys? Give reasons for your answer.

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2) Two histograms were drawn to show the ages of the boys and the girls who
participated in 2009 in an online Census@School questionnaire.
a) Use the two graphs to answer the following questions:
(i) What was the age of the oldest boys who participated?
(ii) What was the age of the youngest boys who participated?
(iii) What was the age of the oldest girls who participated?
(iv) What was the age of the youngest girls who participated?
(v) What age was the mode for the boys?
(vi) What age was the mode for the girls?
b) Write down at least two other observations you can make about the
information in the two graphs.

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Other ways in which errors can be introduced …

Besides graphs being drawn to mislead, errors can be introduced in the following
ways:
1) There can be sampling errors
2) There can be dirty data mixed in with the clean data
3) Misleading conclusions can be made.

Sampling
When you take a sample, you need to make sure that you don’t introduce bias
without thinking, e.g. you might only ask a certain group of people - only white
people or only Indian people. Or you might ask only those people who volunteer to
answer your questions.

Misleading Conclusions
Suppose someone tests Battery A and one Battery B. The batteries are each put in a
torch which is switched on. The time taken for the torch to stop working is
measured.

The conclusion is then made that Battery A lasts for up to twice as long as Battery B.

However, this conclusion is misleading. If one Battery A lasts longer than one
Battery B, we cannot use this fact to make a conclusion about all “Battery A’s”.

Dirty data
9 In a large set of data there are almost certainly going to be errors.
9 Errors may arise in many different ways. They could be caused by human errors
(such as someone misreading or miscounting data), to errors due to poor
sampling.
The following list shows some of the ways that errors can creep into lists of data:
x Mistyping:
A correct value might have been obtained when measuring or counting, but
the incorrect number was written down. This can happen when digits are
interchanged, e.g. writing 1 765 instead of 1 756, or when digits are written
twice, e.g. 772 instead of 72.
x Mistaken answer:
In an interview the respondent may misunderstand a question and so give an
incorrect answer, e.g. giving a yearly income instead of a monthly income.
x Mistaken measurement:
Many people like to write down ‘nice’ numbers, so instead of writing 31 they
might write 30.
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Exercise 5.6

o In this exercise we will look at a data set and find out where errors might have
crept in.
o It has been adapted from a worksheet found at
http://www.censusatschool.ie/en/resources/strand1statistics/118-cleaning-up-
your-data.
o First read the information below carefully.

Hanging out the Dirty Data

When you have got some data, the first thing you need to
do is to check it out and get rid of any obviously wrong or
false data. This is called “Cleaning the Data”.

In the Dirty Spreadsheet on the next page, several playful

pupils have been deliberately tampering with the data.
See if you can spot which rows and cells have been interfered with and by which of
the following characters:
x Pointy Pete: He moves Decimal Points around and adds in unnecessary ones.
x Obvious Olive: She puts in very obvious errors.
x Silly Samantha: She thinks it is funny to answer the question ‘Name?’ as
“Donald Duck”.
x Devious Dave: He thinks up clever ways to change the data.

1) Study the Dirty Spreadsheet on the next page. Find examples of where each of
the following characters tampered with (changed) the data.
a) Pointy Pete
b) Obvious Olive
c) Silly Samantha
d) Devious Dave.
2) The tampered data needs to be corrected. Explain how you will correct the
work of:
a) Pointy Pete
b) Obvious Olive
c) Silly Samantha
d) Devious Dave.
3) Investigate the Spreadsheet further and list any other dirty data on the
spreadsheet.

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Dirty Spreadsheet
Boy or Date of Height Foot Length Favourite Distance to School
Row 1 Grade
Girl Birth in cm in cm Subject in km
Row 2 Boy 12/04/91 5 143 26 Art 1-2km
Row 3 Girl 31/02/92 4 132 22 Science less than 2 km
Row 4 Girl 14/01/91 5,00 14,2 2,3 PE/Sport 2,5423 km
Row 5 Boy 07/09/89 6 136 25 Art 1-2km
Row 6 Boy 13/12/91 4 128 24 PE/Sport 1-2km
Row 7 Boy 14/03/01 5 140 67 PE/Sport less than 1 km
Row 8 Girl 06/05/89 7 142 24 Art 3-5km
Row 9 Girl 15/08/90 6 138 21 Art 85km
Row 10 Boy 20/02/90 6 192 23 PE/Sport 1-2km
Row 11 Girl 19/05/90 6 140 20 Maths 1-2km
Row 12 Neither 29/06/92 7 48 21 Going Home 3 000km
Row 13 Boy 09/10/91 4 128 21 English less than 1 km
Row 14 Girl 18/12/90 5 135 21 Geography less than 1 km
Row 15 Girl 18/07/91 0,5 13,7 20 Art 3-5km
Row 16 Boy 03/06/34 4 129 21 Art less than 1 km
Row 17 Girl 13/02/89 7 148 23 Art 1-2km
Row 18 Girl 15/09/88 7 150 22,5 PE/Sport 1-2km
Row 19 Girl 07/08/89 7 140 24 Art less than 1 km
Row 20 Boy 08/06/89 7 142 24 Computing less than 1 km
Row 21 Boy 31/11/87 11 1 520 22 Computing 5-10km
Row 22 Both 16/07/88 8 142 26 Japanese 2-3 km
Row 23 Girl 28/04/88 8 145 26,5 PE/Sport 1 mile
Row 24 Boy 25/03/92 4,1 132,1 2,4,5 Maths less than 1 km
Row 25 Boy 26/02/92 4 130 21 PE/Sport less than 1 km
Row 26 Girl 08/07/99 6 142 22 Art 2-3 km
Row 27 Boy 23/05/90 6 151 25,5 Maths 2-3 km
Row 28 Boy 01/03/87 9 162 25 PE/Sport less than 1 km
Row 29 Girl 07/08/91 6 150 23 History 2 roads
Row 30 Girl 03/03/92 4 135 21 English less than 1 km

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Chapter

6
Relative Frequency and
Probability

In this chapter you will:

x List all the possible outcomes of simple experiments.
x Determine the relative frequency of actual outcomes for a series of trials.
x Determine the probability of outcomes of simple situations using the definition for probability.
x Predict the frequency of possible outcomes based on their probability.
x Compare relative frequency and probability and explain possible differences.
x Determine probabilities for compound events using two-way tables and tree diagrams

P robability is the part of mathematics that studies chance or likelihood.

For example, when you see on the weather report that there is a 60%
chance of rain, they are talking about a probability.

9 Some things will never happen. We say that they are

impossible.
For example: if you throw an ordinary die – it is impossible
that it will land on 8.

9 Other things will definitely happen and we say that they

are certain to happen.
For example: it is certain for an ordinary die to land on 1,
2, 3, 4, 5, or 6.

9 Some things are not certain and not impossible.

For example:
x It is likely to rain when the sky is very cloudy.
x It is unlikely to snow in Johannesburg in December.
x There is a 50-50 chance of a coin landing on heads when it is tossed.

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9 Rather than using words to describe the chance of an event happening, you can
give probability as a number between 0 and 1.
9 This number can be written as a fraction, percentage or decimal.
x If it is impossible for an event to happen, the probability is 0.
x If an event is certain to happen, the probability is 1.
x All other probabilities are greater than 0, but less than 1.

Listing outcomes of an experiment

9 If we do a probability experiment, for example throw a die, toss a coin, spin a
spinner, we can list all the possible outcomes.
9 An event is a particular outcome or group of outcomes.
9 Favourable outcomes are the outcomes which give the event you are interested
in.
9 If the die is fair, then we can say that the outcomes are equally likely.
In other words all numbers have the same chance of being thrown. One
number doesn’t have a greater chance than the others of being thrown.
9 If a die is NOT fair, we say that it is biased.

Example 6.1

) Suppose you throw a fair die.

1) List all the possible outcomes
2) List the favourable outcomes for the event
a) Getting a 4
b) Getting an even number
Solution
1) The possible outcomes are 1, 2, 3, 4, 5 and 6
2) a) The favourable outcome of “getting a 4” is 4
b) The favourable outcomes of “getting an even number” are 2, 4 and 6

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Exercise 6.1


1) An eight-sided die (like the one alongside) is thrown.
a) List all the possible outcomes.
b) List all the favourable outcomes for the following
events:
Event A: Getting a 2
Event B: Getting an odd number
Event C: Getting a number bigger than 4.

2) A coin is randomly taken from this purse.

Inside the purse are a R5 coin, a R2 coin, two R1 coins, and
a 20c coin.
a) List all the possible outcomes.
b) List all the favourable outcomes for the following
events:
Event D: Getting a R1 coin
Event E: Getting a ‘bronze’ coin
Event F: Getting a coin worth more than R4.

3) Randomly select a card from the following five cards.

M A T H S
a) List all the possible outcomes.
b) List all the favourable outcomes for the following events:
Event G: Getting a T
Event H: Getting a vowel
Event J: Getting a consonant

4) Randomly select a card from this “hand” of

cards.
a) List all the possible outcomes.
b) List all the favourable outcomes for the
following events:
Event K: Getting a 7
Event L: Getting a heart
Event M: Getting a picture card
Event N: Getting a diamond.

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5) The table below is from the South African Census@School 2009 Report.
It shows what percentage of learners use particular modes of transport to
school from home.

Walk/Foot Car Train Bus Bicycle Scooter Taxi Other

South Africa (%) 74,8 10,9 0,6 5,7 0,5 0,1 6,8 0,5

a) The outcomes are the modes of transport from home to school.

List all the possible outcomes.
b) Suppose a South African learner is chosen at random. List all the favourable
outcomes for each of the following events:
Event P: Taking a taxi
Event Q: Using public transport
Event R: Using a two-wheeled mode of transport.

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Finding the relative frequency

9 We find the relative frequency of an experiment by performing the experiment
or by collecting information from past records.

9 Relative frequency can be calculated using the following formulae:

Number of times an outcome occurs in an experiment
Relative frequency =
Total number of trials in the experiment
OR
Number of times an outcome occurs in a survey
Relative frequency =
Total number of observations in the survey

)
Example 6.2
In an experiment a drawing pin is dropped
100 times.
It lands with its point up 37 times.
What is the relative frequency of the drawing pin landing point up?
Solution
Number of times an outcome occurs in an experiment
Relative frequency =
Total number of trials in the experiment
Number of times the drawing pin lands point up
=
Number of times the drawing pin is dropped
37
=
100
= 0,37 or 37%

)
Example 6.3
50 motor cars are observed passing the school gate.
14 of these motor cars are red.
What is the relative frequency of a red motor car passing the school
gate?
Solution
Number of times an outcome occurs in a survey
Relative frequency =
Total number of observations in the survey

Number of red motor cars

=
Total number of motor cars
14
=
50
28
=
100
= 0,28 or 28%

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Exercise 6.2

1) In an experiment a gardener planted 40 daffodil bulbs.

36 of these daffodil bulbs produced flowers.
Use these results to find the relative frequency that a
daffodil bulb will produce a flower.

Use the formula

Number of bulbs producing flowers
Relative frequency =
Total number of bulbs planted

2) Thandi keeps a record of her chess games with

Helen.

Out of the first 30 games, Helen wins 21 games.

a) Use the results to work out

(i) the relative frequency that Helen wins
Use the formula:
Number of games won by Helen
Relative frequency =
Number of games played

(ii) the relative frequency that Thandi wins

b) Use the two relative frequencies to predict whether Thandi will win her next
game of chess with Helen.

3) The results of games of chess played by four children at a chess club are shown
in this table.

Player Games won Games drawn Games lost

Alfred 4 2 6
Busi 8 1 7
Fikile 3 0 1
Pieter 9 2 9

a) Calculate the number of games that each player plays.

b) Calculate the relative frequency of a win for each player.
Use the formula:
Number of games won
Relative frequency of a win =
Number of games played
c) If Alfred plays Pieter, who do you think is more likely to win? Explain your
answer.
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4) A counter is taken at random from a bag of

coloured counters.
Its colour is recorded and the counter is then put
back in the bag.
This is repeated 300 times.
The number of yellow counters taken from the bag after every 100 trials is
shown in the table.

Number of trials Number of yellow counters

100 54
200 101
300 135

a) Calculate the relative frequency of getting yellow after

(i) 100 trials
(ii) 200 trials
(iii) 300 trials

Use the formula:

Number of yellow counters
Relative frequency of getting a yellow counter =
Number of trials

b) Suppose there are 10 coloured counters in the bag. How many of these
counters do you think are yellow? Explain how you decided on your answer.

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Calculating probability
9 In the previous section we looked at how you could estimate probabilities by
doing experiments or making observations.

9 When we have a situations where each outcome is equally likely to occur, we

can calculate the probability of a particular outcome occurring.

Probability can be calculated using the following formula:

Number of favourable outcomes in an event
Probability =
Total number of possible outcomes

)
Example 6.4
An ordinary die is rolled. What is the
probability of getting:
a) a 6?
b) an odd number?
c) a 2 or 3?
Solution
The possible outcomes on an ordinary die are: 1; 2; 3; 4; 5; and 6.
The total number of possible outcomes is 6.

a) The favourable outcome is 6.

The number of favourable outcomes is 1.
1
P(6) = у 0,17 у 16,7%
6

b) The favourable outcomes are 1; 3 and 5.

The number of favourable outcomes is 3.
3
P(even) =
6
1
or 0,5 or 50%
2
c) The favourable outcomes are 2 and 3.
The number of favourable outcomes is 2.
2
P(2 or 3) =
6
1
| 0,33 | 33,3%
3

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Exercise 6.3

1) A bag contains a red counter, a blue counter and a green counter.

A counter is taken from the bag at random.
a) What are the possible outcomes?
b) What is the probability of taking:
(i) A red counter?
(ii) A red or green counter?
(iii) A counter that is not blue?
(iv) A yellow counter?

2) A bag contains 3 red sweets and 7 green sweets.

A sweet is taken from the bag at random.
a) What are the possible outcomes?
b) Are you more likely to get a red or a green sweet?
c) What is the probability of taking:
(i) A red sweet?
(ii) A green sweet?

3) This fair spinner is used in a game.

The player spins the arrow.
Lose a Lose 5
turn points a) What are the possible outcomes?
Have an Win 5 b) What is the probability that the player:
extra go points (i) Loses a turn?
Go back (ii) Wins 5 points or loses 5 points?
to start

4) The letters of the word PROBABILITY are written on separate cards of the
same size.
The cards are shuffled and dealt, face down, onto a table.
A card is selected at random.
a) What are the possible outcomes?
b) What is the probability that the card shows:
(i) The letter T?
(ii) The letter B?

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5) A card is randomly taken from a full pack of 52 playing cards with no jokers.

x Diamonds and Hearts are red cards

x Clubs and Spades are black cards
x The Ace is the same as the number 1
x The picture cards are Jack, Queen and King.

What is the probability that the card that is taken is:

a) A red card?
b) A heart?
c) The Ace of hearts?

6) A bag contains 4 red counters, 3 white counters and 3 blue counters.

A counter is taken from the bag at random.
a) How many possible outcomes are there?
b) What is the probability that the counter is:
(i) Red?
(ii) White or blue?
(iii) Red, white or blue?
(iv) Green?

7) There are twelve numbered discs in a box.

43 44 45 46
Matlaba closes his eyes and takes a disc
from the box at random.
47 48 49 50

What is the probability that Matlaba takes

a disc: 51 52 53 54
a) With at least one 4 on it?
b) That has not got a 4 on it?
c) That has 3 or 4 on it?

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8) Pirates are playing Mamelodi Sundowns.

Victor says that Pirates can either win, draw or lose, so

1
the probability of them winning must be .
3

Explain why Victor is wrong.

9) An 8-sided die has the numbers 1 to 8 painted on its

faces. When thrown, it has an equal chance of landing on
any one of its faces.
a) How many possible outcomes are there?
b) What is the probability that it lands on:
(i) 6?
(ii) A number less than 6?
(iii) An odd number?
(iv) A number more than 6?
(v) A prime number?

10) 240 tickets are sold in a school raffle.

What is the probability of winning first prize if I buy:
a) One ticket?
b) Seven tickets?

11) In a game of Bingo, the numbers 1 to 90 are written on

individual discs and placed in a big box.

The numbers are taken out one by one, and the

numbers are called out.

a) How many possible outcomes are there?

b) What is the probability that the first number chosen at random is:
(i) a 69?
(ii) a number that ends in 0?
(iii)a number that is a multiple of 9?

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12) Below is a data table showing the birth months of South African learners. It is
taken from the 2009 Census@School report.

Month Frequency
January 3 568
February 3 592
March 3 922
April 3 490
May 3 888
June 3 981
July 3 597
August 3 876
September 4 476
October 3 502
November 3 253
December 3 809
TOTAL 44 954

a) If a learner is chosen at random, which month are they most likely to be

born in?
b) If a learner is selected at random, what is the relative frequency that this
learner was born in March?

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)
Example 6.5
Makgoshi does the following experiment with a bag containing
2 red counters and 8 blue counters.

EXPERIMENT
x Take a counter from the bag at random.
x Record the colour then put the counter back in the bag.
x Repeat this for 100 trials.
Makgoshi calculated the relative frequency of getting a red counter after every 10
trials and recorded the results in a table.
Number of throws 10 20 30 40 50 60 70 80 90 100
Relative Frequency 0,3 0,15 0,12 0,24 0,2 0,24 0,21 0,2 0,19 0,21

She then plotted her results on the following graph.

The dotted line shows the

calculated probability.
The solid line shows the
relative frequency.

a) What is the probability that Makgoshi takes a red counter?

b) What do you notice about the relative frequency as the number of trials
increases?

Solution
number of red counters 2 1
a) P(red counter) = 0,2
total number of counters 10 5

b) As the number of trials increases, the relative frequency varies, but eventually
gets close to 0,2. In other words, it gets closer to the calculated probability.

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Example 6.5 shows you that we can use relative frequency to

NOTE
estimate the probability of an event happening.
When a question asks
you to estimate the
probability, it is The larger the number of trials or observations, the closer the
actually asking you to relative frequency is to the probability.
calculate the relative
frequency.


Exercise 6.4

1) Repeat Makgoshi’s experiment in the following way:

a) Cut out 10 circles the same size from a piece of paper.
b) Mark two of them with R (for red) and eight of them B (for blue).
c) Take a counter at random. Record the colour and then put the counter back
in the bag.
d) Repeat this for 200 trials.

2) Draw a graph to show the relative frequency after every 10 trials.

3) Are your results similar to Makgoshi’s results?

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Predicting frequencies based on

probability
9 We can use the probability of an outcome occurring to predict or estimate the
frequency of that outcome occurring in an experiment.

9 We can use the formula to estimate the frequency of an outcome occurring:

Predicted frequency of an outcome
= Probability of that outcome occurring × Number of repeats of the experiment

)
Example 6.6

A fair coin is tossed 10 times.

How many heads could you expect to get?

Solution
1
The probability of getting heads =
2
Predicted frequency of heads
= probability of getting a head × number of times the coin is tossed
1
= × 10
2
=5

)
Example 7
A bag contains 20 counters of different colours.
A counter is randomly taken out, its colour noted and replaced.
The probability of getting a blue counter is 0,4.
How many blue counters are in the bag?
Solution
Number of blue counters
= probability of getting a blue counter × number of counters
= 0,4 × 20
=8

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Exercise 6.5

1) There are 50 cars in a school car park. Five of the cars are black.
a) What is the probability that the first car to leave the car park will be black?
b) If the probability that a red car is the first to leave is 0,2, how many red cars
are there in the car park?

2) 500 tickets are sold for a prize draw.

a) Sam buys one ticket. What is the probability that he wins the first prize?
1
b) The probability of Cynthia winning first prize is . How many tickets did
20
she buy?

3) Mr Sithole buys 300 calculators from a supplier which he intends to sell at his
school.
He is warned that the probability of the calculators being faulty is 0,02.
How many of the calculators could he expect to be faulty?

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Probabilities of Compound events occurring

9 So far, in this chapter, we have been looking at probabilities of single events
occurring.
9 In some problems, we will have to find the probability of compound events
occurring. Events are called compound events when two (or more) activities
take place.
Examples of compound events:
x We spin a spinner and then select a card.
x We throw two dice together
x We toss a coin twice.
9 In situations like these, we can find all the possible outcomes using
(i) A list
(ii) A two-way table
(iii) A tree diagram.

)
Example 6.8
A fair coin is tossed twice.
a) Identify all the possible outcomes
b) Find the probability of getting two heads.
Solution
(ii) Use a two-way table.
(i) List the outcomes systematically. 1st throw
1st throw 2nd throw H T
Head (H) Head (H) H HH TH
throw
2nd

Head (H) Tail (T)

Tail (T) Head (H) T HT TT
Tail (T) Tail (T)
(iii) Use a tree diagram
1st throw 2nd throw Outcome
H Æ HH
H
T Æ HT

H Æ TH
T
T Æ TT

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a) All three of these methods show that there are four possible outcomes, namely:
HH, HT, TH and TT.
b) There is only one favourable outcome in the event of getting two heads, so
1
P(H;H) =
4

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)
Example 6.9
Two dice are thrown together.
1) What is the probability of getting a ‘double 6’?
2) What is the probability of getting any ‘double’?
Solution
o We first need to find how many possible outcomes there are.
o We can list them using a two-way table.

First die
1 2 3 4 5 6
1 1;1 2;1 3;1 4;1 5;1 6;1
2 1;2 2;2 3;2 4;2 5;2 6;2
Second die

3 1;3 2;3 3;3 4;3 5;3 6;3

4 1;4 2;4 3;4 4;4 5;4 6;4
5 1;5 2;5 3;5 4;5 5;5 6;5
6 1;6 2;6 3;6 4;6 5;6 6;6

There are 36 possible outcomes.

1. There is only one favourable outcome for the event “getting double 6”, namely
(6;6)
1
So P(6;6) =
36

2. There are six favourable outcomes for the event “getting any double”, namely
(1;1), (2;2), (3;3), (4;4), (5;5) and (6;6)
6 1
So P(any double) =
36 6

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Exercise 6.6

1) A fair coin is tossed and an ordinary die is rolled.

a) Copy and complete the two-way table to list all the possible outcomes.
Die
1 2 3 4 5 6
H
Coin
T
b) Use the table to calculate the probability of getting:
(i) A head and 5?
(ii) A tail and 6?
(iii) A tail and an even number?
(iv) A tail and an odd number?
(v) A head and a number greater than 3?
(vi) An odd number?

2) Two ordinary dice are rolled and the numbers obtained

are added.
a) Draw a two-way table to show all the possible
outcomes.
b) Use your two-way table to work out:
(i) The probability of obtaining a total of 10.
(ii) The probability of obtaining a total greater
than 10.
(iii) The probability of obtaining a total less than 10.
c) Explain why the probabilities you worked out in (b)
should add up to 1.

3) Bag A contains 2 red balls and

1 white ball. Bag A Bag B Outcome
Bag B contains 2 white balls and W Æ R; W
1 red ball. R W Æ R; W
A ball is drawn at random from each R Æ R; R
bag.
a) Copy and complete the tree
diagram to show all the possible
pairs of colours.
b) Calculate the probability that the
two balls are the same colour.

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4) Tobias has to travel to school in two stages,

Stage 1: He can get a lift with his neighbour, or he can take a bus, or he can
take a train.
Stage 2: He can either take a taxi, or he could walk.
a) Use a tree diagram to help you list all the ways that Tobias could travel to
school.
b) Tobias selects one of the ways at random. What is the probability that he
takes a bus and then a taxi?

5) In a game, the fair spinner shown alongside is

spun twice.
The two amounts that the arrow lands on are
added to calculate the winnings.
a) Draw up a two-way table and use it to show
the amounts that could be won.
b) Calculate the probability of winning:
(i) R20
(ii) R100
(iii) R60
(iv) R15

6) A spinner has an equal probability of landing on red, green, blue, yellow or

white.
The spinner is spun twice.
a) Use a two-way table or a tree diagram to list all the possible outcomes.
b) What is the probability that, on both spins,:
(i) The spinner lands on white?
(ii) The spinner lands on white at least once?
(iii) The spinner lands on the same colour?

7) Each day a blue car (B), a yellow car (Y) and a red car (R) are parked one behind
the other in a narrow driveway. The order in which they park in the driveway is
not fixed and varies from day to day.

Blue Car Yellow Car Red Car

a) List all the possible orders in which the three cars could be parked.
b) What is the probability that, on any particular day, the red car is the first in
the driveway?

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Chapter

7
Answers
CHAPTER 1 – COLLECTING DATA
Note
This chapter is aimed at getting you to think about doing research using data and the many things
you need to think about when you do such research.

Exercise 1.1 (page 7)

The solutions here are illustrative and are given for the example question, “What is the spread of ages of
learners in your school”.
1) The problem is “What is the spread of ages of learners in your school?”
2) Plan:
a) I will draw up a question sheet where I will be able to record the different ages of the
learners at my school.
b) I will collect information about the ages of the learners at my school.
c) I will have responses on questionnaires which I will tally.
3) Data recording
a) I will record my data in tables which I can draw up by hand. If I have access to a computer I
would use an Excel spread sheet to record my data.
b) I would present my data in a table (giving the total numbers of learners with different ages)
and then I would also draw a bar graph to represent my information.
4) Analysis
a) I would look for things like the most common age, the highest age and the lowest age in the
sample.
b) I would think about reasons for why the data looks as it does.
c) If I noticed any relationships between the variables I would think about them a bit more.
Here I might notice that certain ages are more common in certain grades.
5) Conclusion
a) I would write about what I notice about the most common age, the highest age and the
lowest age in the sample.
b) I would write about the reasons for why I think it looks like this.
c) I would explain the relationships between some of the variables that I investigated.

Exercise 1.2 (page 10)

The problem you want to investigate is: What is the favourite tuck shop food for learners in your class?
1) Data about favourite tuck shop food for learners in my class
2) I would draw up a questionnaire which I could use to find out about favourite tuck shop food for
learners in my class. I would make some lists of food that I know are sold at the tuck shop so that they
can tick off their favourites rather than have to write them down in full.
3) I would tally the ticks made by the learners in my class. I would then have raw data. I could make a
table in which I could give the totals of all the tallies.

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4) What form of data collection could you use?

a) I could a census because the population here is my class and I can manage to ask all of them
about their favourite food.
b) No I could not have used a sample survey because if I took a sample from my class, it would be
too small. Data needs to be collected from a big enough sample if you use a sample.
c) No I could NOT have used an experiment because I was not trying to understand cause and effect
relationships.
d) No I could NOT have used an observational study because, again, I was not trying to understand
cause-and-effect relationships.

Exercise 1.3 (page 13)

1) A sample is a smaller group, taken from a population which is used to represent the population.
2) A population is the whole group of people (or things) that you want to speak about.
3) The statement of Thembi’s research problem is: What percentage of learners play soccer at
break time at school?
4) Thembi asked 7 of his friends whether or not they liked to play soccer at break. This sample is
not representative of the whole South African population.

Exercise 1.4 (page 15)

1) Thembi’s school.
2) No.
3) (Thembi wonders whether 57% of South African learners play soccer at break time.) Habib
thinks that Thembi’s sample is not representative. I agree with him. You can’t use 7 learners to
represent a whole school population. You can’t use them to talk about your whole country.
4) Thembi can improve his survey by going to the SA Census at Schools website and finding the
information about learners that they have. If they have answered a similar question to the one
he has thought about, this could be very useful because he could never carry out a census
himself.
5) Answers would vary.
a) A problem that you investigate in your class or church youth group.
b) A problem that you investigate in your school or neighbourhood.
c) A problem when you want to generalise the result for the whole population.
d) A problem when you will not be making generalisations about what you find. You will keep the
results to speak about to the sample that you dealt with only.

Exercise 1.5 (page 16)

1) 217 learners
2) No. She just asked every learner that she saw over a certain period of time.
3) No. Her sample would not have been representative of the school. It was too small and it probably did
not have a representative spread of the learners at the school.
4) Athletics, soccer, netball, softball and cricket.
5) Soccer, netball, athletics, no favourite sport and rugby.
6) Athletics in number one in Jane’s survey and number 3 in the C@S survey. They are both high on the
top 10 list, but not the same.
7) No. her sample is too small and not representative.
8) Yes. It was randomly selected and it is very large (24 172 learners)

Exercise 1.6 (page 18)

Answers will vary
1) A particular question – What are the favourite hobbies of the boys in my Grade 8 class?
2) A comparative question – What is the difference between the favourite hobbies of boys and girls in my
Grade 8 class?
3) A relationship question – Do learners with a lot of extra-mural interests perform better at school?

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Exercise 1.7 (Page 19)

Answers will vary
b) What are the favourite hobbies of the boys in my Grade 8 class? – I will ask all of the boys in my Grade
8 class.
c) What is the difference between the favourite hobbies of boys and girls in my Grade 8 class? – I will ask
all of the boys and girls in my Grade 8 class.
d) Do learners with a lot of extra-mural interests perform better at school? – I will have to ask a
representative sample of the learners at my school about their extra mural interests, and find out
about how they achieve at school from teachers’ records of marks. Then I will be able to see if there is
a relationship between academic performance and number of extramural activities.

Exercise 1.8 (page 21)

Answers will vary. The model answers given here apply to the question: “What types of transport do the
learners in my school use to go to school?”

Questionnaire – types of transport used to travel to school

Circle the correct answer to each of the questions below
1) Do you walk to school?
a) Yes b) No
2) Do you drive in a car to school?
a) Yes b) No
3) Do you drive in a taxi to school?
a) Yes b) No
4) Do you drive in a bus to school?
a) Yes b) No
5) Do you ride a bicycle to school?
a) Yes b) No
6) Do you take a train to school?
a) Yes b) No
7) If you go to school in a car, who drives the car?
a) Your mom/dad b) Your guardian c) One of your siblings d) A relative/friend e) I do not
drive to school
8) Which would be your favourite way to travel to school
a) Car b) Taxi or bus c) Bicycle or by foot d) Train e) No favourite
9) How do you choose your favourite method of transport?
a) It’s the quickest
b) It’s the safest
c) I want to keep fit
d) It’s the easiest to get to from my house
e) I don’t have a favourite
10) Using your form of transport how long do you take to get to school?
a) 0 – 15 minutes
b) 16 – 30 minutes
c) 31 – 45 minutes
d) 46 – 60 minutes
e) More than 60 minutes

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CHAPTER 2 – ORGANISING AND SUMMARISING DATA

Exercise 2.1 (page 22)
1) 136; 138; 139;
140; 140; 140; 146; 147; 148; 148; 148; 148; 148; 148; 148; 148
150; 150; 150; 152; 152; 154; 155; 155; 155; 155; 155; 155; 156; 156; 156; 157; 158; 158; 158;
158; 158; 158
160; 160; 160; 160; 160; 160; 160; 160; 160; 160; 160; 160; 161; 162; 162; 162; 163; 163; 163;
164; 164; 165; 165; 165; 165; 165; 165; 165; 165; 165; 167; 167; 167; 167
170; 170; 170; 170; 170; 170; 170; 170; 171; 172; 175; 175; 177; 178; 178; 179; 179
180
2) The most common height is 160 cm
3) 16 learners are shorter than 150 cm
4) 10 learners are taller than 170 cm

Exercise 2.2 (page 25)

Answers to 1), 2) and 3) are going to vary from class to class.

Exercise 2.3 (page 27)

1)
STEM LEAVES
13 6; 8; 9
14 0; 0; 0; 6; 7; 8; 8; 8; 8; 8; 8; 8; 8
15 0; 0; 0; 2; 2; 4; 5; 5; 5; 5; 5; 5; 6; 6; 6; 7; 8; 8; 8; 8; 8; 8
16 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 1; 2; 2; 2; 3; 3; 3; 4; 4; 5; 5; 5; 5; 5; 5; 5; 5; 5; 7; 7; 7; 7
17 0; 0; 0; 0; 0; 0; 0; 0; 1; 2; 5; 5; 7; 8; 8; 9; 9
18 0
KEY: 13/6 = 136 cm

2) Some conclusions that can be reached are:

x Fifteen of the boys are shorter than 150 cm
x Eighteen of the boys are 170 cm or taller
x The shortest boy is 136 cm and the tallest boy is 180 cm
x Thirty-four of the boys have heights that range from 160 cm to 167 cm
x More than half of the boys (56 of the boys) have heights that range from 150 cm to 167 cm.

Exercise 2.4 (page 30)

1) a)
Mathematics Marks Frequency
30 < m ч 40 6
40 < m ч 50 8
50 < m ч 60 11
60 < m ч 70 4
70 < m ч 80 3
TOTAL 32

b) 6 + 8 = 14 learners got 50 or less for the test.

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2) a)
Heights in metres Frequency
1,50 < x ч 1,55 1
1,55 < x ч 1,60 4
1,60 < x ч 1,65 5
1,65 < x ч 1,70 6
1,70 < x ч 1,75 6
1,75 < x ч 1,80 3
1,80 < x ч 1,85 4
1,85 < x ч 1,90 1
TOTAL 30
b) 3 + 4 + 1 = 8 learners are taller than 1,75 m

3) a)
Stem Leaves
0 2; 3; 3; 5; 6; 6; 6; 7; 8; 8; 9; 9
1 2; 2; 2; 2; 3; 4; 5; 5; 8; 8
2 0; 0; 0; 0; 2; 4; 5
3 0
KEY: 1/2 = 12 hours

b) Some possible conclusions are

x The learners in the class watched between 2 and 30 hours TV in a week
x More than half of the learners (12 + 10 = 22 learners) watch less than 20 hours of TV in
a week
x Four learners watch 20 hours of TV in a week
x Four learners watch 12 hours of TV in a week

4) a)
Number of learners on the bus Number of buses
0 < l ч 10 12
10 < l ч 20 6
20 < l ч 30 9
30 < l ч 40 6
40 < l ч 50 4
50 < l ч 60 6
TOTAL

b) The bus company might want to know the numbers of learners on the bus to know whether
one bus was enough or if maybe more buses were necessary

2 235  2 449  33 4 717

5) a) Fraction having blue eyes =
70 180  76 681  1 144 148 005
b) Number with brown eyes = 60 531 + 66 993 + 789 = 128 313 learners
c) The majority of South African learners are black with brown eyes.

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CHAPTER 3 – MODE, MEDIAN, MEDIAN

Exercise 3.1 (page 33)
1) a) 9, 9, 11, 15, 15, 15, 15, 17, 18, 19
b) Two (2) learners are 9 years old.
c) 19 years
d) Three (3) learners are older than 15.
e) Four (4) learners are 15 years old.
f) The age of 15 years occurs most often.
g) The mode is the age that occurs most often, therefore 15 years is the mode age.

2) a) 121, 138, 140, 142, 160, 161, 165, 170, 170, 182
b) 121 cm is the smallest height.
c) Five (5) learners are taller than 160 cm.
d) Two (2) learners are 170 cm tall.
e) 170 cm is the mode of the heights.

3) a) 2; 3; 4; 5; 6; 7; 7; 8; 10; 14
b) There are ten (10) learners in the data set.
c) The fewest number of people in a home is two (2).
d) The most number of people in a home is fourteen (14).
e) Two (2) homes have 6 people.
f) Two (2) homes have 7 people.
g) There are two (2) modes in this data set: 6 and 7 people.
h) The first seven (7) values are: 10, 14, 6, 8, 3, 2, 6.
Then you rank the values: 2, 3, 6, 6, 8, 10, 14.
It is easy to see that six (6) occurs most often. The mode is 6 people.
i) The first five (5) values are: 10, 14, 6, 8, 3
Then you rank the values: 3, 6, 8, 10, 14.
Each value only occurs once, so there is NO mode in this data set.

4) a) Three (3) learners played cricket

b) Two (2) learners played volleyball
c) The category “No favourite sport” was selected by the most learners
d) Tennis and volleyball were chosen by the fewest learners.
e) “No favourite sport” is the mode. In other words, the most learners indicated this category.
f) 3 + 3 + 6 + 6 + 2 + 2 + 11 = 34. Thirty four learners took part in the survey.
g) If the category, “No favourite sport” is ignored, the modal sport is Soccer.
h) Seven (7) learners chose soccer as their favourite sport.

5) a) Walking is the mode.

b) 10% × 174 learners = 17,4 learners.
You cannot have a 17,4 learners, therefore you round 17,4 down to 17 learners travelling to
school by bus.
c) 4% × 174 learners = 6,96 learners.
You cannot have 6,96 learners, therefore you round 6,96 up to 7 learners travelling to
school by taxi.

Exercise 3.2 (page 37)

1) a) The sum of the ages of the learners ins 72
72
b) The mean of the ages is: 14, 4 years.
5
2) a) There are ten (10) ages of learners in this data set
b) The sum of the ages is 143 years

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143
c) The mean of the ages = 14,3 years
10
9  15  9  15  17 65
d) The mean of the first 5 ages = 13 years
5 5
3) a) The mean height of all the learners
138  161  121  170  170  165  142  160  140 1 367
| 151,89 cm.
9 9
b) The mean height of the first 6 learners
138  161  121  170  170  165 925
= | 154,17 cm
6 6

Exercise 3.3 (page 39)

1) a) There are 5 learners’ ages in this data set
b) Odd number
c) 9, 11, 15, 18, 19
d) 15 years
2) a) Rank the ages from youngest to oldest: 9, 9, 11, 15, 15, 15, 15, 17, 18, 19
Count and see that there are 10 ages, an even number.
Because there is not only one age right in the middle of the data set, you add the two
middle ages, and divide by 2 to calculate the median.
15 +15 30
b) Median = = 15 years
2 2
3) a) 1,13 1,28 1,5 1,53 1,58 1,6 1,65 1,68 1,75 1,77 1,81
b) 11 heights
c) Median = 1,65 m
4) a) 0,50; 0,50; 0,58; 0,58; 0,67; 0,75; 0,83; 1,00; 1,33
b) 9
c) (i) 0,58 h
(ii) 0,83 h
(iii) 1,00 h
d) Learners in positions 1 and 2 as 0,50 h = 30 minutes
e) 6th position
f) The median is in the 5th position
g) 0,67 h
5) a) Median is in the 61st position
b) You have to add the number in the 251st position and the number in the 252nd position
together and divide the answer by 2.
6) If the total number of data values is an even number, the median will be the sum of the values in
the 20th and 21st positions, divided by 2. Therefore, this statement is not valid.

Exercise 3.4 (page 42)

1) a) 9, 11, 15, 18, 19
b) 19 years
c) 9 years
d) 19 – 9 = 10
e) Range = 10 years

2) First arrange the data values from small to large. 9, 9, 11, 15, 15, 15, 15, 17, 18
a) 18 years
b) 9 years
c) Range = 18 – 9 = 9 years

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3) Arrange the data values from small to large: 121, 138, 140, 142, 160, 161, 165, 170, 170, 182
a) 182 cm
b) 121 cm
c) Range = 182 – 121 = 61 cm

4) First order the number of minutes from short to long: 5, 5, 10, 10, 11, 20, 24, 45
a) 5 minutes
b) 45 minutes
c) Range = 45 – 5 = 40 minutes

Exercise 3.5 (page 44)

1) a) 14 values
b) 4, 5, 5, 13, 15, 15, 15, 15, 18, 20, 25, 45, 50, 90
c) Most learners take from 4 minutes to 50 minutes to travel to school.
90 minutes (or 1 ½ hours) is an unexpectedly long time to travel to school. 90 minutes is
therefore an extreme value.

2) a) Household 12 had 12 school-going boys

b) Household 15 had 4 school-going girls
c) (i) Household 9
(ii) 0 children
(iii) Add the number of school-going boys and girls together.
(iv) This answer is not possible, because the learner who answered the question is still
school-going, therefore the answer should be at least 1.
d) (i) Household 12
(ii) 13 children
(iii) Yes, 13 children in a household are possible.
e) 12 boys are extreme, because all the other households only have from 0 to 3 school-going
boys.
f) There are no extreme (unexpected) values among the girls, because the number of girls
only varies from 0 to 4.

3) a) 109; 140; 145; 146; 150; 151; 155; 159; 160; 162; 165; 168; 169; 170; 170; 171; 173; 176;
178; 181
b) 20 heights
c) Modal height = 170 cm
d) There are 20 values in the data set. This is an even number of values. The middle two
numbers of this data set are 162 cm and 165 cm.
162  165 327
The median = 163,5 cm .
2 2
e) Remember, you do not have to rank the data to calculate the mean.
3 198
Mean = 159,9 cm
20
f) Range = 181 – 109 = 72 cm
g) 109 cm is an extreme value. It is a lot less than the rest of the heights.
3 198  109 3 089
h) New mean = = 162,6 cm.
19 19
The new mean is larger than the old mean, because the extreme value which was an
unexpected low value, was deleted.
i) The mode is still 170 cm, but the median is now the 10th value and is now 162 cm.

4) In a data set with no mode, all values occur equally often, or there is not one value that occurs
more often than the rest. In a data set where the number 0 occurs most often, 0 will be the
mode of the data set. For example: in the data set: 0; 1; 2; 3; 4; 5; 6; 7 there is no mode, but in
the data set 0; 2; 4; 0; 6; 8; 0, the mode = 0.
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CHAPTER 4: REPRESENTING DATA

Exercise 4.1 (page 46)
1) Note: Answers will vary, depending on what key you choose. You could choose each picture to
represent 2, 4 or 8 people. In the solution below, each picture represents 8 people.

2) Note: Answers will vary, depending on what key you choose

In the solution below, each picture represents 2 people: = 2 people

Exercise 4.2 (page 54)

1) 2)

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Exercise 4.3 (page 57)

1)

Exercise 4.4 (page 59)

1)

2) Answers will vary, but may include:

x The languages are favourite subjects for both boys and girls.
x Natural Science and Technology overall are the least favourites across both genders
x Natural Science is rated equally by girls and boys in the survey.

Exercise 4.5 (page 62)

1)
Method of travel Number of learners Angle
Walk 14 14 × 12° = 168°
Bus 7 7 × 12° = 84°
Car 6 6 × 12° = 72°
Bike 3 3 × 12° = 36°
Helicopter 0 0 × 12° = 0°

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1) continued

2)
Month No of people Angle Month No of people Angle
January 7 28q July 6 24q
February 4 16q August 8 32q
March 9 36q September 11 44q
April 8 32q October 7 28q
May 7 28q November 10 40q
June 6 24q December 7 28q

NO OF BIRTHDAYS PER MONTH

3) a)
NUMBER OF
FAVOURITE SUBJECT
LEARNERS
History 30
Geography 40
Maths 50
TOTAL 120

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b) Answers will vary. The pie chart show both the difference between the groups and how
each group compares to the whole, so this seems to be a good representation of the data.

EXERCISE 4.6 (page 65)

1)

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2)

Exercise 4.7 (page 68)

1) Except for the people who have measurements (22; 153) and (27; 174), the taller people
generally have wider hand spans.

2) Except for the people who have measurements (20; 120), and (23; 126), the taller people
generally have longer foot lengths.

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CHAPTER 5 – INTERPRETING DATA

Exercise 5.1 (page 72)
1) a) The least learners were born in November
b) The most learners were born in September
c) If numbers were used, the scale would take up too much space
d) The graph may be easier to read, but the table gives the best representation. This is
because we can only read approximate percentages off the graph.
2) The heading is missing so we don’t know what the graph is about
3) They look different because the values on the vertical axes are different.
Graph A: values go from 0 to 1 000 in steps of 200; Graph B: values go from 0 to 2 400 in steps
of 400

Exercise 5.2 (page 75)

1) a) The graph compares the facilities and services at schools from the 2001 Census@School
survey and the 2009 Census@School survey.
b) The first column represents results from the 2001 Census@School survey and the second
column represents the results from the 2009 Census@School survey.
c) Sentences you could write include:
x The percentage of facilities and services at schools has improved in all cases from 2001
to 2009
x The highest improvement has been in the percentages of computers at school
x Only a small percentage of the schools surveyed have access to email and/or internet
x Approximately 70% of the schools surveyed in 2009 had access to a maths teacher.

2) a) Grades 3 to 7: Maths;
Grades 8 to 12: Languages.
b) Maths; Languages; Literacy; Arts & Culture; Technology; Numeracy; Life Orientation
c) Life Orientation

3) a) Some of the sentences you could write include:

x Netball is the girls’ favourite sport for females, whereas the boys’ favourite sport is
soccer
x No girls prefer rugby as their favourite sport, whereas approximately 18% of the boys
prefer rugby as their favourite sport
x Tennis is more popular amongst girls than amongst boys
x There are many more girls who do not have a favourite sport as compared to the boys
b) Yes they have. We know this because the lengths of most of the bars touching each other
are different.

4) Some possible sentences you could write include:

x The largest percentage of boys and girls would prefer to be happy
x The smallest percentage of boys and girls would prefer to be famous
x The percentage of learners who would prefer to be famous is the same for boys as for girls
x There is a higher percentage of boys that would prefer to be rich than girls
x There is a higher percentage of girls that would prefer to be happy than the percentage of
boys.

Exercise 5.3 (page 79)

1) a) The graphs show the percentage of food types eaten by males and females
b) Carbohydrates
c) Fruit and vegetables
d) Most: carbohydrates; least: Protein
e) Females eat 2% more dairy than males

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2) a) The bar graph gives a better visual representation of the data, but the table gives more detail
b) The most learners were in Grade 9
c) There were more girls. We can see this from the table.
d) You will find out approximately how many learners participated in the census by adding up
all the bars. However, the scale makes it impossible to find the exact number represented
by each bar.
You could not use the pie graph to say how many learners participated in the survey. The
pie graph only shows percentages of the whole census.
e) The bar graph shows the percentage of boy and girl learners in each grade as opposed to
the number of learners in each grade shown in the first bar graph.
f) In Grades 3 and 4 there are more boys than girls.
In the rest of the grades there are more girls than boys.

Exercise 5.4 (page 84)

1) They want you to believe that the temperatures in August 2010 were about the same as the
temperatures in October 2010. In reality, the temperatures in August were about 9q lower than
those in October.

2) a) The scale on the vertical axis doesn’t start at 0

a) One possible answer: It looks like there were no visitors to the park on Mondays and
Fridays

Exercise 5.5 (page 86)

1) a) Graph A: 20 cm; Graph B: 60 cm
b) Graph A shows clearly that the heights of most of the learners lie between 160 cm and 200
cm. This is not shown in Graph B

2) a) (i) The oldest boys who participated in the survey were 18 or 19 years old
(ii) The youngest boys were 11 or 12 years old
(iii) The oldest girls were 16 or 17 years old
(iv) The youngest girls were 11 or 12 years old
(v) The mode for the boys is 15 to 16 years
(vi) The mode for the girls is 15 to 16 years
b) Some of the sentences you could write are:
x The boys are older than the girls
x More girls who were 14 or 15 responded than boys of the same age.

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Exercise 5.6 (page 89)

The shaded cells are the one where there are errors.
Dirty Spreadsheet
Boy or Date of Height Foot Length Favourite Distance to School
Row 1 Grade
Girl Birth in cm in cm Subject in km
Row 2 Boy 12/04/91 5 143 26 Art 1-2km
Row 3 Girl 31/02/92 4 132 22 Science less than 2 km
Row 4 Girl 14/01/91 5,00 14,2 2,3 PE/Sport 2,5423 km
Row 5 Boy 07/09/89 6 136 25 Art 1-2km
Row 6 Boy 13/12/91 4 128 24 PE/Sport 1-2km
Row 7 Boy 14/03/01 5 140 67 PE/Sport less than 1 km
Row 8 Girl 06/05/89 7 142 24 Art 3-5km
Row 9 Girl 15/08/90 6 138 21 Art 85km
Row 10 Boy 20/02/90 6 192 23 PE/Sport 1-2km
Row 11 Girl 19/05/90 6 140 20 Maths 1-2km
Row 12 Neither 29/06/92 7 48 21 Going Home 3 000km
Row 13 Boy 09/10/91 4 128 21 English less than 1 km
Row 14 Girl 18/12/90 5 135 21 Geography less than 1 km
Row 15 Girl 18/07/91 0,5 13,7 20 Art 3-5km
Row 16 Boy 03/06/34 4 129 21 Art less than 1 km
Row 17 Girl 13/02/89 7 148 23 Art 1-2km
Row 18 Girl 15/09/88 7 150 22,5 PE/Sport 1-2km
Row 19 Girl 07/08/89 7 140 24 Art less than 1 km
Row 20 Boy 08/06/89 7 142 24 Computing less than 1 km
Row 21 Boy 31/11/87 11 1 520 22 Computing 5-10km
Row 22 Both 16/07/88 8 142 26 Japanese 2-3 km
Row 23 Girl 28/04/88 8 145 26,5 PE/Sport 1 mile
Row 24 Boy 25/03/92 4,1 132,1 2,4,5 Maths less than 1 km
Row 25 Boy 26/02/92 4 130 21 PE/Sport less than 1 km
Row 26 Girl 08/07/99 6 142 22 Art 2-3 km
Row 27 Boy 23/05/90 6 151 25,5 Maths 2-3 km
Row 28 Boy 01/03/87 9 162 25 PE/Sport less than 1 km
Row 29 Girl 07/08/91 6 150 23 History 2 roads
Row 30 Girl 03/03/92 4 135 21 English less than 1 km

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CHAPTER 6 – RELATIVE FREQUENCY AND PROBABILITY

Exercise 6.1 (page 93)
1) a) Possible outcomes: 1; 2; 3; 4; 5; 6; 7; 8
c) Event A: 2; Event B: 1; 3; 5; 7; Event C: 5; 6; 7; 8

2) a) Possible outcomes: R5; R2; R1; R1; 20c

b) Event D: R1; R1; Event E: 20c; Event F: R5

3) a) Possible outcomes: M; A; T; H; S
b) The vowels are a, e, i, o and u. All other letters of the alphabet are called consonants,
Event G: T; Event H: A; Event J: M; T; H; S

4) a) Possible outcomes: JƆ; 7ƅ; Kƅ; 7Ɔ; QƄ

b) Event K: 7ƅ; 7Ɔ; Event L: JƆ; 7Ɔ; Event M: JƆ; Kƅ; QƄ; Event N: None

5) a) Possible outcomes: Walk/Foot; Car; Train; Bus; Bicycle; Scooter; Taxi; Other
b) Event P: Taxi; Event Q: Train; Bus; Taxi; Event R: Bicycle; Scooter

Exercise 6.2 (page 96)

You only have to give one answer, unless your teacher asks specifically for your answer to be a
decimal fraction or a percentage. If you leave your answer as a common fraction – make sure it is in
its simplest form. All three answers are given here.
Number of times a bulb flowers 36 9
1) Relative frequency of flowers= = 0,9 = 90%
Total number of bulbs planted 40 10

2) We could calculate the relative frequency after 10 games, or after 30 games. It is better to
calculate the relative frequency after 30 games, as the more times an experiment is repeated,
the more accurate our prediction will be.
a) (i) Relative frequency of Helen’s wins
Number of games won by Helen 21 7
= = 0,7 = 70%
Number of games played 30 10
(ii) Relative frequency of Thandi’s wins
Number of games won by Thandi 9 3
= = 0,3 = 30%
Number of games played 30 10
b) It is unlikely that Thandi will win her next match against Helen as 0,3 is much lower than 0,7.
We cannot however, say for certain that Thandi will not win.

3) a) Alfred plays 4 + 2 + 6 = 12games

Busi plays 8 + 1 + 7 = 16 games
Fikile plays 3 + 0 + 1 = 4 games
Pieter plays 9 + 2 + 9 = 20 games.
b) Relative frequency that Alfred wins
Number of times Alfred has won
=
Number of games Alfred has played
4 1
у 0,333 (correct to three decimal places) у 33% (correct to nearest %)
12 3
Relative frequency that Busi wins
Number of times Busi has won 8 1
= = 0,5 = 50%
Number of games Busi has played 16 2
Relative frequency that Fikile wins

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Number of times Fikile has won 3

= = 0,75 = 75%
Number of games Fikile has played 4
Relative frequency that Pieter wins
Number of times Pieter has won 9
= = 0,45 = 45%
Number of games Pieter has played 20
c) Pieter is more likely to win than Alfred because the relative frequency of Pieter winning
(0,45) is higher than the relative frequency of Alfred winning (0,333)
4) a) (i) Relative frequency of getting a yellow counter after 100 trials
Number of yellow counters 54
= = 0,54 = 54%
Number of trials 100
(ii) Relative frequency of getting a yellow counter after 200 trials
Number of yellow counters 101
= = 0,505 = 50,5%
Number of trials 200
(iii) Relative frequency of getting a yellow counter after 300 trials
Number of yellow counters 135 9
= = 0,45 = 45%
Number of trials 300 20

b) The most accurate of these relative frequencies is 0,45 because we used the most trials to
calculate this relative frequency.
The number of yellow counters in a bag is 0,45 × 10 = 4,5
As there are no fractions of counters, we can predict that there are 4 or 5 yellow counters in
the bag.

Exercise 6.3 (page 99)

1) a) Possible outcomes: Red; Blue; Green OR R; B; G
b) (i) The number of possible outcomes is: 3; The favourable outcome is: R;
There is 1 favourable outcomes
Number of favourable outcomes in an event
P(R) =
Total number of possible outcomes
1
у 0,333 (correct to three decimal places) у 33% (correct to nearest %)
3
(ii) The number of possible outcomes is: 3; The favourable outcomes are: R; G;
There are 2 favourable outcomes
Number of favourable outcomes in an event
P(RorG) =
Total number of possible outcomes
2
у 0,667 (correct to three decimal places) у 67% (correct to nearest %)
3
(iii) The number of possible outcomes is: 3; The favourable outcome is: R; G;
There are 2 favourable outcomes
Number of favourable outcomes in an event
P(notB) =
Total number of possible outcomes
2
у 0,667 (correct to three decimal places) у 67% (correct to nearest %)
3
(iv) The number of possible outcomes is: 3; There is NO favourable outcomes;
There are 0 favourable outcomes
Number of favourable outcomes in an event 0
P(Y) = = 0 = 0%
Total number of possible outcomes 3

2) a) Possible outcomes: R; R; R; G; G; G; G; G; G; G
b) You are more likely to get a green sweet because there are more of them.

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c) (i) The number of possible outcomes is: 10; The favourable outcome is: R: R; R;
There are 3 favourable outcomes
Number of favourable outcomes in an event 3
P(R) = = 0,3 = 30%
Total number of possible outcomes 10
(ii) The number of possible outcomes is: 10; The favourable outcome is: G: G; G; G; G; G; G
There are 7 favourable outcomes
Number of favourable outcomes in an event 7
P(G) = = 0,7 = 70%
Total number of possible outcomes 10
3) a) Possible outcomes:
Lose a turn; Lose 5 points; Win 5 points; Go back to start; Have an extra go
b) (i) The number of possible outcomes is: 5; The favourable outcome is: Loses a turn
There is 1 favourable outcome
Number of favourable outcomes in an event 1
P(Loses a turn) = = 0,2 = 20%
Total number of possible outcomes 5
(ii) The number of possible outcomes is: 5;
The favourable outcomes are: Lose 5 points; Win 5 points
There are 2 favourable outcomes
Number of favourable outcomes in an event 2
P(Lose or Win 5 pts) = = 0,4 = 40%
Total number of possible outcomes 5
4) a) Possible outcomes: P; R; O; B; A; B; I; L; I; T; Y
b) (i) The number of possible outcomes is: 11; The favourable outcome is: T
There is 1 favourable outcome
1
P(T) = = 0,091 (correct to three decimal places) = 9% (correct to nearest %)
11
(ii) The number of possible outcomes is: 11; The favourable outcomes are: B; B
There are 2 favourable outcomes
2
P(B) = = 0,182 (correct to three decimal places) = 18% (correct to nearest %)
11
5) a) Number of possible outcomes: 52; Number of favourable outcomes: 26
26 1
P(red card) = = 0,5 = 50%
52 2
b) Number of possible outcomes: 52; Number of favourable outcomes: 13
13 1
P(heart) = = 0,25 = 25%
52 4
c) Number of possible outcomes: 52; Number of favourable outcomes: 1
1
P(Ace of hearts) =
52
= 0,019 (correct to three decimal places) = 2% (correct to nearest %)
6) a) There are 10 possible outcomes.
4 2
b) (i) P(red) = = 0,4 = 40%
10 5
6
(ii) P(white or blue) = = 0,6 = 60%
10
10
(iii) P(red or white or blue) = = 1 = 100%
10
0
(iv) P(green) = = 0 = 0%
10
7) a) There are 12 possible outcomes; Favourable outcomes: 43; 44; 45; 46; 47; 48; 49; 54
There are 8 favourable outcomes: 8
P(at least one 4)
8 2
у 0,667 (correct to three decimal places) у 67% (correct to nearest %)
12 3

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b) There are 12 possible outcomes; Favourable outcomes: 50; 51; 52; 53

There are 4 favourable outcomes
4 1
P(no 4) у 0,333 (correct to three decimal places) у 33,3% (correct to nearest %)
12 3
c) There are 12 possible outcomes; Favourable outcomes: 43; 44; 45; 46; 47; 48; 49; 53; 54
There are 9 favourable outcomes
9
P(3 or 4) = = 0,75 = 75%
12
8) The outcomes of winning, drawing or losing are not equally likely.
We can only calculate probabilities using the Probability formula when the outcomes are equally
likely. In this case there are other factors which would influence the chances of Pirates winning.

9) a) There are 8 possible outcomes.

1
b) (i) P(6) = = 0,125 = 12,5%
8
5
(ii) P(number less than 6) = = 0,625 = 62,5%
8
4 1
(iii) P(odd number) = = 0,5 = 50%
8 2
2 1
(iv) P(number more than 6) = = 0,25 = 25%
8 4
4 1
(v) P(prime number) = = 0,5 = 50%
8 2
10) a) Number of possible outcomes: 240; There is 1 favourable outcome
1
P(winning) = 0,00417 (correct to 5 decimal places) = 0,4% (correct to 0,1%)
240
b) Number of possible outcomes: 52; There are 7 favourable outcomes
7
P(winning) = 0,02917 (correct to 5 decimal places) = 2,92% (correct to 0,01 %)
240
11) a) There are 90 possible outcomes.
b) (i) Favourable outcome: 69; there is 1 favourable outcome
1
P(69) = 0,011 (correct to three decimal places) = 1% (correct to nearest %)
90
(ii) Favourable outcome: 10; 20; 30; 40; 50; 60; 70; 80; 90;
There are 9 favourable outcomes
9 1
P(ends in 0) = = 0,1 = 10%
90 10
(iii) Favourable outcome: 9; 18; 27; 36; 45; 54; 63; 72; 81; there are 9 favourable outcomes
9 1
P(multiple of 9) = = 0, 1 = 10%
90 10
12) a) They are most likely to be born in September
b) Number of possible outcomes: 44 954; there are 3 922 favourable outcomes
3 922 1 961
P(March) =
44 954 22 477
= 0,0872 (correct to 4 decimal places) = 8,7% (correct to 0,1 %)

Exercise 6.4 (page 104)

You may find that your results are similar to Makgoshi’s results.
However, because this is an experiment, your results may be very different to Makgoshi’s results.

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Exercise 6.5 (page 106)

5 1
1) a) P(black) = = 0,1 = 10%
50 10
b) Predicted number of red cars = P(red) × Number of cars = 0,2 × 50 = 10
1
2) a) P(win) = = 0,002
500
1
b) Predicted number of tickets = P(win) × Total number of tickets sold × 500 = 25
20
3) Predicted number of faulty calculators = P(faulty calculator) × Total number of tickets sold
= 0,02 × 300 = 6

Exercise 6.6 (page 109)

1) a)
Die
1 2 3 4 5 6
H H; 1 H; 2 H; 3 H; 4 H; 5 H; 6
Coin
T T; 1 T; 2 T; 3 T; 4 T; 5 T; 6
1
i) P(H; 5) = = 0,083 (correct to three decimal places) = 8% (correct to nearest %)
12
1
ii) P(T; 6) = = 0,083 (correct to three decimal places) = 8% (correct to nearest %)
12
3 1
iii) P(T; even) = = 0,25 = 25%
12 4
3 1
iv) P(T; odd) = = 0,25 = 25%
12 4
3 1
v) P(H: number greater than 3) = = 0,25 = 25%
12 4
6 1
vi) P(odd) = = 0,5 = 50%
12 2
2) a)
2nd die
+ 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
1st die

3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
b) (i) P(total of 10)
3 1
= = 0,083 (correct to 3 decimal places) = 8% (correct to nearest %)
36 12
(ii) P(greater than 10)
30 5
= = 0,833 (correct to three decimal places) = 83% (correct to nearest %)
36 6
(iii) P(less than 10)
3 1
= = 0,083 (correct to three decimal places = 8% (correct to nearest %)
36 12
c) The probability is equal to 1 because less than 10 and greater than 10 includes all the
possible outcomes.

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3) a)
Bag A Bag B Outcome
W Æ R; W
R W Æ R; W
R Æ R; R
W Æ R; W
R W Æ R; W
R Æ R; R
W Æ W; W
W W Æ W; W
R Æ W; R
4
b) P(RR or WW) = 0,444 (correct to four decimal places) = 44% (correct to nearest %)
9
4) a)
Stage 1 Stage 2 Outcome
Taxi Æ Lift; Taxi
Lift
Walk Æ Lift; Walk
Taxi Æ Bus; Taxi
Bus
Walk Æ Bus; Walk
Taxi Æ Train; Taxi
Train
Walk Æ Train; Walk
1
b) P(bus; taxi) = 0,167 (correct to four decimal places) = 17% (correct to nearest %)
6
5) a)
First Spin
R10 R5 R5 R50
R10 R20 R15 R15 R60
Second Spin

R5 R15 R10 R10 R55

R5 R15 R10 R10 R55
R50 R60 R55 R55 R100

1
b) (i) P(R20) = = 0,0625 = 6% (correct to nearest %)
16
1
(ii) P(R100) = 0,0625 = 6%
16
2 1
(iii) P(R60) = = 0,125 = 12,5%
16 8
4 1
(iv) P(R15) = 0,25 = 25%
16 4

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6) a) TWO-WAY TABLE:
1st Spin
R G B Y W
R R; R G; R B; R Y; R W; R

2nd Spin
G R; G G; G B; G Y; G W; G
B R; B G; B B; B Y; B W; B
Y R; B G; Y B; Y Y; Y W; Y
W R; W G; W B; W Y; W W; W
TREE DIAGRAM:
1st Spin 2nd Spin Outcomes
R Æ RR
G Æ RG
R B Æ RB
Y Æ RY
W Æ RW
R Æ GR
G Æ GG
G B Æ GB
Y Æ GY
W Æ GW
R Æ BR
G Æ BG
B B Æ BB
Y Æ BY
W Æ BW
R Æ YR
G Æ YG
Y B Æ YB
Y Æ YY
W Æ YW
R Æ WR
G Æ WG
W B Æ WB
Y Æ WY
W Æ WW
1
b) (i) P(W; W) = 0,04 = 4%
25
(ii) The favourable outcomes here are: RW; GW; BW; YW; WR; WG; WB; WY; WW
9
P(at least one W) = 0,36 = 36%
25
(iii) Here the favourable outcomes are: RR; GG; BB; YY; WW.
5 1
P(both the same colour) = = 0,2 = 20%
25 5
7) a) R; B; Y R; Y; B B; R; Y B; Y; R Y; R; B Y; B; R
2 1
b) P(red is first) = у 0,333 (correct to three decimal places) у 33% (correct to nearest %)
6 3

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