Chapter
16
Statistics
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Contents:
A Data collection
B Categorical data
C Numerical data
D Measuring the centre and spread
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STATISTICS
(Chapter 16)
Opening problem
Zach and Ed both enjoy going fishing with their fathers.
They record how many fish they catch each time they
go fishing over the holidays:
5 8 4 6 9 7 9 6 9 9
8 7 10 5 4 8 4 6
Zach:
Ed:
Things to think about:
² By just looking at these values, is it easy to tell
who catches more fish?
² Would it be fair to compare them by finding the
total number of fish caught by each boy?
² How could we determine which boy generally
catches more fish?
When we collect facts or information about something, we call it data.
For example, the data in the Opening Problem are the numbers of fish caught by each boy.
Statistics is the study of solving problems and answering questions by collecting, organising,
and analysing data.
Governments, businesses, sports organisations, manufacturers, and scientific researchers all use
statistics to examine things.
For example, an athletics club may want to know whether a
new training method has improved the speed of its athletes.
The club could collect data about the speed of the athletes
before and after the change in training method. If the
speeds have increased significantly since the change, it
could indicate that the new method is effective.
In statistical work we use tables, graphs, and diagrams to represent data.
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Write a report.
0
Step 6:
5
Analyse the data and make a conclusion.
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Step 5:
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Summarise and display the data.
50
Step 4:
75
Organise the data.
25
Step 3:
0
Collect data.
5
Step 2:
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Examine a problem which may be solved using data. Pose the correct questions.
100
50
Step 1:
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0
5
The process of statistical enquiry or investigation includes the following steps:
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STATISTICS
(Chapter 16)
329
DATA COLLECTION
A
Discussion
Ways of collecting data
Organisations and marketing researchers have many
clever ways of gathering information by tempting us
with offers.
Discuss some of the clever ways information
is collected from you.
Collect samples from
newspapers, magazines, packaging, emails, and letter
box deliveries which invite you to provide data.
Why do you think they want this data?
When a statistical investigation is to be conducted there is always a target population about which
information is required. The population might be the entire population of a country, all the students
at a school, an entire animal species, or the items produced by a machine.
Activity 1
Ratings
1 Use the internet, encyclopedias, or library to find out:
a what a ‘ratings survey’ is
b why radio and television stations want
to know the results of ratings surveys
c how ratings surveys are conducted.
2 Explain why it is important for radio and
television stations to collect this data.
CENSUS OR SAMPLE
One of the first decisions to be made is from whom or what we will collect data. We can collect
data using either a census or a sample.
A census involves collecting data about every individual in the whole population.
The individuals may be people or objects. A census is detailed and accurate but is expensive, time
consuming, and often impractical.
A sample involves collecting data about a part of the population only.
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A sample is cheaper and quicker than a census but is not as detailed or as accurate. Conclusions
drawn from samples usually involve some error.
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STATISTICS
(Chapter 16)
Example 1
Self Tutor
Would a census or sample be used to investigate:
a the length of time an electric light globe will last
b the causes of car accidents in a particular state over one weekend
c the number of people who use White-brite toothpaste?
a Sample. It is impractical to test every light globe produced as there would be
none left for sale!
b Census. An accurate analysis of all accidents would be required.
c Sample. It would be very time consuming to interview the whole population
to find out who does or does not use White-brite toothpaste.
EXERCISE 16A
1 State whether a census or a sample would be used for each of these investigations:
a the number of goals scored each week by a
waterpolo team
b the sizes of paintings in an art gallery
c the most popular radio station in Victoria
d the number of new cars that fail crash tests
e the number of litres of milk bought each week
by a family
f the pets owned by students in a given Year 7
class
g the number of camping holidays Australians
have taken.
2 Give three examples of data which would be collected using a:
a census
b sample.
3 Troy wants to know his classmates’ favourite subject and asks the entire class which subject
they preferred. Is this a census or a sample?
B
CATEGORICAL DATA
Categorical data is data which can be placed in categories.
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For example, the favourite subjects of a class of students is categorical data. The possible categories
may include Mathematics, Art, Science, Music, and English.
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Art
Music
Music
Science
Mathematics
Music
Music
English
Art
Mathematics
331
Science
Art
Mathematics
Music
Mathematics
Music
(Chapter 16)
TALLY AND FREQUENCY TABLES
We can organise data on favourite subjects using a tally and frequency table.
For each student we place a tick mark in the tally for his or her favourite subject.
The frequency of a category is the number of data in that category.
Favourite subject
Tally
Frequency
Mathematics
jjjj
4
Art
jjj
3
2
Music
jj
© j
jjjj
©
English
j
1
Total
16
Science
Each group of five
is represented as
© .
jjjj
©
6
From this table we can identify features of the data.
4
£ 100% = 25% of the students.
16
For example, mathematics is the favourite subject for
THE MODE
The mode is the most frequently occurring category.
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For this data set, the mode is Music.
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STATISTICS
(Chapter 16)
Example 2
Self Tutor
The data below records how students in a class travel to school on
a particular day.
W = walk, Bi = bicycle, Bu = bus, C = car, T = train
The data is:
W Bi Bu T C Bi C W Bi Bu Bi C C Bi Bu W Bu Bu T C
Bi Bi Bu T C C Bi C C C
W W Bu T C
a Draw a frequency table to organise the data.
b Find the mode of the data.
a
Method of travel
Walk
Bicycle
Bus
Car
Train
Tally
©
jjjj
©
© jjj
jjjj
©
© jj
jjjj
©
© jjjj
© j
jjjj
©
©
jjjj
Total
b The mode is ‘car’ as
this category occurs
most frequently.
Frequency
5
8
7
11
4
35
EXERCISE 16B.1
1 Students in a science class obtained the following
levels of achievement:
D C C A A C C D C B C C C D
B C C C C E B A C C B C B C
a Complete a tally and frequency table for this data.
b Use your table to find the:
i number of students who obtained a C
ii fraction of students who obtained a B.
c What is the mode of the data?
2 A class of students were asked which summer sport they wanted to play. The choices were:
T = tennis, S = swimming, C = cricket, B = basketball, and A = athletics.
The data was: A A C T C C S A S T T T B A S A A C S A T A T B C
a Draw a tally and frequency table for the data.
b Find the mode of the data.
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3 Students voted the most popular attractions at the local show to be the side shows (S), the
farm animals (F), the ring events (R), the dogs and cats (D), and the wood chopping (W).
The students in a class were then asked to name their favourite.
The results were: S R W S S W F D D S R R F W S R S R W S S R R R F
a Draw a tally and frequency table for the data.
b Find the mode of the data.
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(Chapter 16)
333
GRAPHS TO DISPLAY CATEGORICAL DATA
Categorical data may be displayed using:
² a vertical column graph
² a horizontal bar chart
Vertical column graph
² a pie chart.
Horizontal bar chart
Pie chart
Mathematics 25%
subject
6
5
4
3
2
1
English
~ 6%
English
Art
~ 19%
Science
English
Music
Art
Mathematics
Science
Music
Art
Mathematics
0
The heights of the columns
indicate the frequencies.
2
4
Science
~ 13%
Music
~ 38%
6
8
frequency
The lengths of the bars
indicate the frequencies.
The angles at the centre
indicate the frequencies.
Example 3
Self Tutor
Recess time drinks
The graph given shows the types of drink
purchased by students at recess time.
a What is the least popular drink?
b What is the mode of the data?
c How many students drink orange juice?
d What percentage of students drink
chocolate milk?
frequency
40
30
20
10
0
ge
ran
ce
jui
O
ft
So
lk
fee
mi
cof
e
t
d
a
e
l
Ic
oco
Ch
nk
dri
type
fshortest columng
a Iced coffee
b ‘Soft drink’ is the mode.
c 27 students drink orange juice.
d The total number of students purchasing drinks = 27 + 35 + 18 + 10
= 90
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So, the percentage of students drinking chocolate milk is
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£ 100% = 20%
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STATISTICS
(Chapter 16)
EXERCISE 16B.2
1 A vet clinic kept a record of the animals they
treated on Wednesday. The results are displayed
in the column graph.
a How many animals were treated?
b How many cats were treated?
c What percentage of the animals treated were
rabbits?
d Find the mode of the data.
8
Animals treated
frequency
6
4
2
0
Dog
2 The 20 players of a football team voted to decide who should be
their captain. The results are given in the table alongside.
a Draw a horizontal bar chart to display the data.
b Which candidate received the:
i most votes
ii least votes?
c What percentage of the team voted for:
i Luke
ii Greg or Steve?
Cat
Bird Fish Rabbit
animal
Candidate
Votes
Cameron
3
Greg
7
Luke
4
Steve
6
3 A randomly selected sample of adults was asked to name the evening television news service
that they watched. The results are shown below.
News service
Frequency
Channel 7
40
Channel 9
45
Channel 10
64
SBS
25
ABC
23
None
3
a How many adults were surveyed?
b Which news service is the most popular?
c What percentage of those surveyed watched the
most popular news service?
d What percentage of those surveyed watched
Channel 9?
e Draw a vertical column graph to display the data.
4 At a school camp, the students selected their favourite icecream flavour out of chocolate (C),
strawberry (S), vanilla (V), and lime (L).
The results were:
CVCSS VLSCV CVSLV SCCVV CSLCV
VCLSC CCVLS SLVCV CLCSC LCVLC
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Organise this data into a tally and frequency table.
How many students chose vanilla?
What percentage of the students chose lime?
Find the mode of the data.
Draw a horizontal bar chart to display the data.
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a
b
c
d
e
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(Chapter 16)
Example 4
335
Self Tutor
The table opposite shows the results when the Year 7 students
at a school were asked “What is your favourite fruit?”
Fruit
Orange
Apple
Banana
Pineapple
Pear
Total
Construct a pie chart to display this data.
There are 60 students in the sample, so each student is entitled to
1
th of 360± is 6± , so we can calculate the
60
Frequency
13
21
10
7
9
60
1
th of the pie chart.
60
sector angles on the pie chart:
13 £ 6±
21 £ 6±
10 £ 6±
7 £ 6±
9 £ 6±
= 78±
= 126±
= 60±
= 42±
= 54±
for
for
for
for
for
the
the
the
the
the
orange sector
apple sector
banana sector
pineapple sector
pear sector.
54°
42°
78°
60° 126°
5 The pie chart shows the different types of traffic fines
handed out by a police officer over one month.
Determine whether the following statements are true
or false:
a The most common fine is for drink driving.
b Fines for not wearing a seatbelt account for about
one quarter of all fines.
c More than half of the fines were either for
speeding or drink driving.
d There were more traffic light offence fines than
expired licence fines.
Traffic light
offence
Expired licence
Not
wearing
seatbelt
Speeding
Drink driving
6 A survey of eye colour in a class of 30 teenagers revealed the following results:
Eye colour
Blue
Brown
Green
Grey
Number of students
9
12
2
7
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a Illustrate these results on a pie chart.
b What percentage of the group have:
i green eyes
ii blue or grey eyes?
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STATISTICS
(Chapter 16)
USING TECHNOLOGY
STATISTICS
PACKAGE
Click on the icon to load a statistical package which can draw a variety of statistical
graphs.
Change to a different graph by clicking on a different tab. You can also change the
labels on the axes and the title of the graph.
Use the software or a spreadsheet to reproduce some of the statistical graphs in the previous
exercise. You can also use this software in any statistical project.
C
NUMERICAL DATA
Numerical data is data which is given in number form.
The number of musical instruments that students in a class can play is an example of numerical
data. It can take the values 0, 1, 2, ....
0
1
2
0
1
0
2
1
0
2
1
0
1
, , , ....
1
3
1
As with categorical data, numerical data can be organised using a tally and frequency table:
Number of instruments
Frequency
0
Tally
©
jjjj
©
1
© jj
jjjj
©
7
2
jjj
3
3
j
1
Total
16
5
GRAPHS TO DISPLAY NUMERICAL DATA
Numerical data can be displayed using:
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² a stem-and-leaf plot.
5
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² a dot plot
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² a column graph
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8
The column graph is the same as for categorical data, but
with numbers on the horizontal axis instead of categories.
(Chapter 16)
337
frequency
6
4
2
0
0
1
2
3
no. of instruments
Dot plots are used when we have a small amount of data,
and not many possible values for the data. Each dot
represents a data value.
0
1
2
3
no. of instruments
STEM-AND-LEAF PLOTS
A stem-and-leaf plot displays a set of data in order of size.
For example, the numbers of photographs taken by tourists
on a bus tour were:
21 33 41 17 24 38 40 12 26 39
15 43 23 35 72 29 19 47 38 21
20 35 12 46 37 40 25 32 18 24
For each data value, the units digit is used as the leaf, and
the digits before it determine the stem on which the leaf is
placed.
So, the stem labels are 1, 2, 3, 4, 5, 6, 7 and they are written under one another in ascending
order.
We now look at each data value in turn. We remove the last digit to find the stem, then write the
last digit as a leaf in the appropriate row.
Once we have done this for all the data values, we have an unordered stem-and-leaf plot.
We can then order the stem-and-leaf plot by writing each set of leaves in ascending order.
Unordered stem-and-leaf plot
1
2
3
4
5
6
7
7
1
3
1
2
4
8
0
5
6
9
3
9
3
5
7
2
2
9
8
6
Ordered stem-and-leaf plot
8
1 0 5 4
5 7 2
0
1
2
3
4
5
6
7
Scale: 1 j 7 means 17
2
0
2
0
2
1
3
0
5
1
5
1
7
3
5
3
8
4
7
6
DEMO
9
4 5 6 9
8 8 9
7
2
So, 4 j 0 0 1 3 6 7 represents the values 40, 40, 41, 43, 46, and 47.
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Notice how the value 72 is separated from the rest of the data. Values such as this are called
outliers.
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STATISTICS
(Chapter 16)
Example 5
Self Tutor
A tennis player has won the following number of matches in tournaments during the last
18 months:
1 2 0 1 3 1 4 2 1 2 3 4 0 0 1 2 2 3 2 1 6 3 2 1
1 1 1 2 2 0 3 4 1 1 2 3 0 2 3 1 4 1 2 0 3 1 2 1
a
b
c
d
Organise the data to form a frequency table.
Draw a column graph of the data.
How many times did the player advance past the second match of a tournament?
On what percentage of occasions did the player win less than 2 matches?
a
Wins
0
1
2
3
4
5
6
Tally
© j
jjjj
©
© jjjj
© jjjj
© j
jjjj
©
©
©
© jjjj
© jjj
jjjj
©
©
© jjj
jjjj
©
b
Frequency
6
16
13
8
4
0
1
48
jjjj
j
Total
Tennis matches won
frequency
20
15
10
5
0
c The player won at least 2 matches on
13 + 8 + 4 + 1 = 26 occasions.
So, the player advanced past the
second match of a tournament 26
times.
1
0
2
3
4
5 6
matches won
d The player won less than 2 matches
on 6 + 16 = 22 occasions.
) percentage =
22
£ 100%
48
¼ 45:8%
EXERCISE 16C
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95
50
75
0
100
0
25
0
Overseas travel
10
5
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1 Workers in an office were asked how many
times they had travelled overseas. The
responses are displayed in the column graph
alongside.
a How many workers were surveyed?
b How many workers have never been
overseas?
c What percentage of workers have been
overseas at least three times?
d Identify the outlier in the data.
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2
3
4
5
6
7
8
no. of times
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STATISTICS
2 The Year 7 students at a school ran as many laps of the
school athletics track as they could in one hour. The results
are recorded on this frequency table.
a Draw a column graph for this data.
b What was the most common number of laps completed?
c How many students completed 12 laps or less?
d What fraction of the students completed at least 14
laps?
(Chapter 16)
Number of laps
10
11
12
13
14
15
16
17
18
19
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Students
1
2
4
6
3
10
17
8
13
2
3 A birthday party was held at an all-you-can-eat pizza
restaurant. The number of slices eaten by each person
is shown in the dot plot below.
1
a
b
c
d
2
3
4
5
6
7
8
9
10
slices
How many people attended the party?
What was the least number of slices eaten?
How many people ate six slices?
Are there any outliers in the data?
4 Yvonne counted the number of chocolate chips in each biscuit of a packet, and obtained these
results:
4, 7, 5, 5, 6, 4, 7, 8, 2, 5, 6, 6, 5, 5, 7, 5, 7, 5, 3, 6
a
b
c
d
e
Draw a dot plot of her results.
What is the most frequent number of chocolate chips?
What is the highest number of chocolate chips in a biscuit?
How many biscuits contained five chocolate chips?
What percentage of biscuits contained less than five chocolate chips?
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0
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50
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25
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5
5 A group of schools in Melbourne were
4 8
surveyed to find how many Year 7
5 447
students they had.
The results are
6 02568
displayed in a stem-and-leaf plot.
7 247
a How many schools were surveyed?
8 011
9 52
b How many schools had 54 Year 7
10 5 8
students?
Scale: 4 j 8 means 48 students
11 1
c What was the highest number of
Year 7 students a school had?
d How many schools had at least 80 Year 7 students?
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STATISTICS
(Chapter 16)
6 The numbers of runs scored by a batsman over a 30 game season were:
27, 7, 12, 74, 30, 11, 42, 19, 29, 51, 62, 14, 49, 22, 2,
35, 43, 12, 62, 22, 28, 37, 59, 40, 5, 13, 69, 32, 16, 21
a Construct an unordered stem-and-leaf plot of the data. Make sure you include a scale.
b Construct an ordered stem-and-leaf plot of the data.
c How many times did the batsman score more than 25 runs?
d Find the batsman’s:
i lowest
ii highest score.
7 Walter recorded the number of pages in the daily newspaper for 4 weeks:
86 94 78 108 96 112 100 122 92 88 100 96 80 112
78 92 104 124 88 160 116 92 86 94 106 82 114 116
a Construct a stem-and-leaf plot to display the data.
b How many newspapers contained at least 100 pages?
c What percentage of the newspapers contained less than 95 pages?
d Are there any outliers in the data?
Discussion
When displaying numerical data, when is it best to use:
² a dot plot
² a column graph
² a stem-and-leaf plot?
Activity 2
Conduct your own survey
What to do:
1 Decide on a question about your class you would like to investigate. For example:
“What is the most common method of travelling to school?”
“What type of pet is most common?”
“What type of TV show is the most watched?”
“How many pets have you owned?”
2 Collect the questions from the students in the class, and use them to make a survey for
everyone to do.
3 Collect the data for your question from each of your classmates.
4 Is your data categorical or numerical?
5 Organise your data into a table.
6 Display your data using an appropriate graph.
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STATISTICS
D
(Chapter 16)
341
MEASURING THE CENTRE AND SPREAD
When we analyse numerical data, we want to be able to describe how the numbers are distributed.
To help us understand a distribution, we want a measure of its centre, and also how the data is
spread on either side of this centre.
MEASURING THE CENTRE
There are three different numbers which are commonly used to measure the middle or centre of
a set of numerical data. These are the mean or average, the median, and the mode.
THE MEAN
The mean or average is the total of all data values divided by the number of data values.
We use the symbol x to represent the mean.
sum of data values
x=
number of data values
THE MEDIAN
The median of a set of data is the middle value of the ordered set of data values.
For an odd number of data values there is one middle value which is the median.
For an even number of data values there are two middle values. The median is the average of
these two values.
THE MODE
The mode is the score which occurs most often in a data set.
For example, the mode of the data set 0, 2, 3, 3, 4, 5, 5, 5, 6, 7, 9 is 5 since 5 occurs most
frequently.
Example 6
Self Tutor
An exceptional footballer scores the following goals for her school during a season:
1 3 2 0 4
2 1 4 2 3
0 3 3 2 2
5 2 3 1 2
Find the:
a mean
b median
c mode for the number of goals she scored.
a mean =
sum of all scores
number of matches
=
45
= 2:25 goals
20
b The ordered data set is:
0 0 1 1 1 2 2 2 2 |{z}
22 233333445
) median =
2+2
= 2 goals
2
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STATISTICS
(Chapter 16)
Example 7
Self Tutor
In the last 7 matches, a basketballer has scored 19, 23, 16, 11, 22, 27, and 29 points.
a Find the mean score for these matches.
b Find the median score for these matches.
c In the next game the basketballer scores 45 points.
i mean
ii median.
Find his new
total of all data values
number of data values
a The mean, x =
=
19 + 23 + 16 + 11 + 22 + 27 + 29
7
=
147
= 21 points
7
b In order, the scores are:
11 16 19 22 23 27 29
) the median = 22 points
fthe middle score of the ordered setg
c
i The new mean
=
147 + 45
8
=
192
8
fthe sum of the first 7 values was 147g
= 24 points
ii The ordered set is now:
11 16 19 22 23 27 29 45
For 8 data values,
the median is the
average of the 4th
and 5th values.
) the median
=
22 + 23
2
= 22:5 points
faverage of the middle scoresg
Discussion
In the Example above, the 8th value of 45 is an outlier.
² Which of the mean or median is more affected by the addition of the outlier?
² Why is the mode useless as a measure of the centre of this set of data?
EXERCISE 16D.1
1 Find the mean of the following data sets:
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b 7, 5, 0, 3, 0, 6, 0, 9, 1
d 5, 2:4, 6:2, 8:9, 4:1, 3:4
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a 3, 1, 5, 4, 4, 7
c 2:1, 4:5, 5:2, 7:1, 9:3
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343
2 Find the median of the following data sets:
a 2, 1, 1, 3, 4, 3, 2, 1, 5, 4, 3, 3, 0
c 1:2, 1:9, 2:2, 2:6, 2:9
b 5, 9, 2, 4, 6, 6, 7, 6, 11
d 0:5, 5:6, 3:8, 4:9, 2:7, 4:4
3 Consider the data set: 7, 8, 0, 3, 0, 6, 0, 11, 1.
a Find the
i mean
ii median
iii mode of this data.
b Is the mode a suitable measure of the ‘middle’ of this data set? Explain your answer.
4 Margaret played 10 games of Scrabble in a tournament, and obtained the following scores:
206 120 108 185 219 168 245 295 195 307
a mean
b median of these scores.
Find the
5 The number of text messages that Jim received each day for the last 15 days were:
2 3 9 13 4 3 12 1 6 15 3 4 10 2 3
a mean
Find the
b median
c mode of the data.
6 List the data represented by the following graphs.
Then find the
i mode
ii mean
iii median
a
b 6
frequency
1
2
3
4
c
4
2
3
4
5
6
0
7
score
35
0127
466
359
Scale: 1 j 3 means 1:3
0
1
2
3
4
5
score
7 The students in a class were asked how many
events they competed in at the school’s
swimming carnival.
The results are displayed on a dot plot.
0
1
2
3
4
5
no. of events
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a How many students did not compete in any events?
b Find the
i mode
ii median
iii mean of the data.
c Copy the graph, and locate on it the mode, median, and mean.
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STATISTICS
(Chapter 16)
8 Roger sells lemonade at a market stall. He has graphed
his sales for the last 12 days on a column graph.
a Calculate the mean, median, and mode of the
data, and locate them on the graph.
b The next day is very cold, and Roger only sells
2 lemonades. Find the new mean, median, and
mode.
c Which of the measures of centre has been most
affected by the addition of the outlier?
5
frequency
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
sales
9 Consider the performances of two groups of students in the same mental arithmetic test out
of 10 marks.
Group X: 7, 6, 6, 8, 6, 9, 7, 5, 4, 7
Group Y: 9, 6, 7, 6, 8, 10, 3, 9, 9, 8, 9
a Calculate the mean mark for each group.
b There are 10 students in Group X and 11 in Group Y. Is it unfair to compare the mean
scores for these groups?
c Which group performed better at the test?
10 Consider the data in the Opening Problem on page 328.
a Calculate the mean and median for each boy.
b Who generally catches more fish?
11 Josh and Eugene each own a hot dog stand. They record the number of hot dogs they sell
every day for two weeks. The results are:
33, 40, 28, 43, 38, 32, 24, 35, 47, 29, 31, 36, 27, 38
39, 47, 32, 51, 48, 55, 61, 35, 49, 58, 52, 67, 55, 43
Josh:
Eugene:
a What was the most hot dogs that Josh sold in one day?
b Calculate the mean and median for each data set.
c Who generally sells more hot dogs?
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12 The heights in centimetres of boys and girls at a party
are:
Boys:
150, 142, 146, 137, 140
Girls: 141, 140, 155, 138, 145, 157
a Calculate the mean and median for each group.
b Are the boys or girls taller? Explain your answer.
c A 185 cm boy joins the party.
i Calculate the new mean and median for the
boys.
ii How has the addition of this outlier affected
the comparison between the two groups?
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STATISTICS
(Chapter 16)
345
MEASURING THE SPREAD
In addition to measuring the centre of a set of data, it is also important to consider how the data
is spread.
The simplest measure of spread is the range.
The range of a data set is the difference between the maximum or largest data value, and the
minimum or smallest data value.
range = maximum value ¡ minimum value
Example 8
Self Tutor
17 students were asked how many days they had been home sick from school so far
this year. The results were:
2, 1, 5, 5, 3, 4, 3, 6, 2, 9, 4, 2, 3, 5, 6, 2, 3
Find the range of this data set.
The minimum value is 1 and the maximum value is 9.
So, the range = 9 ¡ 1 = 8 days.
EXERCISE 16D.2
1 Find the range of the following data sets:
a 2, 4, 4, 5, 6, 8, 9, 10, 11, 11, 13
c 6, 6, 6, 6, 7, 7, 7, 7, 8
b 7, 9, 12, 9, 4, 8, 11, 6, 10
d 8:5, 4:2, 7:6, 7:2, 9:3, 9:1, 5:6
2 Find the range of the data represented by the following graphs:
a
4
b
frequency
3
2
1
0
2
3
4
5
c
1
2
3
4
579
01268
3569
038
5
6
7
8
score
6 7 8 9 10 11 12 13 14 15 16
score
Scale: 5 j 0 means 50
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3 The number of items bought by customers at a convenience store were:
3 5 5 8 5 3 5 9 7 4 5 8 7 7 6
a Draw a dot plot of the data.
b Calculate the mean, median, and range of the data, and indicate these values on your dot
plot.
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STATISTICS (Chapter 16)
4 The table alongside shows the
maximum temperatures, in ± C, in
Australia’s capital cities for one
week:
a Calculate the range for each
city.
b Which city had the:
i most variation
ii least variation in the
maximum temperature?
City
Mon Tue Wed Thu Fri Sat Sun
24
28
25
34
22
26
27
29
Adelaide
Brisbane
Canberra
Darwin
Hobart
Melbourne
Perth
Sydney
22
29
24
33
18
20
30
26
23
29
21
33
19
22
32
24
23
27
23
33
18
22
34
26
30
28
24
31
19
23
25
24
30
28
23
31
23
27
27
24
28
29
27
31
21
30
32
26
Research
With most goods we buy, we can read the amount we are buying on
the packaging. For example, we might buy 35 g of sultanas, 20 m of
alfoil, or 600 mL of water. But how do we know the manufacturer
is telling the truth?
You need to choose a bulk packet that has several of the same item
in it. For example:
² a bag containing 8 balls of wool, each 75 m long
² a packet containing 12 bags of chips, each 22 g
² a 6 pack of fruit juice cartons, each 175 mL.
Your task is to analyse whether the manufacturer has made a truthful claim about how much
is in their product.
What to do:
1 Choose your item to analyse.
a Describe exactly what the problem is, and how you are going to test it.
b What do you expect your results to be?
2
a Measure the mass, length, or volume of each item in your packet. Round your data
as appropriate. Construct a stem-and-leaf plot of your results.
b For your data, find the:
i mode
ii median
iii mean.
c Were the results in b what you expected? Explain your answer.
3
a Calculate the percentage of items that were below the amount stated on the packaging.
b What would you expect this percentage to be?
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4 From your results, can you form any conclusions about the amount of each item in your
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347
Use the bridges to move from the Start circle to the Finish circle.
The mean of the numbers you have landed on must never drop below 5 or rise above 10. You
cannot land on the same island more than once.
2
1
4
40
7
2
8
Start
15
15
1
5
Finish
15
7
30
35
28
ACTIVITY
4
19
12
20
3
1
2
Click on the icon to run this activity as a game.
KEY WORDS USED IN THIS CHAPTER
²
²
²
²
²
²
²
²
²
²
²
²
bar chart
centre
dot plot
mode
sample
stem-and-leaf plot
²
²
²
²
²
categorical data
column graph
mean
numerical data
spread
tally and frequency table
census
data set
median
pie chart
statistics
1 State whether a census or a sample would be used to investigate:
a the number of double-yolk eggs in cartons of
12 eggs
b the ages of visitors to a museum
c the sports played by students in a Year 7 class
d the makes of cars for sale in a particular car
yard
e the weights of new-born lambs.
15
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2 Samantha rolled a die 50 times. The results
are shown in the column graph alongside.
a How many times did Samantha roll a 2?
b What percentage of the rolls were greater
than 4?
c Find the:
i mode
ii median of the data.
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STATISTICS (Chapter 16)
3 A random sample of people were surveyed
about their blood type.
The results are
displayed in the pie chart opposite.
a Use your protractor to measure the size of
each sector angle.
b What percentage of people surveyed have
type A blood?
c If 22 of the people surveyed have type AB
blood, how many people were surveyed in
total?
Blood type
AB
B
O
A
4 A survey of hair colour in a class of 40 students revealed
the results alongside:
a Construct a horizontal bar chart to display this data.
b For this group of students, which was the least
common hair colour?
c Could conclusions be made from this survey about
the hair colour of all students? Explain your answer.
Hair colour
Frequency
Red
Brown
Black
Blonde
4
17
11
8
5 Find the mean, median, mode, and range of the data represented by the following graphs:
a
8
b
6
frequency
4
10
c
2
3
4
5
11
12
788
13445
00067
24
13
14
2
15
0
score
1
2
3
4
5
score
Scale: 3 j 1 means 31
6 During the 24 game netball season, Alyssa played in all games and scored 482 goals. Due
to injury, Stephanie only played 19 matches, and scored 335 goals. Which player received
the award for the highest average number of goals scored per match?
7 A mental arithmetic test out of 10 was given to two groups of students.
The results were:
Group 1:
7, 8, 6, 6, 9, 10, 7, 8, 8, 7
Group 2:
10, 9, 10, 8, 4, 9, 10, 8, 7, 7, 9
a Find the mean and median score for each group.
b Which group performed better?
PRINTABLE
TEST
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Click on the link to obtain a printable version of this test.
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AUS_07
STATISTICS (Chapter 16)
Practice test 16B
349
Short response
1 State whether each of the following is a census or a sample:
a Customers passing through the checkouts at a furniture store are asked for the postcode
of their home suburb.
b All members of a football club were surveyed about membership fees.
2 Find the
a mean
b median
3, 4, 6, 6, 7, 9, 12, 13, 14, 17, 19
c mode of:
3 Melanie is conducting a survey of her classmates about their favourite TV show. Is this
7 frequency
categorical or numerical data?
6
4 The numbers of drinks sold at tables in a
5
café are displayed in the frequency column
4
3
graph opposite.
2
What percentage of customers ordered 4 or
1
more drinks?
0
5 Ruth notes the number of biscuits she eats at
work each day, and records them in the table
opposite.
a Copy and complete the table.
b For how many days did Ruth record data?
0
1
2
3
No. of biscuits
0
4
5
6
number of drinks
Tally
Frequency
j
1
4
3
© jjj
jjjj
©
© j
jjjj
©
4
jj
2
Total
6 A supermarket puts 1 L cartons of milk on sale, and records the number of cartons bought
by each customer over an hour. The results were:
0 0 1 1 1 2 1 0 3 1 2 4 0 1 2 7 1 1 0 2 3
a Draw a dotplot to display this information.
b Are there any outliers in the data?
7 Jillian recorded the number of pages in the weekly
local newspaper over a period of time. The results are
shown in the stem-and-leaf plot opposite.
What percentage of newspapers contained at least
60 pages?
3
4
5
6
7
8
1 1 2 3 6 7 8 8
1 2 7 8 9
0 0 1 2
1
3 j 8 means 38
8 Is the mode a suitable measure of the ‘middle’ of the data set:
12, 0, 7, 4, 6, 0, 4, 9, 7, 0, 7, 0?
Explain your answer.
9 While practising at the driving range, Colin hit golf balls the following distances (in m):
186 229 234 192 235 229.
Find the mean distance of Colin’s shots.
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70 35 25 67 82 53 63 79 41
10 Find the range of the following data set:
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350
STATISTICS (Chapter 16)
Practice test 16C
Extended response
1 In a survey of school students to determine their favourite type of music, the following
data was obtained:
a
b
c
d
e
Music type
Pop
R’n’B
Classical
Dance
Jazz
Other
Number of students
55
46
23
32
39
5
How many students were surveyed?
What percentage of those surveyed favoured Dance music?
Which was the most favoured type of music?
Construct a horizontal bar chart to display the information.
Can conclusions be made from this survey about the favourite type of music of all
students? Give a reason for your answer.
bus
train
walk
car
bicycle
Transport to school
2 50 randomly selected students were asked about their
20
frequency
method of transport to school. The results of the
survey are displayed in the graph opposite.
15
a What sort of graph is this?
10
b Which was the most common method of
transport to school?
5
c How many students travelled to school by bus?
d What percentage of students travelled to school
0
method
by car?
e What percentage of students travelled to school either by bicycle or walking?
3 A group of 20 students played a round of mini-golf.
Their scores were:
43 32 59 35 60 26 39 41 53 67
39 54 28 46 65 30 45 23 32 65
a
b
c
d
Draw a stem-and-leaf plot to display the data.
How many students scored less than 40?
What percentage of students scored more than 55?
Find the:
i mean
ii median
iii range of the data.
4 Some children were asked how much pocket money they receive each week. The results,
in dollars, were:
2, 4, 0, 10, 4, 0, 5, 5, 2, 4, 10, 5, 2, 0, 10, 0, 2, 5, 2, 8, 5, 10, 2, 0, 10
a Draw a dot plot of the data.
b Find the:
i mode
ii mean
iii median
c Indicate the values found in b in your dot plot.
iv range
QUESTION 5
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5 Click on the icon to obtain this question.
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