Post on 25-Nov-2014
Jemma has been asked by
the club president to
analyse the results of her
AFL football team for a
season. The points scored in
22 matches were:
85, 96, 118, 93, 73, 71, 98, 77,
106, 64, 73, 88, 62, 97, 104,
85, 73, 92, 62, 76, 90, 79.
What conclusions can you
draw from these data? The
data as listed are difficult to
work with so we need to
present them in a way that
makes them easier to
analyse.
This chapter looks at
various ways of displaying
data as well as different
measures which describe
aspects of the data.
12Data andgraphs
548 M a t h s Q u e s t 8 f o r V i c t o r i a
READY?areyou
Are you ready?Try the questions below. If you have difficulty with any of them, extra help can be
obtained by completing the matching SkillSHEET. Either click on the SkillSHEET icon
next to the question on the Maths Quest 8 CD-ROM or ask your teacher for a copy.
Reading scales. (How much is each interval worth?)
1 On each of the following scales, state what each interval is worth.
a b c
Reading line graphs
2 The line graph at right shows the height of
a child (Timmy) over 5 years.
a How tall was Timmy at the start of the measurement
period?
b How much did Timmy grow in the first year?
c How much did Timmy grow over the five years?
d How many years did it take for Timmy to grow
10 cm?
Producing a frequency table from a frequency histogram
3 Copy and complete the following frequency table to
show the data represented in the frequency histogram.
Finding the mean
4 a Find the sum of the following data: 6, 3, 5, 4, 5, 4, 6, 7.
b Divide this sum by the number of items in the data set.
Arranging a set of data in ascending order
5 Arrange each of the following sets of data in ascending order.
a 25, 20, 22, 21, 29, 34, 25
b 215, 381, 276, 345, 298, 277, 325, 400, 304
c 4.6, 0.3, 3.6, 5.8, 2.9, 1.8, 3.5, 5.8, 3.1, 2.8, 3.6
Finding the score in a data set that occurs most frequently
6 For each of the following data sets, find the score that occurs most frequently.
a 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 5
b 23, 29, 25, 24, 23, 21, 25, 26, 25, 29
c 7, 12, 8, 3, 5, 11, 8, 4, 2, 1, 6, 10, 13
Score (x) Frequency (f)
20 5
21
22
23
24
12.2
54 8060 0 100
12.3
Years
Hei
ght
(cm
)
100
110
120
130
140
150
160
170
2001 2002 2003 20052004 2006
Increase in Timmy’s height
between 2001 and 2006
12.4
876543210
Fre
quen
cy
20 21 22 23 24Score
12.512.7
12.8
12.812.912.12
C h a p t e r 1 2 D a t a a n d g r a p h s 549
Data collection and organisationInformation or data is constantly being collected.
Different organisations collect different types of data.
For example, at a cricket match, some of the
statistics gathered for a batsman are: time spent
batting, the number of balls faced, the runs off a par-
ticular delivery, where the ball was hit, the number of
4s or 6s hit, and so on. Once the data is collected, it
can be organised, analysed and interpreted.
Data can be collected from existing sources (such
as government records), from experiments or by
observation.
A survey is the process of collecting data. If
every member of a target population is surveyed,
the process is called a census. A census is con-
ducted in Australia every 5 years to obtain an accurate profile of Australians. On
census night each person in Australia is required to complete a detailed booklet con-
taining a series of questions relating to age, marital status, employment, income,
housing, education, modes of transport and so on. This allows the government to ana-
lyse the population and make decisions on how to improve services.
Due to limitations in time, cost and practicality, in many cases a sample of the popu-
lation is selected at random (not in any particular order or pattern) to prevent biased
(leaning in a favoured direction) results.
A sample can give us an indication of what the whole population is like.
Consider these situations:
1. You cook a batch of muffins to take to a party. Naturally, you want to test whether they
turned out well. Do you eat the whole population of cakes as a check?
2. A factory produces 400 cars
per day.
Should there be a crash-test of
every car before it is sold to the
public?
In both cases it is not practical
or viable to test each item. There-
fore, a sample needs to be taken.
The following investigations require you to research different ways to obtain un–
biased samples and conduct surveys.
Find out how samples are chosen and surveys conducted for:
a television program ratings
b top 10 songs, videos and movies.
COMMUNICATION Samples and surveys
550 M a t h s Q u e s t 8 f o r V i c t o r i a
It is important that a sample is chosen randomly to avoid bias.
Consider the following situation.
The government wants to improve sporting facilities in Melbourne. They decide
to survey 1000 people about what facilities they would like to see improved. To
do this, they choose the first 1000 people through the gate at a football match at
the Telstra Dome.
In this situation it is likely that the results will be biased towards improving
facilities for football. It is also unlikely that the survey will be representative of
the whole population in terms of equality between men and women, age of the
participants and ethnic backgrounds.
Questions can also create bias. Consider asking the question, ‘Is football your
favourite sport?’ The question invites the response that football is the favourite
sport rather than allowing a free choice from a variety of sports by the respondent.
Consider each of the following surveys and discuss:
a any advantages, disadvantages and possible causes of bias
b a way in which a truly representative sample could be obtained.
1 Surveying food product choices by interviewing customers of a large
supermarket chain as they emerge from the store between 9.00 am and 2.00 pm
on a Wednesday.
2 Researching the popularity of a government decision by stopping people at
random in a central city mall.
3 Using a telephone survey of 500 people selected at random from the phone book
to find if all Australian States should have Daylight Saving Time in summer.
4 A bookseller uses a public library database to survey for the most popular
novels over the last three months.
5 An interview survey about violence in sport taken at a football venue as
spectators leave.
THINKING Bias
C h a p t e r 1 2 D a t a a n d g r a p h s 551
Great care must be taken when framing questions to collect data by post or by
personal interview. It is common for people to misunderstand the point of a question.
The answers to the questions must be in a form that makes them easy to collate.
Designing a questionnaireWhen designing a questionnaire it is important to keep these key principles in mind.
1 Know exactly what kind of data you are after before you start framing the questions.
2 Adapt the questions to suit the target population.
3 Word each question for clarity and courtesy.
4 Keep the answers’ format as simple as possible. Some of the ways of doing this are:
a circling or recording a number
b ticking a box
c Yes/No/Don’t know
d a scaled rating, such as the one below
e a single word.
5 Arrange all questions in a clear, uncluttered layout.
6 Go for the minimum possible number of questions.
7 Trial your best efforts on several people from the target group and rewrite as
necessary.
The members of the Student Representative Council (SRC) have drafted a
questionnaire regarding their role in the school.
Making your SRC work
1 In your opinion, how has the SRC gone about achieving its goals of the past year?
2 In your opinion, what should the SRC try to achieve this year?
3 If the SRC were to fundraise for a specific school-based project, what would be
some realistic ideas you could offer that the money could go to?
4 To what extent are you prepared to support and assist your leaders this year?
(continued)
DESIGN Collecting data for surveys and questionnaires
Totallydisagree
Totallyagree
Stronglydisagree
Stronglyagree
Moderatelydisagree
Moderatelyagree
Neitheragreenor
disagree
0 1 2 3 4 5 6 7 8 9 10
552 M a t h s Q u e s t 8 f o r V i c t o r i a
The results to a questionnaire can be tabulated using a database. Investigate how you
may be able to do this.
Each of these questions is open-ended. This means that the possible responses
are unlimited and will be difficult to collate.
A more effective questionnaire follows.
1 In your opinion did the SRC achieve its goals last year?
YES / NO / DON’T KNOW
2 On a scale of 1 (very unimportant) to 5 (very important) rate the following goals
of this year’s SRC.
a Improve sporting facilities for the school.
b Establish a senior study room.
c Have a separate canteen line for Year 8 students.
3 Rank the following fundraising projects from 1 to 4.
Royal Children’s Hospital Appeal
Two new computers for the library for student Internet access
Replacement of worn sporting equipment
50 CAS calculators for student use.
4 I am prepared to assist with ALL / SOME / A FEW / NONE of the SRC
activities over the next year.
These questions are much easier to collate and produce a snapshot of school
opinion about the SRC.
Now design a questionnaire of your own for one of the following issues.
1 Are school uniforms needed?
2 Are wages for teenage workers reasonable?
3 More leisure activities are required for teenagers.
4 There is too much violence in computer games, in movies and on television.
5 There is a need for a new Australian flag.
6 Compulsory military service should be introduced for 18-year-olds
7 Speed limits throughout Australia should be changed.
8 Which radio stations throughout Australia are most popular?
C h a p t e r 1 2 D a t a a n d g r a p h s 553
Tables and chartsIt is far better to present data in an organised manner rather than leave them in their raw
form. The simplest way to sort and display information is to form a table or chart. Hor-
izontal rows and vertical columns intersect to form boxes in which the data are given.
This table represents the prices at Franky’s Fast Foods. Use the table to answer the following:a What is the cost of a
large homemade pie?b Which is more
expensive, a family lasagne or a family whole chicken?
c How much more is paid for large fries than for a standard serving?d What is the total cost of a family chicken and family fries?e How much change would there be from
$50 after buying 2 large fries, 3 large nuggets and a standard lasagne?
f Considering all products, about how many times the cost of a standard serving (to the nearest whole number) is the cost of a family serving?
g Based on your answer to part f, if a standard size pizza was $6.00 at this shop, what would you expect a family pizza to cost?
Continued over page
THINK WRITE
a Obtain the cost of a large homemade
pie by reading directly from the
table. Refer to the cell in which the
Homemade pies row intersects with
the ‘Large’ column.
a
Answer the question. The cost of a large homemade pie is $3.00.
Food product Standard Large Family
Fries $1.50 $2.50 $4.00
Chicken nuggets $2.00 $3.50 $5.50
Homemade pies $1.80 $3.00 $5.00
Whole BBQ chicken $4.95 $6.95 $9.95
Lasagne $1.95 $3.95 $6.95
1Product Standard Large Family
Fries $1.50 $2.50 $4.00
Chicken nuggets
$2.00 $3.50 $5.50
Homemade pies
$1.80 $3.00 $5.00
2
1WORKEDExample
554 M a t h s Q u e s t 8 f o r V i c t o r i a
THINK WRITE
b Obtain the price of a family lasagne. b Cost of a family lasagne = $6.95
Obtain the price of a whole family
chicken.
Cost of a whole family chicken = $9.95
Compare the prices. A whole family chicken is more expensive
than a family lasagne.Answer the question.
c Obtain the price of large fries. c Cost of large fries = $2.50
Obtain the price of standard fries. Cost of standard fries = $1.50
Subtract the price of standard fries
from large fries.
Difference = large fries − standard fries
= $2.50 − $1.50
= $1.00
Answer the question. Large fries cost $1.00 more than standard
fries.
d Obtain the cost of a whole family
chicken.
d Cost of whole family chicken = $9.95
Obtain the cost of family fries. Cost of family fries = $4.00
Add the two amounts. Total cost = family chicken + family fries
= $9.95 + $4.00
= $13.95
Answer the question. The total cost of a family chicken and family
fries is $13.95.
e Obtain the cost of 2 large fries. e Cost of large fries = 2 × $2.50
= $5.00
Obtain the cost of 3 large nuggets. Cost of 3 large nuggets = 3 × $3.50
= $10.50
Obtain the cost of a standard
lasagne.
Cost of a standard lasagne = $1.95
Add the 3 amounts. Total = $5.00 + $10.50 + $1.95
= $17.45
Subtract the total amount obtained in
step 4 from $50.
Change = $50.00 − $17.45
= $32.55
Answer the question. The change received from fifty dollars is
$32.55.
f Divide each family product by its
respective standard product.
f =
Round the answer to the nearest
whole number.
= 2.67 (≈ 3)
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
5
6
1Family fries
Standard fries---------------------------------
4.00
1.50----------
2
C h a p t e r 1 2 D a t a a n d g r a p h s 555
Tables and charts
1 This table shows the maximum and minimum daily temperatures in a city over a one-
week period.
Day 1 2 3 4 5 6 7
Maximum (°C) 26 25 27 25 24 22 23
Minimum (°C) 18 18 19 17 17 16 16
THINK WRITE
Note: If the digit in the first decimal
place is greater than or equal to 5,
round the value up.
Note: If the digit in the first decimal
place is less than 5, round the value
down.
=
= 2.75 (≈ 3)
=
= 2.78 (≈ 3)
=
= 2.01 (≈ 2)
=
= 3.56 (≈ 4)
Answer the question. On average, the cost of a family product is
3 times the cost of its respective standard
product.
g To obtain the family price of a
product, multiply the standard price
by 3.
g Family pizza ≈ 3 × $6.00
= $18.00
Answer the question. A family pizza would cost approximately
$18.00.
Family nuggets
Standard nuggets-----------------------------------------
5.50
2.00----------
Family pies
Standard pies--------------------------------
5.00
1.80----------
Family chicken
Standard chicken-----------------------------------------
9.95
4.95----------
Family lasagne
Standard lasagne----------------------------------------
6.95
1.95----------
3
1
2
1. Tables and charts should include:
(a) an appropriate title
(b) clear headings for columns
(c) clear labels for rows
(d) well-spaced data for easy reading.
2. Tables are read by looking at the combination of a row and a column.
remember
12A
WORKED
Example
1
Mathcad
Tables
and
charts
556 M a t h s Q u e s t 8 f o r V i c t o r i a
Use the table to answer the following:
a What was the maximum temperature on day 5?
b Which day(s) had the lowest minimum temperature?
c Which day was the hottest?
d Which day had the warmest minimum temperature?
e What was the temperature range (variation) on day 3?
f Which day had the smallest range of temperatures?
g What would you expect the maximum and minimum temperatures to be on day 8?
2 The cost of entry to Wild and Wet theme park is shown in this table:
Number of
children
(under 15)
Number of
adults
0 1 2 3 4
0 0 $16 $30 $42 $55
1 $30 $46 $60 $72 $85
2 $58 $88 $100 $113
3 $84 $100 $126 $139
4 $100 $116 $130 $155
Parties of over 4 adults and 4 children by special arrangement
C h a p t e r 1 2 D a t a a n d g r a p h s 557
a What is the cost of entry for 3 children with no adults?
b What is the cost of entry for 2 adults and no children?
c Which is more expensive, the cost of entry for 2 adults and 3 children or the cost of
entry for 1 adult and 4 children? What is the difference in cost?
d Fill in the costs left blank.
e How much does each adult save by going in a party of four adults rather than going
alone?
f How much change from $150 would there be after a group of 2 adults and 4 children
paid the admission?
g Complete this sentence describing the overall pattern of costs.
The greater the number of people in a group, the _____________ .
h Four Year 8 students decided to go together to the Wild and Wet. How much did each
student pay?
3 The country is divided into local
telephone zones. If you ring outside
of your area, you are making an
STD (subscriber trunk dialling) call.
These phone calls are charged at
two rates depending on the day and
what time of day you are making
the call (and on what plan you have
with the phone company). Peak
charges apply to STD calls made
between 7 am and 7 pm Monday to
Friday. Off-peak rates apply to STD
calls made at all other times.
The following table shows the cost
of making STD calls for the two
rates.
a Use the table to calculate the cost of a 3-minute call:
iii at 1.00 pm on Monday iii at 8 pm on Tuesday
iii at any time on Sunday iv at 2 am on Wednesday
iv at 6.30 pm on Thursday.
b Use the table to calculate the cost of each of the following calls:
iii 10 min call at 5 pm on Saturday iii 5 min call at 9.30 am on Friday
iii 1 min call at 10 pm on Monday iv 30 min call at 3 pm on Sunday
iv 20 min call at 1.20 pm on Tuesday.
Rate
Call duration
1 min 3 min 5 min 10 min 20 min 30 min
Peak 25c 75c $1.25 $2.50 $5.00 $7.50
Off-peak 18c 54c $1.90c $1.80 $3.60 $5.40
558 M a t h s Q u e s t 8 f o r V i c t o r i a
4 Satri decides to travel by train to Geometric for a holiday with relatives. The timetable
for the Silver Streak Train Line is provided below.
Use the timetable to answer the following.
a If Satri left on Saturday, what time would she arrive at Geometric?
b On what day(s) could she catch an evening train?
c How long does the journey from Mathsville to Geometric take?
d Where does the train stop for a meal? For how long?
e At what time does the Monday evening train pass through Wholetown?
f If Satri took the Wednesday morning train and slept from 1.00 to 2.30 pm, what
towns would she miss seeing?
g If Satri wants to connect with a Friday 1400 hours flight out of Geometric and the
train (leaving Friday morning) is running on time, should she be able to make it?
Explain.
Timetable — Silver Streak Train Line
Mathsville to Geometric
DailyMonday–Friday
EveryMonday
Saturday and Sunday
Mathsville 7.00 am 6.00 pm 7.30 am
Integral 7.45 6.45 8.15
Addsville 9.30 8.30 10.00
Greater Rock 10.30 9.30 11.00
Pronumeral Arrive 11.30 10.30 12.00 noon
Meal Break Leave 12.30 pm no stop 1.00 pm
Halftown 1.20 11.20 1.50
Wholetown 2.25 12.25 am 2.55
Geometric 3.05 1.05 3.35
C h a p t e r 1 2 D a t a a n d g r a p h s 559
Questions 5 and 6 refer to the following table for International Parcel Post to zones
A–C to calculate charges for posting parcels overseas.
Source: Australia Post
Parcel mass and zones
Air
Economy
Air
Sea
EMS
Documents
EMS
Merchandise
Zone A New Zealand
Over 250 g up to 500 g $9.00 $8.00 — $30.00 $37.00
Over 500 g up to 750 g $12.50 $11.00 — $34.50 $41.50
Over 750 g up to 1000 g $15.50 $13.50 — $34.50 $41.50
Over 1000 g up to 1250 g $19.00 $16.00 — $39.00 $46.00
Over 1250 g up to 1500 g $22.00 $19.00 — $39.00 $46.00
Over 1500 g up to 1750 g $25.50 $22.00 — $43.50 $50.50
Over 1750 g up to 2000 g $28.50 $24.50 — $43.50 $50.50
Extra 500 g or part thereof $3.50 $3.00 — $4.50 $4.50
Zone B Asia Pacific
Over 250 g up to 500 g $11.00 $9.50 — $32.00 $39.00
Over 500 g up to 750 g $15.50 $13.00 — $38.50 $45.50
Over 750 g up to 1000 g $19.50 $16.00 — $38.50 $45.50
Over 1000 g up to 1250 g $24.00 $19.50 — $45.00 $52.00
Over 1250 g up to 1500 g $28.00 $22.50 — $45.00 $52.00
Over 1500 g up to 1750 g $32.50 $26.00 — $51.50 $58.50
Over 1750 g up to 2000 g $36.50 $29.00 — $51.50 $58.50
Extra 500 g or part thereof $4.50 $3.50 — $6.50 $6.50
Zone C
USA/Canada/Middle East
Over 250 g up to 500 g $13.00 $11.00 $6.00 $35.00 $42.00
Over 500 g up to 750 g $18.50 $15.50 $12.50 $43.50 $50.50
Over 750 g up to 1000 g $23.50 $19.50 $15.50 $43.50 $50.50
Over 1000 g up to 1250 g $29.00 $24.00 $19/00 $52.00 $59.00
Over 1250 g up to 1500 g $34.00 $28.00 $22.00 $52.00 $59.00
Over 1500 g up to 1750 g $49.50 $32.00 $25.50 $60.50 $67.50
Over 1750 g up to 2000 g $44.50 $36.50 $28.50 $60.50 $67.50
Extra 500 g or part thereof $6.50 $5.50 $3.50 $8.50 $8.50
560 M a t h s Q u e s t 8 f o r V i c t o r i a
5 Use the International Parcel Post table on page 559 to find the total cost to send, from
Australia:
a a 700-g parcel to New Zealand (zone A) by Air Mail
b a 2.3-kg parcel to the USA (zone C) by Economy Air
c a 1.8-kg document to China (zone B) by EMS Documents
d a 4-kg parcel to Malaysia (zone B) by Air Mail
e a 6-kg parcel to Egypt (zone C) by Economy Air
f a 300-g parcel to Japan (zone B) by Air Mail
g a 420-g parcel to Fiji (zone B) by Economy Air
h a 3.9-kg parcel to Canada (zone C) by Sea Mail
i an 8-kg parcel to Papua New Guinea (zone B) by Air Mail
j 11 kg of merchandise to India (zone B) by EMS.
6 How much more does it cost to post a 10-kg parcel to Indonesia (in zone B) by Air
Mail than by Economy Air?
The table at right shows the
comparison of road deaths in
2001 between NSW and other
states.
Source: Australian Bureau of
Statistics and NSW Roads and
Traffic Authority
Use the table to answer the
following questions.
1 Which state or territory had:
a the most road fatalities?
b the least road fatalities?
2 Which state or territory had
165 fatalities?
3 Which state or territory had:
a the highest number of road deaths per 100 000 of population?
b the lowest number of road deaths per 100 000 of population?
4 Which column — ‘Killed’ or ‘Fatalities per 100 000’ — is a more accurate
measure of the risk of death on the road? Why?
5 Therefore, which state or territory is:
a the safest?
b the most dangerous for road deaths?
6 Out of the 8 states and territories, where does NSW rate in terms of road safety
(ACT would be 1), based on the ‘Fatalities per 100 000’ column?
COMMUNICATION Staying alive
State/Territory Killed
Fatalities
per 100 000
population
New South Wales
Victoria
Queensland
Western Australia
South Australia
Tasmania
Northern Territory
Australian Capital
Territory
524
444
324
165
153
61
50
16
7.9
9.2
8.9
10.1
8.6
12.9
25.3
5.1
C h a p t e r 1 2 D a t a a n d g r a p h s 561
Questions 1 to 4 refer to Pete’s pizza price list.
1 What is the cost of a large Hot salami pizza?
2 What is the most expensive medium pizza?
3 What is the cost of a family Chicken pizza and a small Supreme pizza?
4 How much change would you have from $50 after buying a small Vegetarian pizza, a
medium Mexican pizza and a family Supreme pizza?
Questions 5 to 7 refer to the following frequency distribution table, which shows data
collected on magazine sales in one week at a local newsagency.
Answer true or false to the following statements.
5 The magazines Girlfriend and Smash
Hits sold equally well in this week.
6 During this week 40 copies of House
and garden were sold.
7 Dolly was the most popular
magazine sold.
Questions 8 to 10 refer to the following table. A number of people were surveyed as to
their most preferred water sport.
8 List the sports in ascending order
of preference.
9 What fraction of people surveyed
prefer waterskiing?
10 What percentage of people sur-
veyed prefer snorkelling?
Pete’s Pizzas
Pizza Small Medium Large Family
Vegetarian $5.00 $7.00 $9.00 $11.00
Supreme $6.00 $7.50 $9.50 $12.00
Chicken $6.50 $8.00 $10.00 $12.80
Hot salami $5.50 $7.50 $9.80 $12.50
Mexican $5.50 $7.50 $9.00 $11.90
1
Magazine Number of sales
Dolly 50
Girlfriend 60
House and garden 30
Smash Hits 60
Total 200
Water sports survey
Water sport Number of people
Snorkelling 20
Waterskiing 10
Whitewater rafting 35
Diving 15
562 M a t h s Q u e s t 8 f o r V i c t o r i a
Column and bar graphsGraphs are very helpful when displaying and interpreting information. It is generally easier
to analyse the data when it is displayed as a graph rather than in a frequency table.
Column graphsWhen constructing column graphs, they should be drawn on graph paper and have:
1. a title
2. labelled axes which are clearly and evenly scaled
3. columns of the same width
4. an even gap between each column
5. the first column beginning half a unit (that is, half the column width) from the
vertical axis.
The graph at right represents the favourite pets of a
particular Year 8 class.
a How many students preferred a dog as a pet?
b How many students in the class had a favourite
pet?
c Which was the least favoured pet?
d How many times more popular than horses are dogs?
e If there are 28 students in the class, how many do
not have a favourite pet?
THINK WRITE
a Read the ‘Dog’ column of the graph and
answer the question.
a Eight people preferred a dog as a pet.
b Add the numbers corresponding to
the top of each column.
b Number of students = 8 + 6 + 4 + 3 + 2
= 23
Answer the question. In the class, 23 students had a favourite pet.
c Make note of the shortest column and
answer the question.
c The least favoured pet is the horse.
d Obtain the number of students
preferring horses and those
preferring dogs.
d Students preferring horses = 2
Students preferring dogs = 8
Compare the two values. Eight is four times as large as two; that is,
8 = 2 × 4.
Answer the question. Dogs are four times more popular than
horses.
e Subtract the number of students who
have a favourite pet (that is, 23) from
the total number of students.
e Number of students = 28 − 23
= 5
Answer the question. Five students do not have a favourite pet.
Num
ber
pre
ferr
ing (
freq
uen
cy)
Year 8 Blue’s favourite pets
8
6
4
2
0Dog Cat Bird Mouse Horse
Pet
1
2
1
2
3
1
2
2WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 563
Bar graphsWhen constructing bar graphs, they should be drawn on graph paper and have:
1. a title
2. labelled axes that are clearly and evenly scaled
3. horizontal bars of the same width
4. an even gap between each horizontal bar
5. the first horizontal bar beginning half a unit (that is, half the bar width) above the
horizontal axis.
The graph at right represents the favourite television shows of 500 teenagers
aged between 13 and 15.
a What percentage of the teenagers
preferred watching comedy
television shows?
b How many of the teenagers in the
sample preferred to watch science
fiction television shows?
c What was the least favoured
television show?
d What were the two most popular
television shows?
e Which television show is three times
more popular than the news?
f Which television show did 10% of the teenagers watch?
g What scale has been used on the horizontal axis?
Continued over page
THINK WRITE
a Read the ‘Comedy’ bar of the graph and
answer the question.
a 10% of the teenagers preferred comedy
television shows.
b Read the ‘Science Fiction’ bar of the
graph.
b 8% of the teenagers preferred science fiction.
Find 8% of the sample. 8% of 500
=
= 40
Answer the question. Forty teenagers enjoyed science fiction
shows.
c Make note of the shortest bar(s) and
answer the question.
c The least favoured television shows are the
documentaries and lifestyle programs.
d Make note of the two longest bars and
answer the question.
d The two most popular television shows are
the cartoons and police dramas.
Favourite television shows
Comedy
Soaps
Police Drama
News
Documentaries
Cartoons
Science Fiction
Lifestyle
Thriller
Tel
evis
ion s
how
s
0 5% 10% 15% 20% 25%
Percentage favouring
1
2
8
100--------- 500×
3
3WORKEDExample
564 M a t h s Q u e s t 8 f o r V i c t o r i a
THINK WRITE
e Obtain the percentage of teenagers
who preferred watching the news.
e Teenagers preferring the news = 6%
Multiply the news percentage by 3. Required percentage is 3 × 6% = 18%
Find the percentage obtained in step 2
on the horizontal scale of the graph and
see which bar it corresponds to.
18% corresponds to the soaps.
Answer the question. The soaps are three times more popular
than the news.
f Find 10% on the horizontal scale of the
graph and see which bar(s) it corresponds
to.
f Comedy and thriller television shows are
preferred by 10% of the teenagers.
g Read the horizontal scale and
determine how many centimetres
represent each marking.
g 1 cm = 5%
Answer the question. From the graph, each one centimetre
represents 5% favouring.
1
2
3
4
1
2
These results were obtained when a Year 8
class was surveyed on their favourite leisure
activity.
a Select a suitable title and draw a column
graph to display the data. Label the
horizontal axis ‘Leisure activity’ and the
vertical axis ‘Number preferring
(frequency)’. Scale the vertical axis from
0 to 10 so as to include the highest score.
b How many students chose sport as their
favoured leisure activity?
c What is the difference between the most and least favoured activity?
d Which activity received 5 votes?
e If everyone in the class completed the survey, how many students were in the class?
THINK WRITE
a Rule a set of axes on graph paper.
Provide a title for the graph that
relates to the data, for example,
‘Favourite leisure activity’. Label the
horizontal and vertical axes.
a
Leisure activity
Number preferring (frequency)
Reading 3
Television 9
Sport 7
‘Hanging out’ 5
Other 2
1
Num
ber
pre
ferr
ing
(fr
equen
cy)
Favourite leisure activity
10
8
6
4
2
0Reading Television Sport ‘Hanging
out’Other
Leisure activity
4WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 565
Column and bar graphs are used for displaying categorical data, that is, data relating to
eye colour, pets, favourite pastimes and so on.
THINK WRITE
Scale the horizontal and vertical axes.
Note: Leave a half interval at the
beginning and end of the graph; that is,
begin the first column half a unit from
the vertical axis.
Draw a vertical column so that it
reaches a vertical height corresponding
to 3 people. Label the section of the
axis below the column as ‘Reading’.
Leave a gap (measuring one column
width) between the first column and
the second column.
Repeat steps 3 and 4 for each of the
remaining leisure activities.
b Read the ‘Sport’ column of the graph and
answer the question.
b Seven students chose sport as their
favourite leisure activity.
c Make note of the tallest and shortest
columns of the graph.
c Television (most favoured) = 9 students
Other (least favoured) = 2 students
Subtract the number of students
corresponding to the shortest column
from those corresponding to the tallest
column.
Difference = 9 − 2
= 7
Evaluate and answer the question. The difference between the most and least
favoured leisure activity is 7.
d Find 5 on the vertical scale of the
graph and see which column it
corresponds to.
d ‘Hanging out’ received 5 votes.
Answer the question.
e Add the frequencies corresponding to
the top of each column.
e Total = 3 + 9 + 7 + 5 + 2
Total = 25
Answer the question. There are 25 students in the class.
2
3
4
5
1
2
3
1
2
1
2
1. Graphs should be drawn on graph paper for greater accuracy. When
constructing column or bar graphs they must have:
(a) a title
(b) labelled axes that are clearly and evenly scaled
(c) vertical columns or horizontal bars of the same width
(d) an even gap between each column or bar
(e) the first column or bar beginning half a unit from the appropriate axis.
2. Column and bar graphs are used for displaying categorical data, that is, data
relating to eye colour, pets, favourite pastimes and so on.
remember
566 M a t h s Q u e s t 8 f o r V i c t o r i a
Column and bar graphs
1 This column graph represents the Jumpin’ Jeans
company’s profits.
a Which year showed the highest profit? How
much was it?
b In which year did losses start? What was the
loss that year?
c What was the profit or loss for 2005?
d i Find the total profits and the total losses.
ii Calculate the company’s overall profit/loss
over the period shown.
e In which year was the only improvement made? By how much?
2 The graph at right represents the preferred television snacks of 160 Year 8 students at
Mathsville High.
a Which snack is most favoured? What per-
centage favoured it?
b Which snack was preferred by 12% of those
surveyed?
c How much greater was the percentage prefer-
ring corn chips than the percentage preferring
popcorn?
d What must the total of all column percentages
be? Why?
e How many times more popular than nuts are
potato chips?
f What other choices could have been added to the survey?
g What scale is used on the horizontal scale?
3 These results were obtained when a class
voted to elect a captain.
a Select a suitable title and draw a column
graph to display the data. Label the hori-
zontal axis ‘Student’ and the vertical axis
‘Votes received’. Scale the vertical axis
from 0 to 12 so as to include the highest
score.
b Who was elected class captain?
c What was the winning margin over the
next most popular candidate?
d Which two candidates received the same
number of votes?
e If everyone in the class used their one
vote, how many were in the class?
12B
Pro
fit
Loss
Jumpin’ Jeanscompany profits
20
10
0
–10
–20
($m
illi
ons)
Year
’01 ’02 ’03 ’04 ’05 ’06
Mat
hcad
Columnand bargraphs
WORKED
Example
2
EXCE
L Spreadsheet
Columngraphs
EXCE
L Spreadsheet
Columngraphs(DIY)
EXCE
L Spreadsheet
Bargraphs
EXCE
L Spreadsheet
Bargraphs(DIY)
SkillSH
EET 12.1
Reading column graphs
WORKED
Example
3
Preferred television snacks
Lollies
Popcorn
Nuts
Corn chips
Potato chips
10% 20% 30% 40% 50% 60%
Percentage favouring
WORKED
Example
4
Student Number of votes
ImranReneeJulianGannThan
63
1193
C h a p t e r 1 2 D a t a a n d g r a p h s 567
4 The age-group composition of the Australian population in 2005 is shown in the table
below.
a Draw a bar graph to display the data.
b Which age groups are roughly equal in numbers?
c What percentage of the population were 65 or over?
d What percentage were in the income earning 20–64 age group?
Questions 5, 6 and 7 relate to the following table.
A survey of houses in Statistics Street produced the data shown in the table below.
5 Select a suitable title and draw a column graph to display the data. Label the vertical
axis ‘Number of houses’ and the horizontal axis ‘Number of bedrooms’.
6
The most common number of bedrooms in the houses of Statistics Street is:
A 10 B 3 C 2 D 5 E 6
7
The number of houses surveyed in Statistics Street is:
A 14 B 4 C 35 D 7 E 21
Age group Number of millions
0–910–1920–6465 +
2.42.59.53.9
Number of bedrooms Number of houses
2345
310
62
GAME time
Data andgraphs
— 001
multiple choice
WorkS
HEET 12.1multiple choice
568 M a t h s Q u e s t 8 f o r V i c t o r i a
This is a sample data sheet to be filled in by everyone in the class. You or your
teacher may decide to omit some items and include others. The data collected will
be used throughout this chapter.
Tabulating data from the class surveyCollate and tabulate the information from your class’s personal data sheets. Using
this information, create frequency distribution tables with the following categories:
1 Hair colour 2 Eye colour
3 Language(s) spoken 4 Children in family
5 Favourite sport 6 Favourite pet
7 Age 8 Height.
Recall from Year 7 that a frequency distribution table consists of three columns,
headed ‘Score’ (or in his case ‘Category’), ‘Tally’ and ‘Frequency’.
COMMUNICATION Personal data sheet
Name: __________________________ Class: 8 ______ Date: ___/___/___
Age ___ years ___ months
Height ___ cm
Hair colour (red, blond, brown, black)
Eye colour (blue/grey, green, brown)
Left-handed or right-handed
Language(s) spoken
Transport to or from school
Number of children in the family (include yourself)
Shoe size
Favourite sport
Favourite pet
C h a p t e r 1 2 D a t a a n d g r a p h s 569
Use the information from your personal data sheets for the class which has been
organised into frequency distribution tables to complete the following tasks.
1 Draw a column graph to display the class data on the number of children in the
families.
2 Draw a bar graph to display the data on favourite sports.
3 Choose another set of data and display this as either a column or a bar graph.
DESIGN Displaying your data as column or bar graphs
History of mathematicsN I C H O L A S O R E S M É ( 1 3 2 5 – 1 3 8 2)
Little is known of the early life of Nicholas
Oresmé. It is believed that he was born near Caen
in Normandy, France, in 1325. In 1348, there is a
record of him attending the College of Navarre to
study theology at the University of Paris. After the
award of his Master’s degree, Nicholas became the
bursar of the college until 1355 and then a teaching
master until 1362.
In 1358, Nicholas met and became firm friends
with the Dauphin, the future Charles V of France.
The Dauphin was having a terrible struggle to
maintain the kingdom of France. Edward III of
England (the Black Prince) had captured his father,
John, and was holding him for ransom in London.
It was a time of political turmoil and danger. The
Black Death (the bubonic plague) swept through
France, killing up to 800 people every day in the
years 1358–59. Also, Europe was embroiled in the
Hundred Years’ War. Nicholas was to become one
of Charles’ chief advisers as well as a friend until
Charles’ death in 1380.
As well as helping Charles V in the financial
administration of France and taking on the role of
canon in a number of cathedrals, Oresmé spent
years working in the area of mathematics. He is
credited as being the first mathematician to use
pictorial representations, or graphs, to represent the
way elements vary. There is evidence that the first
primitive graphs were produced by an Italian,
Giovanni di Cosali. However, di Cosali’s efforts
lacked the clarity and purpose of those done by
Oresmé.
Oresmé also wrote on many scientific subjects,
producing books such as the Book on the Sky and
the World, where he rejected the theories of
Aristotle that stated that the Sun revolved around
the Earth.
Questions
1. What was dangerous about living in
Europe in 1358 and 1359?
2. Which ruler of France did Nicholas
Oresmé become chief adviser to?
3. Which area of mathematics did he gain
credit in developing?
Research
Find out more about Nicholas Oresmé and
some of his other mathematical work. What
was his advice on the coinage used in France
at that time?
570 M a t h s Q u e s t 8 f o r V i c t o r i a
Line graphsSo far we have looked at graphs that make comparisons of some sort about such things
as favourite leisure activities, heights, weights, eye colour and so on.
We will now look at graphs that display changes over a period of time. These graphs,
called line graphs, are commonly used to display such data as: temperature changes
during the day, the state’s monthly employment figures, a company’s profits and sales
during the year. Line graphs are also used in business and sport to analyse trends or
general patterns that occur over a period of time.
A line graph is simply drawn by joining the given points with a line or smooth curve.
When constructing line graphs they must be drawn on graph paper and include:
1. a title
2. a horizontal axis that is evenly scaled and labelled (usually as time)
3. a vertical axis that is evenly scaled and labelled
4. a line or smooth curve that joins successive plotted points.
Line graphs also give meaningful information about the in-between values of particular
data.
0
10
20
30
40
0
100
200
300
400
DNOSAJJMAMFJ
MILDURAmm°C
Average monthly temperatureand rainfall
max.
min.
The line graph at right
represents the temperature
change during a particular day.
a What is the value of each
subdivision (grid line) on:
ii the horizontal axis?
ii the vertical axis?
b What were the maximum
and minimum temperatures
during the day? At what
times did these occur?
c What was the temperature
at:
ii 8.00 am? ii 1.24 pm?
d At what time was the temperature: i 15°C? ii 27°C?
e What would you expect the temperature to be at 5.00 pm?
THINK WRITE
a i Look at the horizontal axis of the
graph and count how many grid
lines represent 1 hour (60 minutes).
a i 5 subdivisions (grid lines) = 1 hour
That is,
5 subdivisions (grid lines) = 60 minutes.
Determine how many minutes one
subdivision (grid line) represents;
that is, divide 60 minutes by 5
grid lines.
1 subdivision =
= 12 minutes
Answer the question. Each subdivision represents 12 minutes
on the horizontal axis.
Time of day
Tem
per
ature
(°C
)
0
5°
10°
15°
20°
25°
30°
35°
0 6 am 7 8 9 10 11 12 1 2 3 4 5 6 pm
Temperature change during the day
1
260
5------
3
5WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 571
Continued over page
THINK WRITE
ii Look at the vertical axis of the
graph and count how many grid
lines represent 5°C.
ii 5 subdivisions (grid lines) = 5°C
Determine how many degrees one
subdivision (grid line) represents;
that is, divide 5°C by 5 grid lines.
1 subdivision =
= 1°C
Answer the question. Each subdivision represents 1°C on the
vertical axis.
b Look at the graph and read the
maximum (highest graph point) and
minimum (lowest graph point)
temperatures.
b Maximum temperature = 30°C
Minimum temperature = 9°C
Look at the maximum temperature
point on the graph and read vertically
down until the corresponding value
on the ‘Time of day’ axis is obtained.
A temperature of 30°C occurs at noon.
Look at the minimum temperature
point on the graph and read vertically
down until the corresponding value
on the ‘Time of day’ axis is obtained.
A temperature of 9°C occurs at 6.00 am.
Answer the question. A maximum temperature of 30°C occurs at
noon and remains at this level until 1.00 pm.
A minimum temperature of 9°C occurs at
6.00 am.
c i Read vertically up from 8.00 am on
the ‘Time of day’ axis to the point
intersecting the graph and then
horizontally across to the
‘Temperature’ axis to obtain the
corresponding value and answer the
question.
c i At 8.00 am the temperature is 18°C.
ii Repeat the process described in part i
to obtain the corresponding
temperature at 1.24 pm.
Note: From part a each grid line on
the horizontal (time) axis represents
12 minutes; therefore, 1.24 pm is 2
grid lines beyond 1.00 pm.
ii At 1.24 pm the temperature is 29°C.
1
25
5---
3
1
2
3
4
572 M a t h s Q u e s t 8 f o r V i c t o r i a
THINK WRITE
d i Read horizontally across from 15°C on the
‘Temperature’ axis to the point intersecting
the graph and then vertically down until the
corresponding value on the ‘Time of day’
axis is obtained and answer the question.
d i A temperature of 15°C occurs at
7.24 am.
ii Repeat the process described in part i to
obtain the required times.
ii A temperature of 27°C occurs twice,
first at 10.36 am and then at 2.30 pm.
e Extend the line graph so that it intersects
with the vertical grid line corresponding to
5.00 pm.
Note: Assume that the temperature
continues to decrease at approximately the
same rate.
e
Locate the corresponding temperature at
5.00 pm and answer the question.
At 5.00 pm the temperature is
approximately 17–18°C.
1
2
The sunrise times on successive Mondays are shown in the following table.
a Plot a line graph to display the data.
b From your knowledge of the seasons, estimate the time of year covered by the graph.
Explain your answer.
Week 1 2 3 4 5 6
Sunrise (am) 6.30 6.22 6.17 6.08 6.00 5.54
THINK WRITE
a Rule and label a set of axes on graph
paper.
a
Plot each of the points onto the axes.
Begin at 1 on the weeks axis and
follow the vertical axis until the
required sunrise time (that is, 6.30)
is reached; mark the point.
Repeat step 2 for each set of data.
Join the points with a line.
1
Week number
Tim
e of
day
(am
)
6.35
6.30
6.25
6.20
6.15
6.10
6.05
6.00
5.55
5.50
5.45
5.40
1 2 3 4 5 6 7
Sunrise times
2
3
4
6WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 573
Line graphs
1 The line graph at right represents Cynthia’s
pulse rate over a six minute period.
a What is the value of each subdivision
(grid line) on the:
i horizontal axis?
ii vertical axis?
b What was Cynthia’s pulse rate after:
i 2 minutes?
ii 3 minutes?
c What were the maximum and
minimum pulse rates and at what
times did they occur?
d At what times were the pulse rates:
i 120 bpm (beats per minute)?
ii 112 bpm?
e What would you expect the pulse
rate to be after 6 minutes?
f What could have caused Cynthia’s
pulse rate to increase?
THINK WRITE
b Observe the direction of the graph and
make note of what is happening to the
times as each week passes.
b The sunrise values are getting smaller as
each week passes. This means sunrise is
occurring earlier each week and the hours
of daylight are increasing. The time of year
represented by the graph could be the
season of spring, as this is when sunrise
begins to occur earlier.
1. A line graph must be drawn on graph paper and include:
(a) a title
(b) a clearly labelled and evenly scaled set of axes
(c) an appropriate line or smooth curve that joins the plotted points.
2. Line graphs are used when displaying changes over a period of time.
remember
12CSkillSHEET
12.2
Reading scales(How much iseach interval
worth?)
Time (min)
Puls
e r
ate
(beats
/min
)
120
100
80
60
40
20
0
0 1 2 3 4 5 6
Cynthia’s pulse rateWORKED
Example
5
Mathcad
Linegraphs
SkillSHEET
12.3
Readingline
graphs
EXCEL Spreadsheet
Linegraphs
1
2---
EXCEL Spreadsheet
Linegraphs(DIY)
574 M a t h s Q u e s t 8 f o r V i c t o r i a
2 The line graph at right represents the conversion rate of Australian currency (A$) to
American currency (US$) on a particular day.
a What is the value of each subdiv-
ision (grid line) on:
i the ‘A$’ axis?
ii the ‘US$’ axis?
b How much American currency can be
exchanged for:
i A$20?
ii A$100?
c How much Australian currency can
be exchanged for:
i US$40?
ii US$90?
d What would you expect in American
currency for A$200?
3 The line graph at right compares the heights
of Yasha and Yolande over a twenty-year
period.
a How tall was Yasha at age:
i 3? ii 5? iii 14?
b How tall was Yolande at age:
i 4? ii 10? iii 14?
c At what age(s) were they the same
height?
d At what age was Yasha:
i 140 cm tall? ii 176 cm tall?
e Between what ages was Yolande taller
than Yasha?
f Between which two birthdays did each person show fastest growth?
g How long did it take Yasha to double his birth height?
h What do the horizontal intervals mean?
i What was their height difference when they both reached maximum height?
4 The line graph at right displays a family’s
weekly income over an eight-year period.
a What was the family’s weekly income in:
i 1999? ii 2006?
b By how much had the income increased over
that period?
c In which two years did the income remain
the same?
d Between which two years did it show the
biggest increase?
e When did the income reach $650 per week?
A$
US
$
160
140
120
100
80
60
40
20
0
0 25 50 75 100 125 150 175 200 225
$Australian – $US conversion graph
Age in years
Hei
ght
(cm
)
200
170
140
110
80
50
200 5 10 15 20
Yasha’s and Yolande’s heights
Yasha
Yolande
Year
Inco
me
($)
1000
800
600
400
200
0
’99 ’00 ’01 ’02 ’03 ’04 ’05 ’06
Family income per week
C h a p t e r 1 2 D a t a a n d g r a p h s 575
5 The line graph at right displays the traffic flow
on a particular road.
a At what time of day did traffic flow hit its
peak?
b During the period studied, when was the
number of vehicles at the lowest level?
c How many cars used the road at:
i 11.00 am? ii 9.30 am?
d At what times of the day were:
i 300 cars on the road?
ii 225 cars on the road?
e Discuss why the traffic flow shows the pat-
tern displayed.
6 A Year 8 student doing an exercise program
recorded her pulse rate at one-minute intervals, as
shown in the graph at right.
a What was her pulse rate after:
i 2 min? ii 4 min?
b After how many minutes did her pulse rate
reach its maximum?
c After how many minutes was her pulse rate
80 bpm? Explain why there are two answers to
this question.
d What was her pulse rate after 3 minutes?
e For how long was her pulse rate above 80 bpm?
7 Sunset times on six successive Fridays are shown in
the table below.
a Plot a line graph to display the data. Label the
horizontal axis ‘Week’ and the vertical axis
‘Sunset (pm)’.
b Estimate the time of year covered by the graph
for your location. Explain your answer.
8 Weekly CD sales for the pop group The Mathemagics
are shown in the table below.
a Construct a line graph of the data.
b Estimate the sales for week 9.
Week 1 2 3 4 5 6
Sunset (pm) 6.15 6.08 6.00 5.54 5.47 5.40
Week 1 2 3 4 5 6 7 8
Sales 1500 2800 3750 4000 3600 3000 2400 1900
Time of day
Num
ber
of
veh
icle
s (h
undre
ds) 6
5
4
3
2
1
0
6 am 7 8 9 10 11 12 1 pm
Traffic flow on a main road
Time (min)P
uls
e ra
te(b
eats
/min
) 120
80
40
00 1 2 3 4 5 6
Exercise pulse rates
1
2---
WORKED
Example
6
576 M a t h s Q u e s t 8 f o r V i c t o r i a
9 The average daily minimum temperatures for a city are shown in this table.
a Construct a line graph to display the data.
b What would you expect the average daily minimum temperature to be in July?
10 Vani’s times for the 100 m sprint in training sessions are shown below.
a What do you notice about Vani’s times? How
will this affect the graph?
b How might the time axis scale be adjusted to
better highlight the time variations?
c Using the suggestion from part b, show the
information as a line graph.
d What was Vani’s average time over all training
sessions?
11 Jani and Kosmo are twins. These heights were recorded each year on their birthday.
a Display the height records as two lines of different colours on the one graph.
b At which age was the height difference the greatest?
c How tall would you expect each of the twins to be at age 6?
Questions 12, 13 and 14 relate to the following table showing variation in temperature
on a snowfield on one day.
Month January February March April May June
Temp (°C) 29 26 22 19 15 10
Run 1 2 3 4 5 6
Time (s) 12.4 12.1 11.9 12.2 11.8 12.0
Age (year) 1 2 3 4 5
Jani’s height (cm) 49 56 64 74 83
Kosmo’s height (cm) 50 58 65 76 86
Time of day5.00 am
7.00 am
9.00 am
11.00 am
1.00 pm
3.00 pm
5.00 pm
7.00 pm
Temperature (°C) −10 −8 −6 −1 4 3 1 −2
C h a p t e r 1 2 D a t a a n d g r a p h s 577
12 a Draw a line graph to show the temperatures recorded at a snowfield resort during
the ski season.
b Estimate the temperatures at: i 10 am ii 8 pm.
13
The temperature at 8.00 am would be approximately:
A −8°C B −7 °C C −7°C D −6 °C E −6°C
14
At what time(s) was the temperature at freezing point (0°C)?
A 12.00 pm B 11.30 am, 5.30 pm C 11.30 am, 6.30 pm
D 12.30 am, 5.30 pm E 6.00 pm
Histograms and frequency polygonsIn this section we will use our knowledge of column and line graphs to create two
similar types of graphs: the histogram and the frequency polygon.
HistogramsA special type of column graph is called a histogram. It must be drawn on graph paper
and have the following characteristics.
Use a thermometer to measure classroom temperature at hourly intervals from 8 am
to 3 pm. Record the data in a table; then display it as a line graph. Label the
horizontal axis ‘Time’, the vertical axis ‘Temperature’. Scale the axes and add a title
to the graph.
multiple choice
1
2---
1
2---
multiple choice
THINKING Recording temperature
578 M a t h s Q u e s t 8 f o r V i c t o r i a
1. All columns are of equal width.
2. No gaps are left between columns.
3. Each column ‘straddles’ an x-axis score; (value) that is, the column starts and
finishes halfway between scores.
4. Usually a half-interval is left at the beginning and end of the graph. That is, the first
score is one unit in from the frequency (y)-axis.
The histogram at right displays the data collected
in a survey conducted to find the number of
children in a family. The data collected for 20
families is shown in the table below.
Frequency polygonsA special type of line graph, called a frequency
polygon, has the following characteristics.
1. The frequency polygon uses the same scaled
axes as the histogram.
2. The midpoints of the tops of the histogram columns are joined by straight
intervals.
3. The polygon is closed by drawing lines at each end down to the score (x) axis.
The data presented in the histogram above can be used to create a frequency polygon.
Score Tally Frequency
012345
| | | |
| | | |
| | | | | |
| | |
|
457301
Total 20
Score
Fre
quen
cy
8
6
4
2
00 1 2 3 4 5 6
Children in familyf
x
Score
Fre
quen
cy
8
6
4
2
00 1 2 3 4 5 6
Children in familyf
x
Score
Fre
quen
cy
8
6
4
2
00 1 2 3 4 5 6
Children in familyf
x
C h a p t e r 1 2 D a t a a n d g r a p h s 579
The table at right represents the
number of hours of sport played
per week by Year 8 students.
a Draw a histogram which
represents the data in the table.
b Which is the most common
score (that is, the most common
number of hours of sport
played per week)?
c Which is the least common
score (that is, the least common
number of hours of sport
played per week)?
d How many students play at
least 6 hours of sport per week?
e How many students play at
most 3 hours of sport per week?
f How many students were
included in the survey?
g Draw a frequency polygon of
the data.
Continued over page
THINK WRITE
a Rule and label a set of axes on graph
paper. Give the graph a title.
a
Add a scale to the horizontal and vertical
axes.
Note: Leave half an interval at the
beginning and end of the graph; that is,
label the first score one unit in from the
vertical (frequency) axis.
Draw in the first column so that it starts
and finishes halfway between scores and
reaches a vertical height of three units.
Repeat step 3 for each of the other scores.
b Use the frequency distribution table to
determine the largest frequency value and
which score it corresponds to.
b The largest frequency value of 16
corresponds to 5 hours of sport played
per week.
Answer the question. The most common number of hours
of sport played per week is 5 hours.
1
Score (hours of sport played)
Fre
quen
cy
181614121086420
0 1 2 3 4 5 6 7 8
Hours of sport played byYear 8 students
f
x
2
3
4
1
2
7WORKEDExample
Score (hours of sport played) Frequency (f)
1 3
2 8
3 10
4 12
5 16
6 8
7 7
Total 64
580 M a t h s Q u e s t 8 f o r V i c t o r i a
The histogram in worked example 7 can also be obtained on the graphics calculator.
THINK WRITE
c Use the frequency distribution table to
determine the least frequency value
and which score it corresponds to.
c The smallest frequency value of 3
corresponds to 1 hour of sport being played
per week.
Answer the question. The least common hours of sport played per
week is 1 hour.
d Add all of the frequencies that
correspond to at least 6 hours of sport
being played per week (that is, 6 and
7 hours) and answer the question.
d At least 6 hours: 8 + 7 = 15.
There are 15 students who played at least
6 hours of sport per week.
e Add all of the frequencies that
correspond to, at most, 3 hours of sport
being played per week (that is 1, 2 and
3 hours) and answer the question.
e At most 3 hours: 3 + 8 + 10 = 21.
There are 21 students who played at most
3 hours of sport per week.
f Add each of the frequencies to
determine the total number of students
surveyed.
f Total = 3 + 8 + 10 + 12 + 16 + 8 + 7
= 64
Answer the question. 64 students were surveyed.
g Mark the midpoints of the tops of the
columns obtained in the histogram
from part a.
g
Join the midpoints by straight line
intervals.
Close the polygon by drawing lines
at each end down to the score (x)
axis.
The frequency polygon may be left
overlayed on the histogram or may
be transferred to a separate set of
axes.
1
2
1
2
1
Score (hours of sport played)
Fre
quen
cy
181614121086420
0 1 2 3 4 5 6 7 8
Hours of sport played byYear 8 students
f
x
2
3
Score (hours of sport played)
Fre
quen
cy
181614121086420
0 1 2 3 4 5 6 7 8
Hours of sport played byYear 8 students
f
x
4
C h a p t e r 1 2 D a t a a n d g r a p h s 581
1. Clear the editor and turn off any existing plots by pressing
and choosing .
2. Press , select and enter x data in L1 and
frequencies in L2.
(For grouped data use the midpoint of the class
interval for x.)
3. Press then press (or
select ).
Choose settings for a histogram as shown at right
using arrow keys and press .
4. Press and choose ; this will set
up the vertical scale.
5. Press and change the settings as follows:
Xmin = 1, the smallest value in L1.
Xmax = 8, the sum of the largest value in L1 and the
Xscl value; that is, 7 + 1 = 8.
Xscl = 1, the difference between two successive
values in L1.
6. Press .
Grouping data using class intervalsWhen the data are spread over a wide range or there is a large amount of data, it is
helpful to group the scores into class intervals. The size of the class interval is impor-
tant when drawing up a frequency distribution table. In general, the choice for the size
of a class interval should lead to the formation of 5 to 10 groups.
Plotting ahistogramGraphics CalculatorGraphics Calculator tip!tip!
CASI
O
Plottinga
histogram
Y= 2nd [STATPLOT]
4: PlotsOff
STAT 1: Edit
2nd [STATPLOT] ENTER
1
ENTER
ZOOM 9: ZoomStat
WINDOW
TRACE
The following data are the results of testing the lives (in hours) of 100 torch batteries.
20, 31, 42, 49, 46, 36, 42, 25, 28, 37, 48, 49, 45, 35, 25, 42, 30, 23, 25, 26,
29, 31, 46, 25, 40, 30, 31, 49, 38, 41, 23, 46, 29, 38, 22, 26, 31, 33, 34, 32,
41, 23, 29, 30, 29, 28, 48, 49, 31, 49, 48, 37, 38, 47, 25, 43, 38, 48, 37, 20,
38, 22, 21, 33, 35, 27, 38, 31, 22, 28, 20, 30, 41, 49, 41, 32, 43, 28, 21, 27,
20, 39, 40, 27, 26, 36, 36, 41, 46, 28, 32, 33, 25, 31, 33, 25, 36, 41, 28, 33
a Choose a suitable class interval for the given data and present the results in
a frequency distribution table.
b Draw a histogram of the data.
c Add a polygon to the histogram.Continued over page
8WORKEDExample
582 M a t h s Q u e s t 8 f o r V i c t o r i a
THINK WRITE
a Choose a suitable size for the class
interval.
a
Obtain the range for the given data;
that is, subtract the smallest value
from the largest.
Range = largest value − smallest value
= 49 − 20
= 29
Divide the results obtained for the
range by 5 and round to the nearest
whole number.
Note: A class interval of 5 hours will
result in 6 groups.
Number of class intervals: = 5.8
≈ 6
Draw a frequency table and list the
class intervals in the first column,
beginning with the smallest value.
Note: The class interval 20–<25
includes hours ranging from and
including 20 to less than 25.
Systematically go through the data
and determine the frequency of each
class interval.
Calculate the total of the frequency
column.
b Rule and label a set of axes on graph
paper. Give the graph a title.
b
Add scales to the horizontal and
vertical axes.
Note: Leave a half interval at the
beginning and end of the vertical
axis.
Draw in the first column so that it
starts and finishes halfway between
class intervals and reaches a vertical
height of 12 units.
Repeat step 3 for each of the other
scores.
1
2
329
5------
4
5
Lifetime (hours) Tally
Frequency(f )
20–<25
25–<30
30–<35
35–<40
40–<45
45–<50
12
23
20
16
13
16
Total 100
| | | | | | | | | |
| | | | | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | |
| | | | | | | | | | | | |
| | | | | | | | | | |
| | | | | | | | | | | | |
6
1
Lifetime of torch batteries (hours)
Fre
quen
cy
25
20
15
10
5
00
20–<
25
25–<
30
30–<
35
35–<
40
40–<
45
45–<
50
Battery lifef
x
2
3
4
C h a p t e r 1 2 D a t a a n d g r a p h s 583
Histograms are used for displaying numerical data.
Histograms and frequency polygons
1 The following histogram displays the heights
of a group of students.
a Which is the most common height?
b Which is the least common height?
c How many students are taller than 156 cm?
d How many students are shorter than
156 cm?
e Which height occurred 18 times?
f How many students were included in the
survey?
THINK WRITE
c Mark the midpoints of the tops of the
columns obtained in the histogram
from part b.
c
Join the midpoints by straight line
intervals.
Close the polygon by drawing lines
at each end down to the class
interval (x) axis.
1
Lifetime of torch batteries (hours)
Fre
quen
cy
25
20
15
10
5
00
20–<
25
25–<
30
30–<
35
35–<
40
40–<
45
45–<
50
Battery lifef
x
2
3
1. A histogram must be drawn on graph paper and should include:
(a) a title
(b) clearly labelled and evenly scaled axes
(c) columns of equal width with no gaps between them
(d) columns that each straddle a score on the x-axis
(e) a half-interval left at each end of the x-axis.
2. Frequency polygons should include:
(a) the same scaled axes as the histogram
(b) straight lines that join the midpoints of the tops of the columns
(c) lines drawn (that close the polygon) at each end down to the score (x) axis.
3. Class intervals are used when:
(a) data are spread over a wide range
(b) there is a large amount of data.
4. The size of a class interval should lead to the formation of 5 to 10 groups.
5. Histograms are used for displaying numerical data.
remember
12D
Height (cm)
Fre
quen
cy
20
15
10
5
00 150 152 154 156 158 160 162
Student heightsf
x
Mathcad
Histo-gramsand
frequencypolygons
SkillSHEET
12.4
Producing afrequency
tablefrom a
frequencyhistogram
584 M a t h s Q u e s t 8 f o r V i c t o r i a
2 The following frequency distribution table represents the scores obtained by a group
of Year 8 students in a test.
a Draw a histogram that represents the data in the table above, using grid paper. To
decide on scaling for the axes, ask yourself: ‘What is the highest score (x-axis)?’
‘What is the highest frequency (y-axis)?’ Use the title: ‘Student Ratings’.
b Which is the most common score?
c Which is the least common score?
d How many students received a score of at least 5?
e How many students received a score of, at most, 3?
f How many students were included in the survey?
g Draw a frequency polygon of the data.
3 A quality control officer obtained random samples of bags of corn chips from the
production line and weighed them. Here are the data:
Corn chips: Net weight (grams)
252, 247, 249, 250, 248, 246, 251, 248, 250, 249, 246, 249, 247, 248, 247, 248, 249,
248, 250, 249, 250, 246, 247, 251, 248
a Sort the data into a frequency distribution table.
b How many packets of corn chips were in the sample?
c How many packets weighed less than the printed weight of 250 g?
d How many packets weighed more than the target weight?
e Present the data in the table as a histogram, and overlay a frequency polygon on it.
Be sure to label the graph and give it a title.
4 These are the results of a test out of 10 for a Year 8 class:
2, 6, 5, 9, 8, 7, 3, 6, 9, 4, 8, 8, 6, 7, 6, 4, 7, 8, 7, 8, 6, 7, 8, 5, 3, 9, 2, 6, 5, 8
a Present the data in a frequency distribution table.
b How many students were in the class?
c What was the most common test score?
d Give the highest and lowest test scores.
e Display the data as a histogram frequency polygon combination graph.
Questions 5 and 6 refer to the following information.
The number of hours of sleep during school week nights for a Year 8 class are recorded
below:
6, 9, 7, 8, 7, 8 , 6 , 8, 7 , 7 , 8, 8 , 6 , 8, 8, 7, 7 , 8, 9, 8
Score (x) Frequency (f )
1234567
2369
1174
Total
WORKED
Example
7
EXCE
L Spreadsheet
Histograms and frequency polygons
EXCE
L Spreadsheet
Histograms and frequency polygons (DIY)
SkillSH
EET 12.5
Presenting data in afrequency table
GC pr
ogram– TI
Univariatestatistics
GC pr
ogram– Casio
Univariatestatistics
1
2---
1
2---
1
2---
1
2---
1
2---
1
2---
1
2---
C h a p t e r 1 2 D a t a a n d g r a p h s 585
5 a Sort the information into a frequency table.
b Draw a histogram of the data.
c How many students were in the sample?
d How many students slept for at least 8 hours?
e How many students slept for fewer than 7 hours?
6
The number of students who slept between 6 and
7 hours inclusive is:
A 1 B 4 C 6
D 5 E 3
Questions 7 and 8 refer to the following information.
The amount of pocket money (in dollars) available to a random sample of 13-year-olds
each week was found to be as shown below:
10, 15, 5, 4, 8, 10, 4, 15, 5, 6, 10, 6, 5, 10, 8, 10, 5, 10, 10, 6
7 a Compile a frequency distribution table from the data.
b Draw a histogram that represents the given data.
c What was the most common amount of pocket money?
d How many received less than $8 per week?
8
The number of 13-year-olds who received at least $8 per week is:
A 9 B 3 C 6 D 5 E 11
9 The following data give the results of testing the lives (in hours) of 100
torch batteries:
25, 36, 30, 34, 21, 40, 36, 46, 29, 38, 20, 41, 34, 45, 25,
40, 31, 39, 24, 45, 27, 44, 23, 35, 47, 49, 20, 37, 43, 26,
35, 28, 48, 30, 20, 36, 41, 26, 32, 42, 21, 31, 45, 42, 26,
37, 33, 24, 45, 38, 36, 43, 21, 34, 38, 35, 28, 41, 30, 22,
29, 32, 39, 25, 44, 21, 35, 38, 41, 35, 30, 23, 37, 43, 33,
34, 28, 39, 22, 31, 35, 42, 38, 27, 36, 46, 28, 34, 37, 29,
24, 30, 39, 44, 31, 24, 36, 28, 47, 21
a Choose a suitable class interval for the given data and present the results on a
frequency distribution table.
b Draw a histogram of the data.
c Add a polygon to the histogram.
10 For each of the following data, choose a
suitable class interval and represent the result
on a frequency distribution table.
a The data below show the fat content
(%) of 30 types of biscuits selected
from a supermarket’s shelves:
6, 12, 1, 5, 8, 13, 20, 18, 12, 2, 25, 13,
18, 20, 8, 9, 17, 21, 7, 22, 30, 28, 12, 19,
29, 12, 28, 2, 7, 17
multiple choice
multiple choice
WORKED
Example
8
586 M a t h s Q u e s t 8 f o r V i c t o r i a
b The following data give the number of hours of television watched by a group of
28 students in a typical week:
16, 20, 5, 2, 60, 40, 13, 2, 25, 30, 45, 24, 12, 8, 10, 16, 9, 25, 0, 50, 16, 29, 32, 41,
30, 12, 12, 6
c Anna was required to measure the mass (in grams) of a variety of ingredients for
her home economics assignment. The following data represent Anna’s results in the
range of 0 g ≤ mass ≤ 250 g:
8, 29, 110, 56, 74, 128, 160, 205, 227, 16, 5, 61, 27, 130, 92, 35, 50, 230, 80, 160
d Nadia’s duties at the delicatessen require her to weigh out a number of products.
The following data represent Nadia’s results in the range of 250 g ≤ mass ≤ 500 g:
260, 300, 410, 289, 310, 278, 316, 480, 410, 270, 360, 492, 321, 325, 380, 252,
312, 291, 315, 280, 460, 400, 280, 265, 350, 290, 460, 370, 425, 310
Questions 1 to 3 refer to the column graph at right.
1 How many people support Hawthorn?
2 Which two clubs are supported by 50 people?
3 Which football club has the highest number
of supporters?
Questions 4 to 6 refer to the line graph at right.
4 After how much time was Marie furthest away
from her starting point?
5 What was her speed during the first two
minutes? (Remember: Speed = ).
6 What total distance did she travel?
Chris received the following results (out of 10) in his mathematics tests in semester 1:
5, 8, 7, 9, 10, 6, 8, 9, 8, 8, 5, 6, 7, 5
7 Organise the data into a frequency distribution table.
8 Display the data as a histogram frequency polygon combination graph.
9 What was the highest score Chris obtained?
10 Which score has the highest frequency?
WorkS
HEET 12.2
2
Fre
quen
cy
Favourite football clubs
80
60
40
20
0
Haw
thor
n
Car
lton
Ric
hmon
d
Col
lingw
ood
Dis
tance
(m
)
Marie’s trip100
80
60
40
20
00 1 2 3 4 5 6
Time (min)
distance
time-------------------
C h a p t e r 1 2 D a t a a n d g r a p h s 587
Stem-and-leaf plotsWhen displaying data, a stem-and-leaf plot may be used as an alternative to the
frequency distribution table. Each piece of data in a stem-and-leaf plot is made up of
two components: a stem and a leaf.
For example, the value 28 is made up of a tens component (the stem) and the units
component (the leaf) and would be written as:
It is important to provide a key when drawing up stem-and-leaf plots, as the plots may be
used to display a variety of data; that is, values ranging from whole numbers to decimals.
Stem Leaf
2 8
Prepare an ordered stem-and-leaf plot for each of the following sets of data.
a 129, 148, 137, 125, 148, 163, 152, 158, 172, 139, 168, 121, 134
b 1.6, 0.8, 0.7, 1.2, 1.9, 2.3, 2.8, 2.1, 1.6, 3.1, 2.9, 0.1, 4.3, 3.7, 2.6
Continued over page
THINK WRITE
a Rule two columns with the headings
‘Stem’ and ‘Leaf’.a Key: 12 1 = 121
Include a key to the plot that informs the
reader of the meaning of each entry.
Make a note of the smallest and largest
values of the data (that is, 121 and 172
respectively). List the stems in ascending
order in the first column (that is, 12, 13, 14,
15, 16, 17).
Note: The hundreds and tens components
of the number represent the stem.
Systematically work through the given
data and enter the leaf (unit component) of
each value in a row beside the appropriate
stem.
Note: The first row represents the interval
120–129, the second row represents the
interval 130–139 and so on.
Redraw the stem-and-leaf plot so that the
numbers in each row of the leaf column
are in ascending order.
Key: 12 1 = 121
1
Stem
12
13
14
15
16
17
Leaf
9 5 1
7 9 4
8 8
2 8
3 8
2
2
3
4
5
Stem
12
13
14
15
16
17
Leaf
1 5 9
4 7 9
8 8
2 8
3 8
2
9WORKEDExample
588 M a t h s Q u e s t 8 f o r V i c t o r i a
From worked example 9 it is evident that there are some advantages in displaying
grouped data in a stem-and-leaf plot compared with a frequency distribution graph.
All the original data are retained; therefore, it is possible to identify the smallest and
largest values as well as any repeated values. This cannot be done when values are
grouped in class intervals.
Stem-and-leaf plots also give a graphical representation of the data, as they resemble
histograms turned on their side.
Leaf
8
7
1
9
6
6
2
9
8
6
3
1
7
1 3
Stem 0 1 2 3 4
THINK WRITE
b Rule the stem and leaf columns and
include a key.b Key: 0 1 = 0.1
Make a note of the smallest and
largest values of the data (that is, 0.1
and 4.3 respectively). List the stems
in ascending order in the first column
(that is, 0, 1, 2, 3, 4).
Note: The units components of the
decimal represent the stem.
Systematically work through the
given data and enter the leaf (tenth
component) of each decimal in a row
beside the appropriate stem.
Note: The first row represents the
interval 0.1–0.9, the second row
represents the interval 1.0–1.9 and
so on.
Redraw the stem-and-leaf plot so
that the numbers in each row of the
leaf column are in ascending order to
produce an ordered stem-and-leaf
plot.
Key: 0 1 = 0.1
1
Stem
0
1
2
3
4
Leaf
8 7 1
6 2 9 6
3 8 1 9 6
1 7
3
2
3
4
Stem
0
1
2
3
4
Leaf
1 7 8
2 6 6 9
1 3 6 8 9
1 7
3
A stem-and-leaf plot allows:
1. all the original data to be retained
2. a graphical representation of the data to be seen as it resembles a histogram
turned on its side.
remember
C h a p t e r 1 2 D a t a a n d g r a p h s 589
Stem-and-leaf plots
1 The following stem-and-leaf plot gives the age of members
of a theatrical group.
a How many people are in the theatrical group?
b What is the age of the youngest member of the group?
c What is the age of the oldest member of the group?
d How many people are over 30 years of age?
e What age is the most common in the group?
f How many people are over 65 years of age?
2 The stem-and-leaf plot at right represents
data for the height of trees (in cm) in
a nursery.
a Redraw this stem-and-leaf plot as an
ordered stem-and-leaf plot.
b Write the tenth number in the
ordered stem-and-leaf plot.
3 The following data give the number of fruit that
have formed on each of 40 trees in an orchard:
29, 37, 25, 62, 73, 41, 58, 62, 73, 67,
47, 21, 33, 71, 92, 41, 62, 54, 31, 82,
93, 28, 31, 67, 29, 53, 62, 21, 78,
81, 51, 25, 93, 68, 72, 46, 53, 39,
28, 40
Prepare an ordered stem-and-leaf
plot that displays the data.
4 The number of errors made each
week by 30 machine operators is
recorded below:
12, 2, 0, 10, 8, 16, 27, 12, 6, 1,
40, 16, 25, 3, 12,
31, 19, 22, 15, 7, 17, 21, 18, 32,
33, 12, 28, 31, 32, 14
Prepare an ordered stem-and-leaf
plot that displays the data.
5 Prepare an ordered stem-and-leaf plot
for each of the following sets of data:
a 132, 117, 108, 129, 165, 172, 145, 189,
137, 116, 152, 164, 118
b 131, 173, 152, 146, 150, 171, 130, 124, 114
c 196, 193, 168, 170, 199, 186, 180, 196, 186, 188,
170, 181, 209
d 207, 205, 255, 190, 248, 248, 248, 237, 225, 239, 208, 244
e 748, 662, 685, 675, 645, 647, 647, 708, 736, 691, 641, 735
12EKey: 2 4 = 24
Stem123456
Leaf7 8 8 9 92 4 7 91 3 3 80 2 2 2 6 65 74
Key: 23 7 = 237
Stem2021222324
Leaf7 4 2 92 0 79 3 3 8 60 2 1 2 1 65
SkillSHEET
12.6
Presenting dataas a stem-and-leaf
plot
WORKED
Example
9a
590 M a t h s Q u e s t 8 f o r V i c t o r i a
6 Prepare an ordered stem-and-leaf plot for each of the following sets of data:
a 1.2, 3.9, 5.8, 4.6, 4.1, 2.2, 2.8, 1.7, 5.4, 2.3, 1.9
b 2.8, 2.7, 5.2, 6.2, 6.6, 2.9, 1.8, 5.7, 3.5, 2.5, 4.1
c 7.7, 6.0, 9.3, 8.3, 6.5, 9.2, 7.4, 6.9, 8.8, 8.4, 7.5, 9.8
d 14.8, 15.2, 13.8, 13.0, 14.5, 16.2, 15.7, 14.7, 14.3, 15.6, 14.6, 13.9,
14.7, 15.1, 15.9, 13.9, 14.5
e 0.18, 0.51, 0.15, 0.02, 0.37, 0.44, 0.67, 0.07
Mean, median and modeWe collect data in order to find out what is going
on now in our area of interest. Then we can
interpret the results to make decisions and pre-
dictions such as: Where should the new school
be built? What do we expect its enrolment to be
by 2010? When do most teenagers watch tele-
vision? What food should be sold at the school
canteen? If sales continue to rise at this rate,
what profits can we expect next quarter?
Simple calculations based on collected data can help give us typical values, or values
that show how the data cluster. These typical values are commonly referred to as aver-
ages. We will look at 3 different types of averages used in interpreting data: mean,
median and mode.
Mean
The mean or average of a set of scores is the sum of all the scores divided by the
number of scores. It is denoted by the symbol (pronounced x bar).
WORKED
Example
9b
x
Jan’s basketball scores were: 18, 24, 20, 22, 14, 12.
What was his mean score? Calculate your answer,
correct to 1 decimal place.
THINK WRITE
Calculate the total of the basketball
scores.
Total score = 18 + 24 + 20 + 22 + 14 + 12
= 1101
10WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 591
THINK WRITE
Count the number of basketball scores. Number of scores = 6
Define the rule for the mean. Mean =
Substitute the known values into the rule. =
Evaluate, rounding to 1 decimal place.
Note: The mean is often not one of the
given scores.
= 18.333 33 …
≈ 18.3
2
3total score
number of scores-----------------------------------------
4 x110
6---------
5
Calculate the mean of the frequency distribution data given below.
Score (x) 1 2 3 4 5 6
Frequency (f ) 3 2 4 0 1 5
THINK WRITE
Rearrange the rows as columns and
include an extra column headed:
‘Score × frequency (x × f )’.
Enter the information into the third
column; that is, the score of 1 occurred 3
times. Therefore, x × f = 1 × 3 = 3.
The score of 2 occurred 2 times.
Therefore, x × f = 2 × 2 = 4.
Continue this process for each pair of
data.
Determine the total of the ‘Frequency’
column. This shows how many scores
there are altogether.
Determine the total of the ‘Score ×
frequency’ column. This shows the
overall value of all the scores.
Define the rule for the mean. Mean =
Substitute the known values into the rule. =
Evaluate the answer to 1 decimal place.
Note: The mean is often not one of the
given scores.
= 3.6
1
Score(x)
Frequency(f )
Score × frequency
(x × f )
1 3 1 × 3 = 3
2 2 2 × 2 = 4
3 4 3 × 4 = 12
4 0 4 × 0 = 0
5 1 5 × 1 = 5
6 5 6 × 5 = 30
Total 15 54
2
3
4
5total of score frequency values×
total frequency values------------------------------------------------------------------------------
6 x54
15------
7
11WORKEDExample
592 M a t h s Q u e s t 8 f o r V i c t o r i a
Median
The median is the middle score for an odd number of scores and the average of the
two middle scores for an even number of scores.
Alternatively, if a set of data contains n scores, the median is given by the th
score.
To obtain the median, the scores must be arranged in numerical order.
Note: For sets of data containing an odd number of scores, the median will be one of
the actual scores, while for the sets with an even number of scores, the median will be
positioned halfway between the two scores.
n 1+
2------------
Find the median of the scores:
a 10, 8, 11, 5, 17 b 9, 3, 2, 6, 3, 5, 9, 8.
THINK WRITE
a Arrange the values in ascending order. a 5, 8, , 11, 17
Select the middle value.
Note: There are an odd number of scores, that
is, 5. Hence, the third value is the middle
number or median.
Alternatively, use the rule , where n = 5
gives the position of the median. The location
of the median is ; that is, the 3rd
score.
Answer the question. The median of the scores is 10.
b Arrange the values in ascending order. b 2, 3, 3, , , 8, 9, 9
Select the middle values.
Note: There are an even number of scores that
is, 8. Hence, the fourth and fifth values are the
middle numbers, or median.
Again the rule could be used to locate
the position of the median.
Obtain the average of the two middle values. Median =
=
= 5 (or 5.5)
Answer the question.
Note: The median in this case is not one of the
actual scores.
The median of the score is 5 or
5.5.
1 10
2
n 1+
2------------
5 1+
2------------ 3=
3
1 5 6
2
n 1+
2------------
35 6+
2------------
11
2------
1
2---
41
2---
12WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 593
To find the median of a list of numbers press
, arrow across to and select .
Press and enter the scores separated by a
comma. Press to close the brackets and then
press .
You can find the mean easily by following the same
steps but selecting rather than .
When determining the median, recall:
1. there are as many scores above the median as there are below it.
2. for an even number of scores, the median may not be one of the listed scores.
CASI
O
Findingthe
median(or themean)
Finding the median(or the mean)
Graphics CalculatorGraphics Calculator tip!tip!
2nd
[LIST] MATH 4: Median(
2nd [ ]
2nd [ ]
ENTER
3: Mean( 4: Median(
Find the median of the data presented in the following stem-and-leaf plots.
a Key: 14 5 = 145 b Key: 25 3 = 253
Continued over page
Stem1415161718
Leaf2 7 8 82 41 3 9 90 2 2 2 6 65
Stem2122232425
Leaf3 0 64 3 2 4 9 79 3 1 52 6 2 0 6 75 7
26 4 7 3 1 1
THINK WRITE
a Check that the given stem-and-leaf
plot is ordered.
a The stem-and-leaf plot is ordered.
Count the pieces of data and
determine the middle value.
There are 17 pieces of data. Therefore, the
middle value is the ninth term.
Answer the question. The median is 169.
b Check if the stem-and-leaf plot is
ordered. It is not.
b The stem-and-leaf plot is not ordered.
Order the stem-and-leaf plot. Key: 25 3 = 253
1
2
3
1
2
Stem
21
22
23
24
25
26
Leaf
0 3 6
2 3 4 4 7 9
1 3 5 9
0 2 2 6 6 7
5 7
1 1 3 4 7
13WORKEDExample
594 M a t h s Q u e s t 8 f o r V i c t o r i a
Mode
The mode is the most common score in a set of data. It is the score with the highest
frequency. It measures clustering of scores.
Some sets of scores have more than one mode or no mode at all; that is, there is no
score that corresponds to the highest frequency, as all values occurred once only.
THINK WRITE
Count the pieces of data and
determine the middle values.
There are 26 pieces of data. Therefore, the
two middle values are the thirteenth and
fourteenth terms.
Add the two middle terms and divide
by 2; that is, obtain the average of
them.
Median =
=
= 239.5
Answer the question. The median is 239.5.
3
4239 240+
2------------------------
479
2---------
5
Find the mode of the following scores:
a 5, 7, 9, 8, 5, 8, 5, 6 b 10, 8, 11, 5, 17 c 9, 3, 2, 6, 3, 5, 9, 8.
THINK WRITE
a Look at the set of data and circle any
values that have been repeated.
a , 7, 9, , , , , 6
Choose the values that have been
repeated the most.
The number 5 occurs 3 times.
Answer the question. The mode for the given set of values is 5.
b Look at the set of data and circle any
values that have been repeated.
b 10, 8, 11, 5, 17
No values have been repeated.
Answer the question.
Note: No mode is not the same as
having a mode which equals 0.
The following set of data has no mode, since
none of the scores correspond to a highest
frequency. Each of the numbers occur only
once.
c Look at the set of data and circle any
values that have been repeated.
c , , 2, 6, , 5, , 8
Choose the values that have been
repeated the most.
The number 3 occurs twice. The number 9
occurs twice.
Answer the question. The modes for the given set of values are 3
and 9.
1 5 8 5 8 5
2
3
1
2
1 9 3 3 9
2
3
14WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 595
To enter data into the TI–83 graphics calculator, press and select 1:Edit. Type
your data in the L1 column. Press after each entry. (You can type over existing
numbers or press to delete.) To find the mean and median of your data, press
; then arrow across to select CALC and 1:1 – Var Stats and press . The
mean is the value given for . Scroll down to find the median. This is shown as Med=.
You will also be able to view the lowest value, minX, and the highest value, maxX, of
the data.
To assist you in finding the mode, you can sort the data list into ascending order.
Press select 2:Sort A(; then press L1 (to sort List 1) and press .
(Note that selecting 3:Sort D( sorts the data in descending order.) You can view the
sorted data by pressing 1:Edit. (Highlighting L1 shows the full list of the data
across the bottom of the screen for easier viewing.)
The screens below show the summary statistics for the data: 3, 4, 8, 4, 5, 6.
Mean, median and mode are all types of averages.
1 Provide two examples of situations where each type of average is the best one to
use.
2 At times, these measures are used to mislead people. Describe some situations
where this may occur.
3 Write a paragraph to describe the difference between the three types of average.
COMMUNICATION What is the difference?
Finding the mean, medianand mode
Graphics CalculatorGraphics Calculator tip!tip!
CASI
O
Findingthe
mean,median
andmode
STAT
ENTER
DEL
STAT ENTER
x
STAT 2nd ENTER
STAT
Mean = 5
Median = 4.5
Mode = 4
596 M a t h s Q u e s t 8 f o r V i c t o r i a
Mean, median and mode
1 Caroline’s basketball scores were: 28, 25, 29, 30, 27, 22. What was her mean score?
Give the answer correct to 1 decimal place.
2 Find the mean (average) of each set of the following scores. Give the answers correct
to 2 decimal places.
a 1, 2, 3, 4, 7, 9
b 2, 7, 8, 10, 6, 9, 11, 4, 9
c 3, 27, 14, 0, 2, 104, 36, 19, 77, 81
d 4, 8.4, 6.6, 7.0, 7.5, 8.0, 6.9
3 Francesca’s soccer team has the following
goals record this season:
2, 0, 1, 3, 1, 2, 4, 0, 2, 3
a What total number of goals have they
scored?
b How many games have they played?
c Find the team’s average score.
4
Frisco’s athletics coach timed 5 consecutive 200 m training runs. He recorded times
of 25.1, 23.9, 24.8, 24.5 and 27.3 seconds. His mean 200 m time (in seconds) is:
A 24.60 B 25.20 C 25.12 D 25.42 E 26.12
5 An Olympic figure skater was given these scores by the panel of judges:
4.8, 4.6, 4.5, 4.7, 4.8, 4.9, 4.2, 4.0, 4.8.
Find the average score correct to 1 decimal place.
6 Two Year 8 groups did the same mathematics test. Their results out of 10 were:
Group A: 5, 8, 7, 9, 6, 7, 8, 5, 4, 2
Group B: 5, 6, 4, 5, 9, 7, 8, 8, 9, 7
a Which group had the highest mean?
b Compare the spread of the marks for the groups.
1. To determine the mean, , of values in a list, obtain the total of all the scores
and divide by the number of scores.
2. To determine the mean of values in a table, add the (x × f ) column, and divide
by the total of the frequency column ( f ).
3. For an odd number of scores arranged in numerical order, the median is the
middle score. For an even number of scores arranged in numerical order, the
median is the average of the two middle scores. There are as many scores
above the median as there are below it.
4. The median can be located using the rule , where n represents the number
of data.
5. The mode is the most common score.
x
n 1+
2------------
remember
12F
SkillSH
EET 12.7
Findingthemean
WORKED
Example
10
GC pr
ogram– TI
Univariate statistics
GC pr
ogram– Casio
Univariate statistics
Mat
hcad
Mean
multiple choice
C h a p t e r 1 2 D a t a a n d g r a p h s 597
7 A third Year 8 group had the following results in the same test as in question 6:
5, 7, 8, 4, 6, 8, 5, 9, 8
a What is the average score of this group?
b What must a tenth student (who was originally absent) score to bring this group’s
average to 7?
8 Calculate the mean of this frequency distribution.
9 Calculate the mean of this frequency distribution.
10 A survey of the number of occupants in each house in
a street gave the following data:
2, 5, 1, 6, 2, 3, 2, 4, 1, 2, 0, 2, 3, 2, 4, 5, 4, 2, 3, 4
Prepare a frequency distribution table with an x × f
column and use it to find the average number of
people per household.
11 These scores show the number of people in each apartment in a block of flats. Use
a frequency table to calculate the mean number of people per unit, correct to 1
decimal place.
1, 3, 2, 4, 2, 1, 3, 5, 3, 2, 4, 1, 3, 2, 1, 2
12 The mean of 10 scores is 8. What is the total of all the scores?
13 The mean of 5 scores is 7.2.
a What is the sum of the scores?
b If four of the scores are 9, 8, 7 and 5, what is the fifth?
14 Find the median of the following scores:
a 5, 5, 7, 12, 13 b 28, 13, 17, 21, 18, 17, 14
15 Find the median of the following scores:
a 2, 52, 46, 52, 48, 52, 48 b 4, 1.5, 1.7, 2.0, 1.8, 1.5, 1.7, 1.8, 1.9
16 Find the median of the data presented in the following stem-and-leaf plots.
a Key: 1 5 = 15 b Key: 24 7 = 247 c Key: 17 4 = 174
Stem Leaf Stem Leaf Stem Leaf
1 1 2 7 8 9 24 2 7 15 6 2 4
2 2 8 25 2 4 6 6 8 16 8 6 1 3 9
3 1 3 7 9 26 0 1 3 5 9 17 0 2 1 8 6 7 3 4
4 0 1 2 6 28 5 6 6 8 18 4 1 5 2 7 1
17 For each set of scores in questions 14 and 15, find the mode.
Score (x) 1 2 3 4 5
Frequency (f ) 4 3 2 1 0
Score (x) 6 7 8 9 10
Frequency (f ) 2 8 3 4 2
EXCEL Spreadsheet
Meanfrom a
frequencytable
EXCEL Spreadsheet
Mean froma frequencytable (DIY)
WORKED
Example
11
SkillSHEET
12.8
Arranging aset of data
in ascendingorder
SkillSHEET
12.9
Findingthe location
of themedian
SkillSHEET
12.10
Findingthe middlescore of aset of data
SkillSHEET
12.11
Finding the middlescore of dataarranged in a
stem-and-leaf plot
EXCEL Spreadsheet
Median(DIY)
WORKED
Example
12a
WORKED
Example
12b
WORKED
Example
13
EXCEL Spreadsheet
Median
EXCEL Spreadsheet
Mode(DIY)
WORKED
Example
14
EXCEL Spreadsheet
Mode
SkillSHEET
12.12
Finding the score in a dataset that occurs most frequently
598 M a t h s Q u e s t 8 f o r V i c t o r i a
Questions 18 and 19 refer to the following set of scores:
1, 1, 1, 4, 4, 5, 5, 6, 3, 3, 7, 6, 5, 4, 6, 2, 1, 8
18
The median of the given scores is:
A 1 B 4.5 C 4 D 5 E 8
19
The mode of the given scores is:
A 5 B 6 C 4 D 3 E 1
20 Over 10 matches, a soccer team scored the following number of goals:
2, 3, 1, 0, 4, 5, 2, 3, 3, 4.
a What was the most common number of goals scored?
b What was the median number of goals scored?
c In this case, does the mode or the median give a score that shows a typical
performance?
21 Here are Tiger Woods’s scores (numbers of strokes) hole by hole for the first 9 holes
of a major golf tournament.
a How many strokes were most commonly hit?
b What was his median score?
c As Woods prepares to tee off towards the next hole, how many strokes could the
crowd expect him to take to complete the hole? Discuss factors which could
influence the outcome.
22 A small business pays the following annual wages (in thousands of dollars) to its
employees: 18, 18, 18, 18, 26, 26, 26, 40, 80.
a What is the mode of the distribution?
b What is the median wage?
c What is the mean wage?
d Which measure would you expect the employee’s union to use in wage negotiations?
e Which might the boss use in such negotiations?
Hole number 1 2 3 4 5 6 7 8 9
Score 4 4 3 2 4 3 3 2 4
multiple choice
multiple choice
GAM
E time
Data and graphs — 002
WorkS
HEET 12.3
MA
TH
S Q
UEST
CHALL
EN
GE
CHALL
EN
GE
MA
TH
S Q
UEST
1 Find five numbers that have a mean of 10 and a median of 12.
2 The mean of 5 different test scores is 15. What are the largest and
smallest possible test scores, given that the median is 12? All test scores
are whole numbers.
3 The mean of 5 different test scores is 10. What are the largest and smallest
possible values for the median? All test scores are whole numbers.
4 The mean of 9 different test scores that are whole numbers and range
from 0 to 100 is 85. The median is 80. What is the greatest possible range
between the highest and lowest possible test scores?
C h a p t e r 1 2 D a t a a n d g r a p h s 599
Two important inventions of 1862
12
24 14 16 13 11 3 15
5
18 23 23 7 14 16 23 128
16 13 22 4 12 8
11 18 25 13
16 12 19 12 27
26 11 14 13 16
20
11
A = mean =
C = median =
D = mode =
17 15 16 12
1417
18
19
20
21
22
23
24
1
1
0
1
0
1
3
4
Age of university students (yrs)
Age Frequency14 14 22 22
15 15 21 21 16
20 20 18 19 19
9
11
22
12 8 13 3 7 23 8 12 18 23 2 13 4
4 15 6 18 23 8 5 16 13
6 8 22 3 14 19 18 23 7
11 17 25 2
2 20 21 20 1
2 9 10 25 21
I = mean =
K = median =
L = mode = M = mean =
N = median =
P = mode =
S = mean =
T = median =
U = mode =
10 3 5 2
3 4 5 12
3 5 3
W = mean =
A = median =
I = mode =
10 6 4 9 12
6 10 5 6 7
11 12 6
E = mean =
G = median =
H = mode =
Calculatethe mean, median
and mode for each setof data to find the
puzzle’s code.
600 M a t h s Q u e s t 8 f o r V i c t o r i a
Measures of spreadIn analysing a set of scores, it is helpful to see not only how the scores tend to cluster,
or how the middle of the set looks, but also how they spread or scatter. Two classes may
have the same average mark, but the spread of scores may differ considerably.
Range
The range of a set of scores is the difference between the highest and lowest scores.
The owner of a clothing store sells denim jeans in the
following sizes.
12 12 14 16 18 12 14 10 8 12
12 14 8 12 18 10 10 10 12 14
What is the most typical size of denim jeans sold in this
store? Which size jeans should the store stock most of? Use
the following questions to guide you to an answer.
1 Place the data in a frequency distribution table including
a column for f × x and cumulative frequency.
2 Use the table to find the mean size of denim jeans sold.
3 Use the table to find the median size of denim jeans sold.
4 Which size of denim jeans sold is the mode?
5 Which of the mean, median and mode is most useful to
the owner of the clothing store? Explain your answer.
COMMUNICATION Denim blues
Find the range of the following sets of data.
a 7, 3, 5, 2, 1, 6, 9, 8.
b x 7 8 9 10
f 1 3 5 2
THINK WRITE
a Obtain the highest and lowest values. a Highest value = 9
Lowest value = 1
Define the range. Range = highest value − lowest value
Substitute the known values into the
rule.
= 9 − 1
Evaluate. = 8
Answer the question. The set of values has a range of 8.
1
2
3
4
5
15WORKEDExample
C h a p t e r 1 2 D a t a a n d g r a p h s 601
Mean absolute difference
When the range and mean of two data sets is exactly the same, information regarding
the measure of spread is limited. Another way to measure the spread of results is to com-
pare how far values are from the mean. This shows whether the range of values has been
caused by a few extreme values or whether values are spread consistently over the range.
The mean absolute difference can be obtained by following the given steps:
1. Subtract the mean from each data value.
Note: The calculation for mean absolute difference is performed in a pair of vertical
bars and is called the modulus. It means that you ignore the sign of the number. For
example, −7 = 7 and 7 = 7; that is, the absolute value (or modulus) of −7 is 7 and
the absolute value (or modulus) of −7 is 7.
2. Calculate the mean (average) of the absolute differences; that is,
(a) add each of the absolute differences
(b) divide the sum of the absolute differences by the number of data values.
The larger the mean absolute difference, the more spread out the data is.
THINK WRITE
b Obtain the highest and lowest values.
Note: Consider the values only, not the
frequencies.
b Highest value = 10
Lowest value = 7
Define the range. Range = highest value − lowest valueSubstitute the known values into the rule. = 10 − 7Evaluate. = 3Answer the question. The frequency distribution table data
have a range of 3.
1
2
3
4
5
Calculate the mean absolute difference (correct to 1 decimal place) of the following data:
2, 2, 3, 4, 5, 5, 7; the mean is 4.
THINK WRITE
Subtract the mean from each data value, and
calculate the modulus or absolute value of
this number.
Absolute difference: 2 – 4=– 2= 2
Absolute difference: 2 – 4=– 2= 2
Absolute difference: 3 – 4=– 1= 1
Absolute difference: 4 – 4=0= 0
Absolute difference: 5 – 4=1= 1
Absolute difference: 5 – 4=1= 1
Absolute difference: 7 – 4=3= 3Calculate the mean (average) of the absolute
differences; that is,
(a) add each of the absolute differences
(b) divide the sum of the absolute
differences by the number of data values.
Mean absolute difference
=
=
Evaluate and simplify. = 1.428 571 429Round the answer to 1 decimal place. ≈ 1.4
1
2
2 2 1 0 1 1 3+ + + + + +
7-----------------------------------------------------------
10
7------
3
4
16WORKEDExample
602 M a t h s Q u e s t 8 f o r V i c t o r i a
Calculate the mean absolute difference (correct to 2 decimal places) of the following data;
the mean is 3.5.
Score (x) Frequency (f)
0 1
1 1
2 8
3 13
4 16
5 11
THINK WRITE
Draw a table with 4 columns. The
column headings are, in order,
‘Score (x)’, ‘Absolute difference,
x − ’, ‘Frequency (f)’ and
‘Absolute difference × frequency,
x − × f ’.
Note: Remember is the mean of
the data set.
Subtract the mean ( ) from each
score value (x) leaving the answer as
a positive number.
Enter the absolute difference values
into the appropriate column.
Multiply the absolute difference
values by the corresponding
frequencies.
Enter the values obtained in step 4
into the appropriate column.
Determine the total for the frequency
column.
Determine the total for the x −
× f column.
Define the rule for the absolute mean
difference.
Mean absolute difference
=
Substitute the known values into the
rule.=
Evaluate and simplify, rounding
answer to 2 decimal places.
= 0.98
1
x
x
x
2 x
3
4
5
6
7 x
8
total absolute difference frequency values×total frequency values
---------------------------------------------------------------------------------------------------------
9 49
50------
10
17WORKEDExample
Score(x)
AbsolutedifferenceΩx - Ω
Frequency(f)
Absolutedifference
×frequency
Ωx - Ω ¥ f
0 0 – 3.5 = 3.5 1 3.5 × 1 = 3.5
1 1 – 3.5 = 2.5 1 2.5 × 1 = 2.5
2 2 – 3.5 = 1.5 8 1.5 × 8 = 12
3 3 – 3.5 = 0.5 13 0.5 × 13 = 6.5
4 4 – 3.5 = 0.5 16 0.5 × 16 = 8
5 5 – 3.5 = 1.5 11 1.5 × 11 = 16.5
Total 50 49
x x
C h a p t e r 1 2 D a t a a n d g r a p h s 603
The results of a test marked out of 20, given to three Year 8 classes, were collated and
organised in the following table.
a What were lowest and highest marks in each class?
b Can the range be used to indicate which class had the most consistent marks? Explain
your answer.
c Which class had the most consistent results? Explain your answer.
Class
A B C
Mean test score 10 10 10
Range 20 20 20
Mean absolute difference 2 10 5
THINK WRITE
a Read the question carefully and
highlight any relevant information.
a The test is marked out of 20 and the range is
equal to 20.
Range = highest score − lowest value
= 20 − 0
= 20
Answer the question. For each class, the lowest mark was 0 and
the highest mark was 20.
b Compare the range values for each class
and answer the question.
b In this case the range cannot be used to
distinguish between the three groups as each
class had the same mean and range values.
c Compare the mean absolute
difference values for each class.
c
Select the class with the smallest
mean absolute difference value and
answer the question.
Note: The smallest mean absolute
difference value corresponds to more
consistent results, because the data
are spread out less.
Class A has the smallest mean absolute
difference value. The smallest mean absolute
difference value corresponds to more
consistent test results, because the scores are
spread out less. Therefore, Class A’s test
results are the most consistent, followed by
Class C. Class B’s test results are the least
consistent.
1
2
1
2
18WORKEDExample
1. The range is the difference between the highest and lowest scores.
2. The mean absolute difference is the average amount of variation of spread from
the mean.
remember
604 M a t h s Q u e s t 8 f o r V i c t o r i a
Measures of spread
1 Find the range of the following scores:
a 5, 5, 7, 12, 13 b 28, 13, 17, 21, 18, 17, 14
c 2, 52, 46, 52, 48, 52, 48 d 4, 1.5, 1.7, 2.0, 1.8, 1.5, 1.7, 1.8, 1.9
2 Find the range of the following sets of data.
3 Find the range of the data presented in the following stem-and-leaf plots.
a Key: 1 5 = 15 b Key: 24 7 = 247 c Key: 17 4 = 174
Stem Leaf Stem Leaf Stem Leaf
1 1 2 7 8 9 24 2 7 15 6 2 4
2 2 8 25 2 4 6 6 8 16 8 6 1 3 9
3 1 3 7 9 26 0 1 3 5 9 17 0 2 1 8 6 7 3 4
4 0 1 2 6 28 5 6 6 8 18 4 1 5 2 7 1
4 Calculate the mean absolute difference (correct to 1 decimal place) of the following
data: 4, 5, 6, 6, 7, 8; the mean is 6.
5 The number of sunny days per week over a 10-week period were: 4, 4, 2, 2, 1, 5, 6, 6,
7, 3.
a Find the mean number of sunny days per week.
b Find the mean absolute difference.
6 For each group of numbers calculate the mean and the mean absolute difference
(correct to 2 decimal places where appropriate).
a 16, 17, 18, 18, 20
b 1, 4, 5, 6, 11, 12
c 95, 120, 115, 122, 114, 113, 112
7 Complete the following table and calculate the mean absolute difference.
a x 6 7 8 9 10 b x 1 2 3 4 5 6
f 1 5 10 7 3 f 7 9 6 8 10 10
c x 5 10 15 20 d x 110 111 112 113 114
f 1 5 10 7 f 2 2 2 3 3
Score(x)
AbsolutedifferenceΩx - Ω
Frequency(f)
Absolute difference¥ frequencyΩx - Ω ¥ f
4 4 – 5.8 = 11 1.8 × 11 = 3.5
5 5 – 3.5 = 0.8 13 2.5 × 13 = 10.4
6 3.5 – 5.8 = 1.5 0.2 × 10 = 12
7 3.5 – 3.5 = 0.5 7 1.2 × 13 = 8.4
8 9 2.2 × 16 = 8
Total 50 60.4
12GWORKED
Example
15a
WORKED
Example
15b
Mat
hcad
Median,mode andrange
GC pr
ogram– TI
Univariatestatistics
GC pr
ogram– Casio
Univariatestatistics
WORKED
Example
16
x x
C h a p t e r 1 2 D a t a a n d g r a p h s 605
8 Calculate the mean absolute difference of the following data (correct to 2 decimal
places where appropriate).
a The mean is 9. b The mean is 3.47.
9 For each of the following tables of values, find:
i the mean ii range iii mean absolute difference.
Answers should be correct to 2 decimal places where appropriate.
a
b
c
d
10 The results of a test marked out of 30, given to three Year 8 classes, were collated and
organised in the following table.
a What were the lowest and highest marks in each class?
b Can the range be used to indicate which class had the most consistent marks?
Explain your answer.
c Which class had the most consistent results? Explain your answer.
Score (x) Frequency (f)
2 2
3 1
6 3
8 8
10 37
x 8.2 10.4 11.2 11.9 13.7 14.8
f 8 4 11 16 4 7
x 107.1 113.7 123.5 128.3 136.4 149.5
f 4 3 5 8 3 2
x 0.08 0.17 0.36 0.66 0.73
f 23 17 18 28 14
x 40.2 40.3 40.9 41.9 42.4
f 10 7 6 2 5
Class
A B C
Mean test score 15 15 15
Range 30 30 30
Mean absolute difference 1.5 6.4 4.2
WORKED
Example
17
WORKED
Example
18
Score (x) Frequency (f)
1.5 14
2.2 4
3.6 10
4.2 13
5.9 9
606 M a t h s Q u e s t 8 f o r V i c t o r i a
11 Samples of 10 icy-pole sticks made by three separate machines were collected and
entered in the following table. All lengths are given in millimetres.
a For each machine find:
i the mean length of icy-pole sticks
ii the range
iii the mean absolute difference.
Answers should be correct to 2 decimal places where appropriate.
b Comment on the range of lengths produced.
c If the length of icy-pole sticks is required to be approximately 126 mm, which
machine is the most consistent? Explain your answer.
12 The following data represent the mathematics exam results (as percentages) for 28 Year
8 students:
65, 70, 67, 82, 71, 25, 83, 78, 58, 72, 94, 66, 86, 73, 71, 31, 71, 87, 65, 76, 86, 66, 98,
74, 84, 96, 100, 73
a Present the data as an ordered stem-and-leaf plot.
b Find the median result.
c Find the mode.
d Find the range.
e Comment on the results obtained by the class.
Analysing dataTo understand what information the data give, and perhaps to draw conclusions from it,
we must appreciate what each statistical measure does.
Machine A 129.9 124.9 127.4 122.1 126.2 129.7 124.9 120.3 124.7 128.8
Machine B 127.7 121.8 127.5 120.4 128.5 130.0 127.4 129.9 125.2 124.5
Machine C 129.6 122.8 126.3 125.8 120.2 129.8 123.8 127.9 129.2 127.6
Statisticalmeasures Definition and purpose
Mode The most common score or category. It tells us nothing about the
rest of the data. Data may have no mode, one mode or more than
one mode.
Median The score in the exact middle of the values placed in numerical
order. It tells us nothing about the rest of the data. It is unaffected
by exceptionally large or small scores.
Mean The sum of all the scores divided by the number of scores. It is
affected by exceptionally large or small scores.
Range The difference between the highest score and the lowest score. It
shows how far the scores are spread apart. It is particularly useful
when combined with the mean or the median.
Mean absolute
difference
The mean of the absolute value of the difference between each data
value and the mean. It shows us the average spread of data from the
mean of the data. It is useful when combined with the mean.
C h a p t e r 1 2 D a t a a n d g r a p h s 607
Explain which statistical measure is referred to in these statements.
a The majority of people surveyed prefer Activ-8 sports drink.
b The ages of fans at the Rolling Stones concert varied from 8 to 80.
c The average Australian family has 2.5 children.
THINK WRITE
a Write the statement and highlight the
key word(s).
a The majority of people surveyed prefer
Activ-8 sports drink.
Relate the highlighted word to one
of the statistical measures.
Majority implies most, which refers to the
mode.
Answer the question. This statement refers to the mode.
b Write the statement and highlight the
key word(s).
b The ages of fans at the Rolling Stones
concert varied from 8 to 80.
Relate the highlighted word to one
of the statistical measures.
The statement refers to the range of fans’
ages at the concert.
Answer the question. This statement refers to the range.
c Write the statement and highlight the
key word(s).
c The average Australian family has 2.5
children.
Relate the highlighted word to one
of the statistical measures.
The statement deals with surveying the
population (census) and finding out how
many children are in each family.
Answer the question. This statement refers to the mean.
1
2
3
1
2
3
1
2
3
19WORKEDExample
Elio’s batting scores in last year’s cricket series were 65, 30, 0, 0, 0, 80. Gaetano’s scores
were 0, 30, 30, 80, 25, 20 in the same matches.
a Calculate the range for each. Does this show that both had equal results?
b Find the mean absolute difference for each. Does this give a measure of their abilities?
c What combination of statistics is needed to give a better measure of their abilities?
Continued over page
THINK WRITE
a Calculate the range for each person’s
batting scores by subtracting the
lowest score from the highest score.
a Range = highest value – lowest value
Elio: Range = 80 – 0
Elio: Range = 80
Gaetano: Range = 80 – 0
Gaetano: Range = 80
Compare the ranges and answer the
question.
Elio’s and Gaetano’s range values are the
same. This does not show that they have
equal results as the range gives no
information of their other scores.
1
2
20WORKEDExample
608 M a t h s Q u e s t 8 f o r V i c t o r i a
THINK WRITE
b Calculate the mean of Elio’s batting
scores:
b
(a) Calculate the total of the batting
scores.
Elio’s total score = 65 + 30 + 0 + 0 + 0 + 80
Elio’s total score = 175
(b) Count the number of batting
scores.
Number of scores = 6
(c) Define the rule for the mean. Mean =
(d) Substitute the known values into
the rule.
=
(e) Evaluate, rounding the value to
1 decimal place.
≈ 29.2
Find the absolute difference of each
of Elio’s batting scores; that is,
subtract the mean from each data
value, leaving the answer as a
positive number.
Difference: |65 – 29.2| = 35.8
|30 – 29.2| = 0.8
|0 – 29.2| = 29.2
|0 – 29.2| = 29.2
|0 – 29.2| = 29.2
|80 – 29.2| = 50.8
Calculate the mean (average) of the
absolute differences; that is,
Mean absolute difference
=
=
(a) add each of the absolute
differences
(b) divide the sum of the absolute
differences by the number of
data values.
Evaluate and simplify. = 29. 166 666 67
Round the answer to 1 decimal place. = 29.2
Repeat steps 1 to 5 for Gaetano’s
mean.
Gaetano’s total score:
= 0 + 30 + 30 + 80 + 25 + 20
= 185
Number of scores = 6
=
≈ 30.8
Difference: |0 – 30.8| = 30.8
|30 – 30.8| = 0.8
|30 – 30.8| = 0.8
|80 – 30.8| = 49.2
|25 – 30.8| = 5.8
|20 – 30.8| = 10.8
1
total scores
number of scores-----------------------------------------
x175
6---------
2
3
35.8 0.8 29.2 29.2 29.2 50.8+ + + + +
6--------------------------------------------------------------------------------------------
175
6---------
4
5
6
x185
6---------
C h a p t e r 1 2 D a t a a n d g r a p h s 609
Analysing data
1 Explain which statistical measure is referred to in these statements.
a There was a 15° temperature variation during the day.
b Children at this school are absent 3.4 days per semester, on average.
c Most often you have to pay $79.95 for those sports shoes.
d The average Australian worker earns about $470 per week.
e A middle-income family earns about $35 000 per annum.
2 Frank scored 5, 7, 6, 8, 7 in a series of spelling tests. Erik scored 8, 8, 6, 8, 9 in the
same tests.
a Calculate the range for each. Does this show that both had equally good results?
b Find the mean absolute difference for each. Does this give a measure of their
abilities? Answers where appropriate should be correct to 2 decimal places.
c What combination of statistics is needed to give a better measure of their abilities?
THINK WRITE
Mean absolute difference
=
=
= 16. 366 666 67
≈ 16.4
Compare the mean absolute
difference values, and answer the
question.
Gaetano’s results were more consistent than
Elio’s. However, as a measure by itself, it
does not give a measure of their abilities.
c Answer the question. c When both their batting means and mean
absolute difference values are compared, it
can be seen that Gaetano’s batting average is
slightly higher and that he is more consistent
than Elio.
30.8 0.8 0.8 49.2 5.8 10.8+ + + + +
6--------------------------------------------------------------------------------------
98.2
6----------
7
This is a summary of what each statistical measure does.
Mean: Uses all the scores as a total, divided by the number of scores.
Median: The score in the exact middle of values placed in numerical order.
Mode: The most common value or category.
Range: The highest score minus the lowest score.
Mean absolute difference: Shows the average spread of data from the mean of the
data.
remember
12HMathcad
Summarystatistics
GC program
– TI
Univariatestatistics
WORKED
Example
19
GC
program–
Casio
Univariatestatistics
WORKED
Example
20
610 M a t h s Q u e s t 8 f o r V i c t o r i a
3 The following scores were made by four teams in sports matches.
Jackals: 4, 0, 5, 9, 4, 8 Panthers: 7, 10, 10, 11, 10, 9
Wallabies: 2, 15, 1, 17, 10, 3 Tigers: 9, 10, 20, 25, 0, 14
a Which team has the highest mean?
b Compare modal scores for Jackals and Panthers.
c Find the median score for each team.
d Which team shows the greatest range of scores?
4 End of semester tests produced the following results in mathematics for a class:
a What is the mode?
b What is the median rating?
c Is it possible to calculate the mean and mean absolute difference? Explain.
5 Jennifer’s batting scores in indoor cricket were: 28, 35, 31, 29, 37, 30, 34, 40, 28, 33.
a What measure shows the degree of consistency in her performances? Calculate it.
b Find the mean, mean absolute difference, median and mode.
c Which of the measures in b would selectors use to evaluate her performance? Why?
d If you were Jennifer, would you use the mode to describe your record to others?
Explain your reasoning.
6 Here are Mark’s scores in the same matches as Jennifer in question 5:
57, 14, 68, 0, 22, 80, 9, 49, 16, 62
a Find range, mean, mean absolute difference, median and mode.
b Compare each measure with Jennifer’s from question 5.
c If you were the team selector, whom would you choose? Give reasons.
d Which measure would Mark use in talking about his performance? Explain.
Rating A B C D E
Number of students 3 8 10 5 2
You will need: Personal data sheets
Analyse your class statistically by using the measures range, mode, median, mean
and mean absolute difference for the attributes for which you gathered data: age,
height, and so on. Develop a profile of the ‘average’ student in your class. You may
like to compare the results from your class with another class’s results.
DESIGN Analysing your class data
C h a p t e r 1 2 D a t a a n d g r a p h s 611
Let us look at the data presented at the start of this chapter. Jemma obtained the
following data for the results of her AFL football team over a season. The points
scored in 20 matches were:
85, 96, 118, 93, 73, 71, 98, 77, 106, 64, 73, 88, 62, 97, 104, 85, 73, 92, 62, 76, 90,
79
1 Decide on an appropriate class interval and complete a frequency distribution
table.
2 Display these data as a histogram and overlay it with a frequency polygon.
3 What is the modal class of these data?
4 Display the data as a stem-and-leaf plot.
5 Calculate a the mean b the mode c the median.
6 What is the range of the data?
7 What is the mean absolute difference of the data?
8 What conclusions can you draw from analysing these data?
Michael has obtained a similar set of data for his local football team. The points
scored in 20 matches were:
83, 75, 93, 67, 62, 105, 118, 96, 84, 99, 92, 81, 88, 93, 100, 98, 87, 104, 84, 76,
115, 80
9 Repeat the analysis described in 1 to 8 above.
10 Compare the two sets of data. Write a summary of your findings.
11 At the start of the season, the president of Jemma’s football club offered an
incentive to the team. If the team’s average over the season was 90 or more,
they would be rewarded with a night out at a restaurant. Does the team receive
their reward?
12 If Michael’s football team had the same incentive plan, would they receive the
reward? Explain your answer.
1 Use the Internet or library to obtain two sets of data related to an area of interest
to you. (Try the Australian Bureau of Statistics’ website, www.abs.gov.au.)
Write a report on the analysis of these data. Explain whether the data have been
obtained from a sample or from the entire population.
2 Collect two sets of data for yourself by surveying students in your class on a
question of your choice. For the first set, survey the whole population of your
class (census). For the second set, work out a way of randomly choosing
10 students (sample).
Write a report on your findings. How do the two sets of data compare? What
are the advantages and disadvantages of using a sample compared to surveying
a whole population?
Note: When comparing the data, make use of the mean, mean absolute difference,
mode, median, range and graphs.
COMMUNICATION Footy season
COMMUNICATION Obtaining your own data
612 M a t h s Q u e s t 8 f o r V i c t o r i a
Copy the sentences below. Fill in the gaps by choosing the correct word or
expression from the word list that follows.
1 Statistics involves collecting data using a survey of samples of
the target , or of the whole population (census).
2 The data are then sorted into a frequency table, which shows
each score and the number of times it occurs.
3 To display the results we can use and charts, consisting of
data arranged in columns and rows; column graphs, bar graphs or
graphs.
4 Column and bar graphs should have these features:
i an appropriate
ii labelled and clearly scaled
iii all columns or bars of the width.
5 Histograms are special graphs showing scores on the hori-
zontal (x) axis and (f ) on the vertical axis. No gaps are left
between columns, which straddle the scale marks on the x-axis.
6 Frequency are special line graphs joining the midpoints of
the top of the columns, and starting and finishing on the
x-axis.
7 A and plot resembles a histogram turned on its
side.
8 Data analysis uses the (average) = total of all scores ÷ number
of scores, median ( score), and (most common
score) as measures of how scores cluster.
9 The is defined as the highest score minus the lowest score
and gives an overall impression of how scores tend to .
10 The mean and range are both used to measure
how the data is spread.
summary
W O R D L I S T
mode
distribution
axes
middle
leaf
spread
random
mean
population
title
range
column
absolute
frequency
tables
same
line
polygons
stem
difference
histogram
C h a p t e r 1 2 D a t a a n d g r a p h s 613
1 A random sample of 24 families was surveyed to
determine the number of vehicles in each household.
Use the frequency distribution column at right to answer
the following questions.
a How many families have no vehicles in their
household?
b How many families have 2 or more vehicles in their
household?
c Which score has the highest frequency?
d Which is the highest score?
e What fraction of families had 2 vehicles in their
household?
2 The following table represents the statistics for the various Rugby League teams.
Key: P = matches played W = number won
D = number drawn L = number lost
F = points for A = points against
Pts = competition points awarded
a How many matches had most teams played?
b Which teams had won 3 matches and lost 2?
c Which team shows the narrowest gap between F and A?
d Why is Brisbane placed ahead of Newcastle if they have both gained the same number
of competition points?
e How many points are awarded for a win?
CHAPTERreview
12ACars Frequency
0 6
1 4
2 8
3 5
4 1
Total 24
12A
Standings
P W D L F A Pts
Brisbane 5 5 — — 166 58 10
Newcastle 5 5 — — 146 64 10
Wests 5 4 — 1 114 56 8
Parramatta 5 4 — 1 112 68 8
Melbourne 5 4 — 1 102 78 8
North Queensland 5 4 — 1 78 95 8
Sydney 5 3 — 2 110 85 6
St George–Illawarra 5 3 — 2 105 102 6
North Sydney 4 2 — 2 84 58 4
Penrith 5 2 — 3 126 119 4
Canberra 5 2 — 3 90 86 4
Auckland 5 2 — 3 87 111 4
Cronulla 5 2 — 3 68 102 4
Canterbury 5 1 — 4 76 98 2
South Sydney 4 1 — 3 46 88 2
Manly–Warringah 5 1 — 4 62 106 2
RUGBY LEAGUE
614 M a t h s Q u e s t 8 f o r V i c t o r i a
3 This table shows the maximum and minimum daily temperatures in a city over a one-week
period.
Use the table to answer the following questions.
a What was the maximum temperature on day 3?
b Which day had the lowest minimum temperature?
c Which day was the coldest?
d Which day had the warmest minimum temperature?
e What was the temperature range (variation) on day 2?
f Which day had the smallest range of temperatures?
4 The table below shows the components that make up the body mass of a typical teenager.
a What is the teenager’s total mass?
b What fraction of the teenager’s
body mass is muscle?
c How much more mass of blood
than fat does he have?
d What percentage of the total
mass is the brain?
e Draw a column graph to display
the data.
5 This table below shows the weekly
expenses of a typical Australian family.
a Select a suitable title and draw a bar graph.
b Which item is the least expensive?
c Which item is half as expensive as food and drink?
d What fraction of the family’s expenses goes on other expenses?
e What percentage of the family’s expenses goes on rent and mortgage?
Day 1 2 3 4 5 6 7
Maximum (°C) 12 13 10 11 9 10 8
Minimum (°C) 3 3 2 1 0 4 2
Bone Brain Fat Blood Muscle Other
12.5 kg 2.5 kg 5 kg 10 kg 25 kg 5 kg
Item Amount ($)
Food and drinkRent or mortgageTransportClothingRecreationOther
120180
604060
140
12A
12B
12B
C h a p t e r 1 2 D a t a a n d g r a p h s 615
6 The graph at right represents
the acre–hectare conversion.
a Convert to hectares:
i 5 acres
ii 7.4 acres.
b Convert to acres:
i 4 hectares
ii 2.4 hectares
iii 3.5 hectares.
7 The table below represents the estimated insect population in a particular region of Victoria.
a Draw a line graph to display the data in the table of insect population growth. Label the
horizontal axis ‘Year’ and the vertical axis ‘Population’. Choose your own scale for the
horizontal axis. Use the scale 1 cm = 50 000 for the vertical axis, starting at 100 000.
b i Which decade showed the smallest increase?
ii Which decade showed the largest increase?
c i What should be the approximate population in 2009?
ii What should be the approximate population in 2015?
d If the trends continue, in what year should the insect population reach:
i 250 000 ii 370 000?
8 A number of people were asked to rate a video on a scale of 0 to 5. Here are their scores:
1, 0, 2, 1, 0, 0, 1, 0, 2, 3, 0, 0, 1, 0, 1, 2, 5, 3, 1, 0
a Sort the data into a frequency distribution table.
b Display the data as a histogram with a frequency polygon overlay.
9 The number of hours spent watching TV on a Friday night by students in a selected Year 8
class is: 1 , 2, 0, , 1, 2, 1 , 3, 0, , 1, 2, 2, 3, 3 , 0, 1, 4, 2, , 1, 0, 2, 1 , 0, 1 .
a Compile a frequency distribution table of the data.
b How many students were surveyed?
c What was the most common time that students spent watching TV?
d How many students spent more than 2 hours watching TV?
e How many students watched less than 1 hour of TV?
f Display the data as a histogram with a frequency polygon overlay.
10 The following data give the speed of 30 cars recorded by a roadside speed camera along a
stretch of road where the speed limit is 80 km/h.
75, 90, 83, 92, 103, 96, 110, 92, 102, 93, 78, 94, 104, 85, 88, 82, 81, 115, 94, 84, 87, 86, 96,
71, 91, 91, 92, 104, 88, 97
Present the data as an ordered stem-and-leaf plot.
Year Population
1980
1990
2000
2010
2020
120 000
160 000
230 000
330 000
460 000
12C
Acres
Hecta
res
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10
Acre–hectare conversion graph
12C
12D
12D1
2---
1
2---
1
2---
1
2---
1
2---
1
2---
1
2---
1
2---
12E
616 M a t h s Q u e s t 8 f o r V i c t o r i a
11 Comment on the data recorded by the roadside speed camera in question 10.
12 Calculate the mean of the following scores 1, 2, 2, 2, 3, 3, 5, 4 and 6.
13 The mean of 10 scores was 5.5; nine of the scores were: 4, 5, 6, 8, 2, 3, 4, 6 and 9; what was
the tenth score?
Questions 14 and 15 refer to the following distribution table.
14 Calculate the mean of the given frequency distribution table.
15 For the given frequency distribution table, determine:
a the mode b the median c the range.
16 a Determine the mode of the following values: 3, 2, 6, 5, 9, 8, 1, 7. Explain your answer.
b Determine the median of the following values: 10, 6, 1, 9, 8, 5, 17, 3.
c Calculate the range of the following values: 1, 6, 15, 7, 21, 8, 41, 7.
17 Find the mode, median and range of the data in question 8.
18 A frozen goods section manager recorded the following sales of chickens by size during a
sample week:
16, 14, 13, 12, 15, 14, 13, 11, 12, 14, 14, 16, 15, 13, 11, 12, 14, 13, 15, 17, 13, 12, 14, 16,
13, 11, 15, 14, 12, 11, 15, 12, 13, 12, 12, 15, 13, 11, 11, 13, 16, 13, 12, 15, 17, 13, 14, 16,
12, 15
a Construct a frequency distribution table showing x, f, and x × f columns. You may
include a tally column if you wish.
b Draw a histogram to display the data.
c Identify the mode of the distribution.
d Calculate the mean and median sizes of the chickens sold.
e Of which size should the manager order most? Explain.
f What is the range of sizes?
g What percentage of total sales are in the size 12 to 14 group?
h Is the mean a useful measure to the manager? Explain.
19 The following table displays the results of the number of pieces of mail delivered in a week
to a number of homes.
a What is the most common number of pieces of mail delivered?
b What is the mean number of pieces of mail delivered?
c Calculate the range.
d Calculate the mean absolute difference.
e Interpret the data.
x 2 3 5 6
f 3 2 8 2
Number of pieces of mail 0 1 2 3 4 5 6 7 8 9
Frequency 7 25 34 11 8 2 4 5 3 1
12E12F12F
12F
12F,G
12F,G
12F,G12H
testtest
CHAPTER
yourselfyourself
12
12F,G,H