chap 12

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


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


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


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?


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


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


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


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


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


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


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

Mail

Economy

Air

Sea

Mail

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


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


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


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


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


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


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


c The least favoured television shows are the

documentaries and lifestyle programs.

d Make note of the two longest bars and


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


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


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


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


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


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


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


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?


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


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


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


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)


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


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


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


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


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


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


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


Fre

quen

cy

181614121086420

0 1 2 3 4 5 6 7 8


f

x

2

3


Fre

quen

cy

181614121086420

0 1 2 3 4 5 6 7 8


f

x

4


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


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


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


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


1

2---

1

2---

1

2---

1

2---

1

2---

1

2---

1

2---


5 a Sort the information into a frequency table.


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.


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


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-------------------


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


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


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


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


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


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)


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


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


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


b 10, 8, 11, 5, 17

No values have been repeated.


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


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


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


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


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


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?


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.


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


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: 4 – 4=0= 0



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.


=

=

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


Calculate the mean absolute difference (correct to 2 decimal places) of the following data;

the mean is 3.5.


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.


=

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


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


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


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


GC pr

ogram– Casio


WORKED

Example

16

x x


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.


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


1.5 14

2.2 4

3.6 10

4.2 13

5.9 9


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.


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.



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.



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


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,


=

=

(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---------


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


=

=

= 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


WORKED

Example

19

GC

program–

Casio


WORKED

Example

20


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


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


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


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


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


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


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