Statistics - Maths and Science at Al Siraat -...

C H A P T E R

12Statistics

What you will learn

12.1 Types of data

12.2 Collecting data

12.3 Column graphs for categorial data

12.4 Histograms

12.5 Dot plots and stem-and-leaf plots

12.6 Back-to-back stem-and-leaf plots

12.7 The mean, mode and range of ungrouped data

12.8 The mean and mode from frequency tables

12.9 Statistical measures: quartiles

12.10 Box plots

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VELS

Measurement, chance and dataStudents generate data using surveys,

experiments and sampling procedures.

Students calculate summary statistics

for centrality (mode, median and

mean), spread (box plot, inter-quartile

range, outliers).

Working mathematicallyStudents choose, use and develop

mathematical models and procedures

to investigate and solve problems set in

a wide range of practical, theoretical

and historical contexts.

Students judge the reasonableness of

their results based on the context under

consideration.

Students select and use technology in

various combinations to assist in

mathematical inquiry, to manipulate

and represent data, to analyse

functions and carry out symbolic

manipulation.

The censusOn the last census night families sat

down to fill in their census forms. This

was a well-designed booklet full of

questions from which the Bureau of

Statistics could collect data about the

state of Victoria and its people.

They were able to establish facts

such as:

2 279 061 males and 2 365 889

females were counted which

represented a 6.2% increase since

the 1996 census

the three most common ancestries

are Australian (33%), English

(30.3%) and Irish (10.2%)

the three most common languages

spoken are English, Italian and

Greek

the median weekly individual

income range was $300—$399

13.7% of the population were

between 15 and 24 years of age

with more males than females in

that age range.

Facts such as these and trends shown

from previous censuses are vital for

planning for the future.

All of this information is gathered

and represented in frequency tables

and graphs so that it is available for

analysis.

If you wish to look at these figures

visit the Australian Bureau of Statistics

website at <www.abs.gov.au>.


444 Essential Mathematics VELS Edition Year 9

Do now

444

Skillsheet

T EACHE R

1 One card was randomly selected from a pack ofplaying cards. The suit of the card was noted, thecard was replaced and the pack was shuffled. Thefrequency of each suit is shown in the columngraph.

a How many times was a heart selected?b How many times was a card selected in total?c In what fraction of the trials was a diamond selected?

2 Organise the data set out below into the frequency table that has been started for you.

15 18 19 13 12 13 10 13 18 17 19 19

16 14 13 12 20 17 16 15 12 10 20 15

3 Organise the data below into a grouped frequency table using the intervals 130�139,140�149, etc. The table has been started for you.

198 195 136 186 165 186 170 187 167 195 186 153

206 200 192 178 173 185 178 206 155 162 169 145

Answers1 a 8 times b 20 times c 2 3

320

Interval Tally Frequency130�139 1140�149 1150�159 2| |

||

Scores Frequency10 211 012 313 414 115 316 217 218 219 320 2

Total 24

Interval Frequency130–139 1140–149 1150–159 2160–169 4170–179 4180–189 5190–199 4200–209 3Total 24

8

6

4

2

0

Cards

Freq

uenc

y

Essential Mathematics VELS Edition Year 9

Scores Tally Frequency10 211 0

| |


Statistics is an area of study concernedwith the collection, organisation,presentation and analysis of informationfrom which conclusions can be drawn.Business people, scientists and economicanalysts use such information to makedecisions and predictions.

445445Chapter 12 — Statistics

12.1 Types of data

Key ideas

Types of data

NNuummeerriiccaall CCaatteeggoorriiccaall ((nnoonn--nnuummeerriiccaall))

Each element in the data is a number, Each element of the data can be placed

e.g. {2.7, 36, �4, 58} in a category, e.g. Adelaide, Brisbane,

Collingwood

DDiissccrreettee

Is obtained by

counting

(can answer the

question `how many?’

The answer is exact

and is a countable

number, e.g. the

number of children in

a family)

CCoonnttiinnuuoouuss

Is obtained by

measuring

(can take on any

value within an

interval, e.g. heights

of students in a Year

9 class. Length, area,

volume, temperature

and speed are other

continuous

quantities.)

NNoommiinnaall

Has no natural order

(can only be

classified by the

name of the category

from which it arises,

e.g. favourite colour)

OOrrddiinnaall

Has a natural order

(can be ordered in

some way, e.g. poor,

satisfactory, good)

Example 1

Classify each of the following as numerical or categorical data.

a the age of members of a sports clubb the eye colour of students in your class

ExplanationSolution

a numerical Age is measured using numbers.b categorical Eye colour is described in words and is a category of colour.

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Example 2

Classify each of the following numerical dataas discrete or continuous data.

a the heights of basketballers in a teamb the number of peas in a podc the number of babies born in a particular

hospital each month

Example 3

Classify each of the following categorical data as nominal or ordinal.

a the results of a survey of students as to whether their school canteen food is ‘poor’,‘good’ or ‘excellent’

b the number of different coloured cars passing through an intersection

ExplanationSolution

a ordinal The data has a natural order.b nominal The data has no natural order.

12AExercise

1 Classify each of the following as numerical orcategorical data.

a the number of children per family in your streetb the weight of the children in a Year 9 classc the favourite types of toys of a group of kindergarten

childrend the quality of music (good, bad, etc.) at a dancee the pulse rates of athletes after a 100-m racef the types of pets Year 9 students haveg the amount of money spent on groceries each week

by 10 different families

1Example

ExplanationSolution

a continuous data Height is obtained by measuring.b discrete data Number of peas is obtained by counting.c discrete data Number of babies is obtained by counting.


447Chapter 12 — Statistics 447

2Example

3Example

h the types of transportation used by students to travel to and from schooli the number of people who attended a youth club over the past 10 weeksj the favourite brands of ice-cream of Year 9 studentsk the age of cars in a particular car lotl the number of lollies of each colour in a 250 g packet of lolliesm the most popular pop starsn the blood types of 25 blood donors

2 Classify each of the following numerical data as discrete or continuous data.

a the heights of students in your classb the shoe sizes of students at your schoolc the weights of newborn babies at a Melbourne hospitald the number of cars per familye the number of tickets purchased at a cinema in a weekf the height of a seedling over a two-week periodg the time spent exercising each weekh the number of children who attend a theme park in a day

3 Classify each of the following categorical data as nominal or ordinal data.

a the favourite sports of students in your classb the favourite colours of students in your classc the quality of food at a take-away shopd the brand of soft drink preferred by teenagerse the result of a survey of TV viewers as to whether a new show is ‘poor’, ‘good’ or

‘excellent’f the most popular types of carg the hair colours of 30 different peopleh the result of a survey that has a four-point scale to show how much you agree with

a statement

4 Andrew has collected the following data about the players at his favourite football team.Classify each of the types of data he has collected as:

i numerical or categoricalii discrete, continuous, ordinal or

nominal

a players’ heightsb players’ weightsc players’ ages measured in a whole

number of yearsd which type of jumper they prefere number of goals kicked during the

seasonf number of handballs made during

the seasong number of games played during the season



5 Joanne collects the following information from her small apple orchard.

Classify each of the types of data she has collected as:

i numerical or categoricalii discrete, continuous, ordinal

or nominal

a age of treeb height of treec colour of applesd number of apples on each treee quality of fruit (poor, satisfactory

or good)

6 Think of at least two examples of possible data that have not been discussed in thisexercise already which are:

a discreteb continousc ordinald nominal

7 Think of at least two examples of data that have not been discussed in this exercisealready which are:

a numerical, discrete datab numerical, continuous datac categorical, nominal datad categorical, ordinal data

Enrichment

8 Whenever data involves the collection of measurements we must decide if it is meantto be discrete or continuous. This is often done by qualifying our statements withsome extra words.

For example, the ages of people is continuous but data about the ages of people ina whole number of years is treated like a discrete variable.

Qualify the following statements about data to allow the data to be treated as if itwere discrete.a the heights of treesb the time it takes to showerc the capacity of dams in Victoriad the amount of grass needed to cover a front lawne the weights of bags of potatoesf the mass of planets

Th



If you are conducting a survey, you need to decide how you are going to ask the

questions, for example whether it will be by interview, mail or some other means.You will also need to design a questionnaire, making sure that the questions are relevant,

clear and unambiguous.

To ensure that responses can be readily collected make sure the survey does not take toolong to complete. It is a good idea to provide a list of possible responses such as‘yes/no/undecided’ or ‘very good/fair/poor’, and so on.

449

12.2 Collecting data

In order to make general statements and draw conclusions statisticians collect raw data thenorganise and analyse the data in numerous ways.

Key ideas

Data may be collected in a variety of ways.

MMaakkiinngg oobbsseerrvvaattiioonnss involves watching, counting and recording.

MMeeaassuurreemmeenntt involves the use of instruments such as tape measures, scales,

speedometers, light sensors and voltmeters.

EExxaammiinniinngg rreeccoorrddss involves searching for reliable sources such as the Bureau of Statistics,

local councils, newspapers and some Internet sites.

IInntteerrvviieewwss and qquueessttiioonnnnaaiirreess involve asking questions and recording the responses. They

are designed to obtain specific information.

Gender:Year level

Movie survey

male female

7 89 10

11 12How often do you go to the movies?Never

Where do you go to the movies?

What kind of movies do you like to watch?

1 to 2 times a month3 to 4 times a month5 to 6 times a monthOther

City theatreSuburban theatreOther

ActionComedyRomanceAnimationScience FictionOther



Example 4

Decide whether a survey should be carried out or whether the data might already beavailable to obtain the following information.

a the number of students in your class who support a particular football teamb the number of goals scored by basketballers playing in the National Basketball League.

ExplanationSolution

a Carry out your own survey. Data would not already be available.b Use data that is already available. Record books would provide the data.

Example 5

The following questions were part of aquestionnaire about whether volleyball should beincluded as part of the school’s sports program.

Decide:a which questions are not relevantb which suitable questions you would

categorise to make data collection easier.Rewrite these questions.

i Why do you like volleyball?ii How often do you play volleyball?iii What are the most important rules in volleyball?iv Do you play volleyball?v Which is your favourite volleyball team?vi Would you like to play volleyball at school?

ExplanationSolution

a i, iii and v are not relevant They are not relevant to the survey.b ii How often do you play

volleyball?Select one of the following It would be easier to categorise the question categories: rather than collect too many mixed answers.0–2 times per week2–4 times per week4–6 times per week, etc.



4Example

5Example

12BExercise

1 For each of the following decide whether a survey (S) should be carried out or whetherthe data might already be available (D) to obtain the following information.

a the exam results of Year 9 students at a particular schoolb the runs scored by players at a national cricket matchc the number of students doing a particular subjectd the rainfall, in millimetres, over a three-week periode the time taken, in minutes, by students to travel to school

2 Decide whether each of the following sets of data should be collected by observation(O), by measurement (M) or by using a questionnaire (Q).

a the number of hours of TV watched by Year 9 students at your schoolb the proportion of red lollies in a packetc the number of Australian-made cars in a particular car parkd the heights of Year 9 students at your school.e the types of transport used by Victorian workers to travel to workf the response of viewers to a new TV programg the most popular rock group

3 Below is a copy of a survey questionnaire on the cost of listening to music.

a Which questions are not suitable?b Which suitable questions would you

categorise to make data collection easier? Rewrite these questions.

Music survey

i How many CDs do you buy in a year?

___________________________________

ii Explain which types of music are important to you.

___________________________

____________________________________________________________

iii Who is your favourite singer? ___________________________________

iv How much money do you spend on CDs? _________________________

v Are CDs expensive? __________________________________________

vi What kind of music do you listen to? _____________________________



4 Below is a survey questionnaire on students’ television viewing habits.

a Which questions are not relevant?b Which suitable questions would you categorise to make data collection easier?

Rewrite these questions.

5 A bank wants to find out what type of home loan it should offer to its customers.Design a questionnaire to find out:

the type of loan customers want, either as fixed or variablethe time length of the loanhow much people want to borrow

6 A school teacher wants to survey students to help them determine a summer sportsprogram. Design a questionnaire to find out:

the most popular summer sportsthe sport in which most students have experiencethe new sport students would like to learn

Enrichment

7 Find out more about your favourite sport by:

a choosing a sportb listing the questions you want to askc designing a questionnaire d carrying out the survey on some friends or classmatese collating and tabulating the data f writing a brief report

Television survey

i How many hours of television do you watch? _____________________

ii What types of programs do you prefer to watch? __________________

iii Do you like to watch commercials? Yes No

iv Which free-to-air channel do you watch most frequently?

2 7 9 10 SBS

Wv hat is your age? __________________________________________

vi How much did your television cost? __________________________

Th



Raw data can be organised by using a frequency table. The information from the frequencytable can be displayed using a column graph. These graphs are commonly used forcategorical data.

12.3 Column graphs for categorical data

Key ideas

Example 6

ExplanationSolution

This column graph shows the responses of students when askedfor their favourite type of television show.

a How many students were surveyed?b How many students like game shows?c What percentage of students surveyed like game shows?

a 120 studentsb 12 students

c

10% of students like game shows � 10

12

120� 100 �

1200120

A frequency table shows the different categories or

data values and the number of times they each occur.

The frequency, f, denotes the number of times a

particular category or data value occurs.

CCoolluummnn ggrraapphhss,, also called bbaarr ggrraapphhss or bbaarr cchhaarrttss,

are usually used for a frequency distribution of

non-numerical data.

They are drawn with vertical or horizontal bars.

The bars are of equal width and equally spaced.

The height of the column or length of the bar represents the frequency.

Freq

uenc

y

red blue purple green

1

2

3

4

5

Column graph

Favourite colourColour FrequencyRed 2Blue 5Purple 3Green 1

3024181260

new

s/do

cum

enta

ries

com

edie

s

mov

ies

gam

e sh

ows

spor

ts

soap

ope

rasNum

ber

of s

tude

nts

Favourite television show

Add all the frequencies: 18, 6, 30, 12, 24 and 30The frequency of game shows is 12.

� 100%no. of students who like game shows

total number of students surveyed

1 2 3 4 5Frequency

Horizontal bar graph

red

blue

purple

green

Col

our



Example 7

This frequency table shows the responses of 30 students when asked about their favourite sport.

a Draw a column graph of the data in thefrequency table.

b Which sport is the most popular?c What percentage of the students surveyed like football best?

ExplanationSolution

a

b Football is the most popular sport.

c

30% of the students like footballbest.

� 30

9

30� 100 �

90030

Type of sport Frequencybasketbal 7cricket 4netball 5tennis 5football 9

Include a heading.The categories are on the horizontal axis andtheir order is not important. The bars are not drawn together. The frequenciesare on the vertical axis.

The highest column is football.9 students chose football out of 30.Multiply the fraction by 100 to convert it to apercentage.

Favourite sports

bask

etba

ll

cric

ket

netb

all

tenn

is

foot

ball

0123456789

10

Num

ber

of s

tude

nts

12CExercise

1 This column graph shows the number ofinternational tourists expected to visitAustralia over the period of six years from2004 to 2010.

a How many more tourists are expectedin Queensland than are expected inVictoria?

6Example

International tourists expectedbetween 2004 and 2010

State or territory

NS

W

Qld

Vic

WA

NT

SA

Tas0

400 000

800 000

1 200 000

1 600 000

Num

ber

of to

uris

ts



b What state, apart form NSW, expects the most tourists?c What percentage of tourists will visit Queensland?

2 This column graph shows the height ofsome of the world’s tallest buildings.

a What is thedifference inheight betweenthe tallestbuilding and theshortestbuilding?

b How much tallerare the PetronusTowers than theEmpire StateBuilding?

3 This frequency table shows the responsesof 40 adults when asked about the type ofcar they drive.

a Draw a column graph for the data in the frequency table.

b Which type of car is the most popular?c What percentage of the adults surveyed

have a 4WD?

4 This table shows the favourite types of music of 140 Year 9 students.

a Draw a column graph of the data inthe table.

b What type of music is most popular?c What type of music is least popular?d How many students like rock music?e What percentage of students like

classical music?

5 The combined column graph on the left shows thenumbers of cats and dogs looked after by The DogSanctuary in 2004 and 2005.

a What is the increase in admissions of dogsbetween 2004 and 2005?

b Is the increase in admissions higher for dogs orfor cats?

455

500

400

300

200

100

0

242290 304

380443 450

Hei

ghts

(m

)

Ria

lto

Tow

er,

Mel

bour

neE

iffe

l Tow

er,

Pari

sS

ydne

y To

wer

,S

ydne

yE

mpi

re S

tate

,N

ew Y

ork

Petr

onas

Tow

ers,

Kua

la L

umpu

r

Sea

rs B

uild

ing,

Chi

cago

The world's tall buildings

7Example

Type of music Number of studentsRock 30Pop 65Classical 15Techno 10Hard rock 15Rap 5

Car type FrequencySedan (large) 20Sedan (small) 10Wagon 2Hatch 34WD 4Other 1

5000

4000

3000

2000

1000

0

Num

ber

of a

nim

als

cats dogs2004

cats dogs2005

Pets looked afterby The Dog Sanctuary



6 The city of Melbourne hosts many major events including the five shown here. Use itto draw a column graph.

Event Number of people attendingMelbourne International Arts Festival 170 000Melbourne Grand Prix 250 000Australian Open tennis 270 000Spring Racing Carnival 300 000AFL finals 320 000

Enrichment

a Draw a frequency table of the information shown on the graph.b What is the increase in sales in June 1995 compared to March 1994? c Find the percentage increase between the following dates. Remember:

percentage increase is calculated by:

i March 1994 to February 1996ii June 1995 to December 1996iii December 1996 to July 1998

d Using the trend in percentage increase predict how many mobile phones will besold in 2006.

% increase �increase

starting value�

1001

7 This column graph shows the number of mobile phones sold in Australia in certainmonths between 1986 and 1998.

456

65.5

54.5

43.5

32.5

21.5

10.5

0Mob

ile

subs

crib

ers

(mil

lion

s)

2000

1986

Mar

ch 1

994

June

199

5

Feb

1996

Dec

199

6

Oct

199

7

July

199

8

Th



Numerical data is more useful sorted and arranged in numerical order in a table or a graph.

12.4 Histograms

Key ideas

A ccoolluummnn ggrraapphh is often used for discreteand continuous numerical data. Therectangular columns are placed next toeach other so that there is no spacebetween them. This type of column graph iscalled a hhiissttooggrraamm

A ffrreeqquueennccyy ppoollyyggoonn is a line graph. Foreach data value a single point is plotted atthe appropriate frequency. The points arethen connected with straight-linesegments. It is often drawn over ahistogram.

00

1 2 3 4 5 6Score

Fre

quen

cy

00

1 2 3 4 5 6Score

Fre

quen

cy

0 60 65 70 75 80 85

Class intervals

This break may be usedwhen our intervals do notstart at zero.

Fre

quen

cy

When a set of data is either continuous or

contains a large number of discrete values,

it is arranged into groups called ccllaassss

iinntteerrvvaallss.. Class intervals have the same

width and we use between 5 and 12 classes

so that trends in the data are noticeable.

Example 8

The data below shows the number of videos watched by a group of 40 Year 9 students overa one-week period.

3 0 2 4 0 3 0 0 0 1 3 1 2 1 1 1 5 2 1 31 1 2 2 3 4 2 2 2 2 3 3 3 2 2 2 4 2 2 5

a Draw a frequency table of this data. b Draw a histogram.c Draw a frequency polygon on your histogram.d How many students watched one, two or three videos that week?



ExplanationSolution

The tallies are convenientlyrecorded in groups of 5 with thefifth tally mark drawn as a strokethrough the other 4 .

For discrete data the data valuesare located along the axis in thecentres of the appropriate columns

The polygon is drawn by joiningthe midpoints of the tops of thecolumns.

Add the frequencies of the ‘1’,‘2’and ‘3’ columns.

(��)

Number ofvideos, x Tally Frequency, f

0 |||| 51 |||| ||| 82 |||| |||| |||| 143 |||| ||| 84 ||| 35 || 2

a

b

c

d Number of students � 8 � 14 � 8� 30

0

2468

101214

1 2 3 4 5Number of videos

Fre

quen

cy

002468

101214


Fre

quen

cy

Example 9

The data below shows the number of hamburgers sold each hour by a 24-hour fast-foodstore during a 50-hour period.

1 10 18 14 20 11 19 10 17 21 5 16 7 15 21 15 1022 11 18 12 12 3 12 8 12 6 5 14 14 14 4 9 1517 19 6 24 16 17 14 11 17 18 19 19 19 18 18 20

a Set up and complete a grouped frequency table, using class intervals 0�, 5�, 10�, etc.b Construct a histogram and a frequency polygon.

ExplanationSolution

Since the minimum is 1 and themaximum is 25, a class interval of width5 is appropriate.

a Class Tally Frequency (f )0� ||| 35� |||| ||| 8

10� |||| |||| |||| 1515� |||| |||| |||| ||| 1820� |||| | 6



12DExercise

459

Grouped frequency polygons areconstructed so that the plottedpoint is in the middle of each class. You could place this polygon overthe existing histogram or draw it onits own.

b18

15

12

9

6

3

00 5 10 15 20 25

Hamburger sales

Freq

uenc

y

8Example 1 The data below shows the number of musical instruments played by 20 students.

2 1 1 2 1 1 3 1 2 11 1 3 2 1 4 1 1 1 2

a Draw a frequency table for this data.b Draw a histogram.c Draw a frequency polygon on your column graph.d How many students play two, three or four

instruments?

2 The data below shows the results of a survey of a class of 50 students regarding the number of children in their families.

1 3 4 1 2 2 3 6 1 1 5 2 2 3 1 1 1 2 2 2 4 3 3 3 23 1 1 1 2 2 2 4 3 3 3 2 1 3 4 1 2 2 3 6 1 1 5 2 2a Draw a frequency table for this data. b Draw a histogram.c Draw a frequency polygon on your histogram.d How many families have two or three children?

3 Set up and complete a grouped frequency table with class frequencies shown and thendraw a histogram and frequency polygon.

a (Use the class intervals 0�, 10�, etc.)0 5 0 35 14 15 18 21 21 36 45 2 8 2 2 3 173 7 28 35 7 21 3 46 47 1 1 3 9 35 22 7 1836 3 9 2 11 37 37 45 11 12 14 17 22 1 2 2 3

b (Use the class intervals 50�, 60�, etc.)50 67 68 89 82 81 50 50 89 52 60 82 52 60 87 89 71 73 75 8386 50 52 71 80 52 87 87 87 74 60 60 61 63 63 65 82 86 76 8850 64 87 64 64 72 71 72 88 86 89 69 71 80 89 52 89 89 60 83

4 The number of goals kicked by a footballer in each of his last 30 football matches isgiven below.

8 9 3 6 12 17 8 3 4 5 2 5 6 4 158 9 12 19 7 12 14 10 9 8 12 10 11 4 5

9Example



a Organise the data into a grouped frequency table using a class interval width of 3.b Draw a histogram and a frequency polygon for the data.c In how many games did the player kick fewer than six goals?d In how many games did he kick more than 11 goals?

5 The data below shows the heights of 42 students, in centimetres.

166 177 168 169 161 147 148 154 146 150 169 168 169 152155 154 173 151 150 164 155 164 163 162 174 164 161 162173 172 171 162 163 172 170 160 171 174 159 160 168 171

a Organise the data into a grouped frequency table using a class interval width of 5.b Draw a histogram and a frequency polygon for the data.c How many students had a height of between 151 cm and 160 cm inclusive?

6 The data below shows the length of STD phone calls (in minutes) made by a particularhousehold over a six-week period.

1.5 1 1.5 1 4.8 4 4 10.1 9.5 1 3 8 5.9 66.4 7 3.5 3.1 3.6 3 4.2 4.3 4 12.5 10.2 10.3 4.5 4.53.4 3.5 3.5 5 3.5 3.6 4.5 4.5 12 11 12 14 14 12

13 10.8 12.1 2.4 3.8 4.2 5.6 10.8 11.2 9.3 9.2 8.7 8.5

a Organise the data into a grouped frequency table using a class interval width of 5.b Draw a histogram and a frequency polygon for the data.c What percentage of phone calls were more than 3 minutes in length? (Answer to

one decimal place.)

Enrichment

7 The data below shows the weekly wages ($) of 50 people.

400 500 552 455 420 424 325 204 860 894 464 379 563230 940 384 370 356 345 380 720 540 654 678 628 656670 725 740 750 730 766 760 700 700 768 608 576 890920 874 860 450 674 725 612 605 600 548 670

a What are the minimum weekly wage andthe maximum weekly wage?

b i Organise the data into about 10 classintervals.

ii Draw a histogram and a frequencypolygon for the data.

c i Organise the data into about fiveclass intervals.

ii Draw a histogram and a frequency polygon for the data.

d Discuss the shapes of the two graphs. Which graph represents the data better and why?

Th



Example: Use the data in this frequencytable to draw a histogram

From the STATmenu select Edit.

Enter the scores (x values) into L1

(List 1) and thefrequencies into L2

(List 2).

From the STATPLOT menu select‘Plot 1’ and turn it‘On’. Then select thecolumn type graphand make ‘X list:’ L1

and ‘Freq:’ L2.

Press WINDOW.Enter values relativeto the score:Xmin�0.5Xmax � 6.5Xscl � 1

Press GRAPHTRACE and arrowleft or right to viewthe data values.

Use the APPS menu andselect the Stats/ListEditor.

Enter the data into list1and list2.

Select F2 Plots followedby 1: Plot Setup.Select F1 Define and forPlot Type selectHistogram.For x type list1(2nd A-LOCK).Select YES for Use Freqand Categories?Type list2 for Freq.Press ENTER twice.

Press F5 ZoomData andadjust the WINDOW tosuit.

Using F3 Trace and pressarrows to the left andright to see the values foreach column.

Using technology to draw histograms

TI 84 family TI 89 family

Score (x) Frequency (f )1 42 103 124 145 26 6



Enter the data into a spreadsheet as shown on the right.

Next select make chart from the OPTIONS menu oruse the chart wizard and choose the appropriate typeof graph.Use the numbers in the first column as the labels onthe horizontal axis and give the graph a title.

A B1 Score Frequency2 1 43 2 104 3 125 4 146 5 27 6 6

Exercise1 For each of the following frequency tables:

i Use a graphics or CAS calculator or spreadsheet to draw a histogram.

ii Describe the shape of the graph.

a b cScore (x) Frequency1 72 83 64 125 116 10

Score (x) Frequency1 32 43 124 155 46 3

Score (x) Frequency1 72 123 44 25 36 1

Score Frequency1 42 103 124 145 26 6

Using a spreadsheet

Freq

uenc

y

16

14

12

10

8

6

6

4

4 5

2

2 310

Game 1

Score



For small sets of data a dot plot or a stem-and-leaf plot is an efficient way of representingthe data.

12.5 Dot plots and stem-and-leaf plots

Key ideas

A ddoott pplloott is a graph formed by placing dots along a horizontal scale to represent the data. It

is useful because:

It gives a clear picture of what the distribution of scores looks like.

Clusters and outliers can easily be identified. A cclluusstteerr is a grouped set of data and an

oouuttlliieerr is an isolated element of the data.

The graph is similar in shape to a histogram or column graph.

A sstteemm--aanndd--lleeaaff pplloott uses a stem number and leaf number to represent the data.

The data is shown in two parts, a stem and a leaf.

The stem should generally have one digit, but there may be many ‘leaf’ digits.

The graph is similar to a histogram on its side or a bar graph with class intervals but

there is no loss of detail of the original data.

Like a histogram or bar graph it clearly shows how the data is distributed and groups

(clusters) and outliers can easily be identified.

OOrrddeerreedd sstteemm--aanndd--lleeaaff pplloott

Example 10

a Construct a dot plot for the following set of data.4 1 6 1 6 7 2 4 4 5 5 2 5 5 9 54 4 6 7 1 2 7 1 1 6 6 5 5 6 10 7

b Identify any clusters or outliers.

clusters

outlieroutlier

A key is added to show

the place value of the

stems and leaves.

Stem Leaf

1 2 6

2 2 3 4 7

3 1 2 4 7 8 9

4 2 3 4 5 8

5 7 9

2|4 represents 24 people



ExplanationSolution

a There are two clusters.b There are two outliers: 9 and 10.

Draw a horizontal scale.Place a dot above the line at the appropriatemark for each data value. The frequency of a particular score is thenumber of dots above that score on the scale.

(1 to 2) and (4 to 7)9 and 10 are isolated from the others

1 2 3 4 5 6 7 8 9 10

Example 11

ExplanationSolution

For this set of data:

0.3 2.5 4.1 4.7 2.0 3.3 4.8 3.3 4.6 0.1 4.1 7.5 5.7 2.33.4 3.0 2.3 4.1 6.3 1.0 7.6 4.4 0.1 7.8 1.4 2.4 5.2 1.0

a organise the data into an unordered stem-and-leaf plotb organise the data into an ordered stem-and-leaf plot

a

b

The lowest number is 0.1 and has a 0 in theunits place.Use 0–7 in the stem.Work across the data table and place eachdecimal part on the leaves.

Rearrange each leaf in numerical order.Add a key.

Stem Leaf

0 3 1 1

1 0 4 0

3 3 3 4 0

4 1 7 8 6 1 1 4

5 7 2

6 3

7 5 6 8

Stem Leaf

0 1 1 3

1 0 0 4

2 0 3 3 4 5

3 0 3 3 4

4 1 1 1 4 6 7 8

5 2 7

6 3

7 5 6 8

4|6 represents 4.6



12EExercise

1 i Construct a dot plot for each of the following sets of data.ii Identify any outliers or clusters.a 6 4 5 3 4 4 1 6 5 4 5 6 5 5 3

3 1 4 1 4 4 4 4 4 1 1 5 5 7 5b 40 41 37 41 49 40 41 42 40 40 41 41

41 42 39 42 43 39 42 43 44 39 40 38

2 The data below gives the maximum temperature each day for a three-week period inspring.

18 18 15 17 19 17 21 20 15 17 15 18 19 19 20 22 19 17 19 15 17a Draw a dot plot of this data.b On how many days was the temperature

over

3 The number of vacant rooms in a motel each week over a 20-week period are shownbelow.

12 10 11 10 12 12 14 11 12 11 14 12 15 15 11 12 17 12 14 11a Draw a dot plot of this data.b In how many weeks were there fewer

than 12 vacant rooms?

4 For each of these data sets:

i organise into an unordered stem-and-leaf plotii organise into an ordered stem-and-leaf plota 31 33 23 35 15 23 48 50 35 42 45 15 21 45

51 31 34 23 42 50 26 30 45 37 39b 205 167 159 159 193 161 161 164 167 157

158 175 177 185 177 202 185 187 193 196159 189 167 159 175 200 184 173 198 200

18� C?

10Example

11Example



5 The data below shows the mass, in grams, of chips in 34 different bags of chips.

162 165 146 150 162 147 151 161 162 144 152 161 155 156 153 154 157150 159 158 150 163 164 144 146 149 154 160 152 160 157 150 165 155

a Draw an ordered stem-and-leaf plot.b What percentage of bags of chips have a mass more than or equal to 150 grams?

Round your answer to one decimal place.

6 The stem-and-leaf plot below shows the time taken, in seconds, by Helena to run 100 min her last 25 races.

a Between which values are the race times concentrated?b Are there any outliers?

Enrichment

7 This data shows the span (in metres) of a list of 30 bridges.

59 69 52 85 57 49 56 64 81 73 79 76 86 80 7448 54 68 51 69 83 71 70 79 89 90 57 117 72 68a Draw both an unordered and ordered stem-and-leaf plot.b Draw a dot plot for the data.c How many bridges have a span of less than 60 metres?c Find any outliers?d What percentage of bridges have a span greater than 80 metres? Round to one

decimal place.e Explain the benefit of the stem-and-leaf plot over the dot plot in this case.

Stem Leaf

14 9

15 4 5 6 6 7 7 7 8 9

16 0 0 1 1 2 2 3 4 4 5 5 5 7 7

17 2

14|9 represents 14.9 seconds

Th


Key ideas

BBaacckk--ttoo--bbaacckk sstteemm--aanndd--lleeaaff pplloottss can be used to compare two sets of data. The stem is drawn in the

middle with the leaves on either side.

Scores for the last 40 football games

Winning scores Losing scores

7 4 5 6 6

8 0 0 3 3 4 4

6 6 3 3 9 0 1 2 2 6 6

9 9 8 5 4 10 0 4 5 6 6 9

5 5 0 0 11

5 5 0 0 12 1

10/9 represents 109 points

The lowest

winning

score is 93.

For both sides, the

smallest leaf is next to

the stem.

The highest losing score

is 121 points.


12.6 Back-to-back stem-and-leaf plots

To compare two sets of data we can show them on the same stem-and-leaf plot by lettingthem share the same stem.

Example 12

A shop owner has two jeans shops. The daily sales in each shop over a 16-day period aremonitored and are recorded as follows.Shop A3 12 12 13 14 14 15 15 21 22 24 24 24 26 27 28Shop B4 6 6 7 7 8 9 9 10 12 13 14 14 16 17 27a Draw a back-to-back stem-and-leaf plot with an interval of 5.b Compare and comment on differences between the sales made by the two shops.

ExplanationSolution

a

b Shop A has the highest number of dailysales. Its sales are generally between 12and 28, with one day of very low sales of 3. Shop B sales are generally between 4and 17 with only one high sale day of 27.

For both sides, the smallest leafis next to the stem.

*0 allows us to split the data inhalf, that is, from 0�4 and 5�9.

Look at both sides of the plot forthe similarities and differences.

Shop A Shop B

3 0 4

0* 6 6 7 7 8 9 9

4 4 3 2 2 1 0 2 3 4 4

5 5 1* 6 7

4 4 4 2 1 2

8 7 6 2* 7

1/2 represents 12 sales



12Example

12FExercise

1 For each of the following sets of data:

i Draw a back-to-back stem-and-leaf plot.ii Compare and comment on the difference between the two data sets.

a Set A: 46 32 40 43 45 47 53 54 40 54 33 48 39 43Set B: 48 49 31 40 43 47 48 41 49 51 44 46 53 44

b Set A: 1 43 24 26 48 50 2 2 36 11 16 37 41 3 364 23 23 6 8 9 10 17 22 10 11 17 29 30 35

Set B: 9 18 19 19 20 21 23 24 27 28 31 37 37 38 39 39 3940 41 41 43 44 44 45 47 50 50 51 53 53 54 54 55 56

c Set A: 0.7 0.8 1.4 8.8 9.1 2.6 3.2 0.3 1.7 1.9 2.5 4.1 4.3 3.3 3.43.6 3.9 3.9 4.7 1.6 0.4 5.3 5.7 2.1 2.3 1.9 5.2 6.1 6.2 8.3

Set B: 0.1 0.9 0.6 1.3 0.9 0.1 0.3 2.5 0.6 3.4 4.8 5.2 8.8 4.7 5.32.6 1.5 1.8 3.9 1.9 0.1 0.2 1.2 3.3 2.1 4.3 5.7 6.1 6.2 8.3

2 The back-to-back stem-and-leaf plot below showsthe birthweight in kg of babies born to smokingand non-smoking mothers.

a What percentage of babies born to motherswho smoke have a birthweight of less than 3 kg?

b What percentage of babies born to motherswho don’t smoke have a birthweight of lessthan 3 kg?

c Compare and comment on the differencesbetween the birthweights of babies born to mothers who smoke and those born tomothers who don’t smoke.

Birthweight of babies

Smoking mothers Non-smoking mothers

4 3 2 2 2 4

9 9 8 7 6 6 5 5 2* 8 9

4 3 2 1 1 1 0 0 0 3 0 0 1 2 2 3

6 5 5 3* 5 5 5 6 6 7 7 8

1 4

4* 5 6



3 a Draw back-to-back stem-and-leaf plots for the final scores of Collingwood andCarlton in the 24 games given here.

Collingwood: 100 107 89 11598 102 89 108

109 114 98 116120 110 125 128132 128 129 128134 136 127 135

b In what percentage of games did each team score more than 100 points?c Compare the two sets of data. Comment on any clusters or outliers.

4 Two brands of batteries were tested to determine their lifetime in hours. The data belowshows the lifetime of 20 batteries of each brand.

Brand A: 7.3 8.2 8.4 8.5 8.7 8.8 8.9 9.0 9.1 9.29.3 9.4 9.4 9.5 9.5 9.6 9.7 9.8 9.9 9.9

Brand B: 7.2 7.3 7.4 7.5 7.6 7.8 7.9 7.9 8.0 8.18.3 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.8 9.8

a Draw a back-to-back stem-and-leaf plot for this data.b How many batteries from each brand lasted more than 9 hours?c Compare the two sets of data. Comment on any clusters or outliers.

Enrichment

5 The data below shows the test results of a class of Year 9 students in English andmaths.

Maths: 45 52 55 64 65 68 68 69 72 75 7878 79 80 82 85 90 94 100 100

English: 48 55 65 66 68 75 76 78 84 85 8688 90 92 94 95 96 100 100 100

a Draw a back-to-back stem-and-leaf plot for this data.b Describe the overall spread of the two sets of data.c How many students scored more than 70% in:

i English? ii maths?d Compare the two sets of data and write a paragraph that analyses the main

differences between the results. e Collect your own class data for two tests and use statistics to make comparisons.

Carlton: 85 99 100 8989 90 92 9598 108 110 98

104 106 108 8688 112 115 125

128 116 120 128

Th



The mean, mode and range are three numbers that help summarise and analyse data. Themean is also called the average. Using the mean, mode and range of the data it is possible tocomment on the centre and spread of a distribution.

Example 13

ExplanationSolution

For the data 8, 8, 15, 11, 16, 7, 7, 8, calculate the:

a mean b mode c range

a

b The mode is 8.c Range � 16 � 7 � 9

� 10

�808

�8 � 8 � 15 � 11 � 16 � 7 � 7 � 8

9

x �a x

nAdd all the data together and divideby the number of scores.

Simplify the numerator.

8 is the most common valueThe highest score is 16 and the lowestscore is 7.

12.7 The mean, mode and range ofungrouped data

Key ideas

MMeeaann oorr aavveerraaggee ((xx))

If there are n values, x1, x

2, x

3, . . . x

n, then the mean is calculated as follows:

Mode

The mode of ungrouped data is the most frequently occurring ssccoorree..

Note: There may be more than one mode.

Range

Range � highest score � lowest score

� ©x

n

� x1 � x2 � x3 � �� xn

n

x � sum of all the values

number of scores



Example 14

ExplanationSolution

The hours a shop assistant spends cleaning the store in eight successive weeks are: 8, 9, 12,10, 10, 8, 5, 10

a Calculate the mean for this set of data.b Determine the score that needs to be added to this data to make the mean equal to 10.

a Mean

� 9b Let a be the new score.

a � 18The new score would need to be 18.

72 � a � 90

72 � a

9� 10

72 � a

8 � 1� 10

�8 � 9 � 12 � 10 � 10 � 8 � 5 � 10

8Sum of the data.

72 � a is the total of the new data.8 � 1 is the new total number of scores.10 is the new mean.

Solve for a.Write the answer.

12GExercise

1 For the following sets of data determine the mean, mode and range.

a 9 9 7 8 5 8 4 9 7 4b 6 3 9 5 6 7 2 5 8 9 8 4 6c 16 24 14 12 30 42 18d 42 38 43 63 63 72 63 49e 7.3 8.2 9.6 6.4 5.2 5.8 7.2 6.4f 26.4 18.6 18.6 7.5 25.6 14.8 17.2 16.4g 32.1 23.4 67.2 56.3 89.2 45.2h 120 234 345 543 234 123

2 In three races Paul recorded the times 35.1 seconds, 34.8 seconds and 34.1 seconds.What is the mean time of the races?

3 The heights (in cm) of the members of a basketball team are:

160 165 168 169 167 168 170 171 170 168 172a What is the mean height of the basketballers?b What is the range of heights?

13Example



4 A netball player scored the following numbers of goals in her 10 most recent games:

12 15 16 14 16 15 16 17 15 14a What is her mean score?b What is the range of the scores?c What number of goals does she need to score in the next game for the mean of

her scores to be 16?5 Skevi obtained the following scores on her first five maths tests:

84 89 93 92 82a What is her mean test score?b If there is one more test left to complete, and she wants to achieve an average of

at least 85, what is the lowest score Skevi can obtain for her final test?6 Seven numbers have a mean of 8. Six of the numbers are 9, 7, 6, 4, 11 and 10. Find

the seventh number.

7 Four positive integers, a, b, c and d, have a mean of 10. Find different possible valuesfor a, b, c and d. Note any patterns that emerge.

Enrichment

8 The range can be misleading if it is not interpreted carefully. Consider the data belowwhich shows the number of goals scored over a five-week period by two netballers.

Player A: 14 11 10 13 9Player B: 14 13 14 3 13a Determine the mean and range for each player.b Who do you think is the most consistent player?c If we remove the week four result from the data for player B as an outlier and

recalculate the mean and range who is the most consistent player?

9 Science students, using the same set of scientific scales, obtained the followingresults for the mass (in mg) of a particular object.

980.6 982.3 980.7 981.1 980.7 981.3 762.2 981.0 980.8 981.1981.0 980.8 981.1 980.7 981.3 981.4 980.9 980.9 981.2 980.8a Find the mean of the data.b Examine the data carefully. Are any of the results inconsistent?c Omit any suspect results and find the mean, using only the consistent values.d What do you notice about the two means?

14Example

Th



Example: Determine the mean for this set of numbers: 2 4 6 8 10

Select EDIT fromthe STAT menuand enter thescores into one ofthe lists (such asL1).

Select 1 – VarStats; from theSTAT CALCmenu.

Press ¸.

Enter the data intolist1 using theStats/List Editorfound in theAPPS menu.

Use F4 Calc andSelect 1-Var Statsand press ¸.

For list type list 1(2nd A-LOCK).Press ¸twice. See themean � 6. Scrolldown if you wantmore information.

Using technology to find the mean


Enter the data into a spreadsheet as shown.

Enter the formula � AVERAGE (A2:A6) incell A8 to obtain the mean (which is 6).

A1 Scores2 23 44 65 86 107 AVERAGE8 6

Exercise 1 Use a calculator or spreadsheet to determine the mean of the following sets of data.

a 8 8 8 9 3 10 6 6 3 8c 73 84 74 74 85 86 92 93 95e 56 38 23 63 68 54 47 28g 9.8 6.2 8.8 5.4 5.6 6.2 7.2 8.5

b 94 84 76 34 98 73d 8.5 8.3 9.2 9.8 9.6 7.4 7.3 7.4f 42 54 78 34 56 88 35 59 59h 42.4 28.2 18.6 9.5 14.6 23.8 22.2

Spreadsheet

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APPE NDIX



This frequency table shows 16 scores. Written out in full, the numbers are:

The mean would therefore be:

� 7.44 (rounded to two decimal places)

Alternatively, since there are 5 sixes, 3 sevens, 4 eights and 4 nines we can group each settogether and determine the mean as:

� 7.44 (rounded to two decimal places)

�11916

�30 � 21 � 32 � 36

16

Mean �(6 � 5) � (7 � 3) � (8 � 4) � (9 � 4)

16

�11916

x �6 � 6 � 6 � 6 � 6 � 7 � 7 � 7 � 8 � 8 � 8 � 8 � 9 � 9 � 9 � 9

16

(x)

12.8 The mean and mode fromfrequency tables

Score (x) Frequency (f )6 57 38 49 4

6 6 6 6 6 7 7 78 8 8 8 9 9 9 9

Key ideas

When you have a frequency table and

wish to calculate the mean:

Add an extra column called xf.

For grouped data x is the midpoint

of each class interval.

Total the xf column and divide it by

the total of the frequency column.

i.e. Mean �

Mean (rounded to two decimal places)

The mmooddee is the score with the highest frequency.

For grouped data the class interval with the highest frequency is called the mmooddaall ccllaassss.

� ©xf

©f �

119

16 � 7.44

©xf

©f

Score (x) Frequency (f ) xf6 5 6 � 5 � 307 3 7 � 3 � 218 4 8 � 4 � 329 4 9 � 4 � 36

Total � 16 Total � 119



Example 15

ExplanationSolution

For the frequency distribution in this table find:a the mean to two decimal places b the mode

a

� 8.48b The mode is 8.

x �26331

Create the score � frequency column(xf ).

Total the xf column.Total the frequency column.

Divide the total of xf by the total of f.

8 has the highest frequency.

Score (x) Frequency (f )7 88 129 4

10 211 5

Score (x) Frequency (f ) xf7 8 7 � 8 � 568 12 8 � 12 � 969 4 9 � 4 � 36

10 2 10 � 2 � 2011 5 11 � 5 � 55

Total � 31 Total � 263

Example 16

ExplanationSolution

For the data in this table, find the:a the mean b modal class

a

b The modal class is 15–19.

�27525

� 13

x �©xf

a f

Add a midpoint column and findthe midpoint of the class interval.

Add an xf column.Total xf and f.

Calculate the mean.

Simplify the fraction.

The class interval 15–19 has thehighest frequency.

Class Frequency Midpoint,interval (f ) (x) xf0–4 7 2 145–9 3 7 21

10–14 5 12 6015–19 8 17 13620–24 2 22 44

©xf � 275©f � 25

Class interval Frequency (f )0–4 75–9 3

10–14 515–19 820–24 2



12HExercise

1 For the frequency distribution in each of these tables find:

i the mean, rounding to two decimal places where necessary ii the modea b

c d

e f

2 For the data in the following tables find:

i the mean, rounding to two decimal places where necessary ii the modal classa b

3 The heights of 50 teachers were recorded in this table.

a Find the modal class.b Find the mean height of the teachers.

Score (x) Frequency (f )3 45 27 89 6

11 2

Score (x) Frequency (f )10 620 330 540 450 2



Score (x) Frequency (f )3.6 53.8 74.0 34.2 84.4 2

Height (cm) Frequency 165– 8170– 14175– 23180–185 5


Class Midpoint ofinterval f class interval, x xf8–10 8 9 72

11–13 514–16 317–19 220–22 1

Class Midpoint ofinterval f class interval, x xf50–59 7 54.560–69 870–79 1080–89 1290–99 6

15Example

16Example


4 The times spent by 50 people waiting in a bank queue were recorded in this table.

a Find the modal class.b Find the mean time spent waiting in the

bank queue, correct to two decimal places.

5 This table shows the weekly rent payable for units in the metropolitan area.

a Find the modal class.b Find the mean cost of renting a flat

in the metropolitan area, correct to two decimal places.

Enrichment

6 The heights in cm of two sets of 10 people were collected. They both had a mean of160 cm but one had a modal class of 120–124 cm and the other a modal class of135–139 cm. Comment on what you think the original data would have looked likefor both sets of data.

Time (min) Frequency 0– 72– 124– 146– 108–10 7

Rent ($) Frequency 80– 7

120– 12160– 14200– 10240– 7280–320 4


Th

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Using technology to determine the mean of a frequencydistribution

Example: Determine the mean for the following set of data.

From the STATmenu select EDIT;and then enter thescores (x) into L1

and the frequencies(f ) into L2.

From the STATCALC menu, select1-Var Stats andspecify that the x values andfrequencies are inlists L1 and L2

respectively.

Press ¸.

Enter the data intolist1 and list2 usingthe Stats/List Editorfound in the APPSmenu.

Use F4 Calc Select1-Var Stats andpress ¸.

For List type list1and for Freq typelist2 (use 2nd A-LOCK). Press ¸twice.

See the mean �24.13 to 2 decimalplaces. Scroll downif you want moreinformation.


x f15 720 1225 1430 835 5

x f3 45 27 89 6

11 2

x f3.6 53.8 74.0 34.2 84.4 2

x f30 1732 2234 1136 1438 20

Exercise 1 Use a calculator to determine the mean of each of the following sets of data.

a b c

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Another useful way of analysing data arranged in order is to divide the data into four groupseach containing the same number of values.

12.9 Statistical measures: quartiles

Key ideas

If a set of numerical data is placed in order, from smallest to largest, then:

the middle number of the lower half is called the lower quartile (Q1)

the middle number of the data is called the median (Q2)

the middle number of the upper half is called the upper quartile (Q3)

the difference between the upper quartile and lower quartile is called the interquartile

range (IQR) ... IQR � Q3

� Q1

These three statistical values can help us to understand the centre and spread of a set of data.

The median, called the second quartile, Q2, should be found before finding Q

1and Q

3.

The three quartiles divide the data into four groups, each containing the same number of values.

IQR

25% 25% 25% 25%Q1 Q2 Q3

Example 17

ExplanationSolution

The following data values are the results for a school mathematics test.

67 96 62 85 73 56 79 19 76 23 68 89 81

a List the data in order, from smallest to largest.b Find:

i the median (Q2) ii the lower quartile (Q1)iii the upper quartile (Q3) iv the interquartile range (IQR)

a Q1 Q2 Q3

19 23 56 62 67 68 73 76 79 81 85 89 96b i Q2� 73

ii

iii

iv IQR � 83 � 59 � 24

Q3 �81 � 85

2� 83

Q1 �56 � 62

2� 59

Order the data and locate Q2, Q1 andQ3.The middle number is 73.

Q1 is halfway between 56 and 62.

Q3 is halfway between between 81and 85.IQR is the difference between Q1

and Q3.



Example 18

ExplanationSolution

Here is a set of measurements, collected by measuring the lengths, in metres, of 10 longjump attempts.

6.7 9.2 8.3 5.1 7.9 8.4 9.0 8.2 8.8 7.1

a List the data in order, from smallest to largest.b Find:


c Interpret the IQR.

Q1 Q2 Q3

a 5.1 6.7 7.1 7.9 8.2 8.3 8.4 8.8 9.0 9.2

b i

� 8.25 mii Q1 � 7.1 miii Q3 � 8.8 miv IQR � 8.8 � 7.1

� 1.7 mc The middle 50% of jumps differed by less

than 1.7 m.

Q2 �8.2 � 8.3

2

Order the data and locate Q2,

Q1 and Q3.Q2 is halfway between 8.2 and8.3.

Q1 is 7.1.Q3 is 8.8.IQR is the difference betweenQ1 and Q3.The IQR is the middle 50% ofthe data.

1 For each of the following sets of data:

i list the set of data in order, from smallest to largestii find the median (Q2)iii find the lower quartile (Q1)iv find the upper quartile (Q3)v find the interquartile range (IQR)a 3 7 5 2 9 6 1 0 11 5 8 7 9b 253 316 197 228 346 219 183 180 176c 27 36 21 18 38 41 29 37 30 26 35 21 30d 1.8 1.2 1.3 1.9 2.1 0.9 0.8 1.7 1.2e 23 163 28 76 854 256 367 343 3 787 12 43 520 64 28f 0.02 23 0.1 35 2.3 10 12

17Example

12IExercise



2 The time lengths, in minutes, of 16 recently rented videos were as follows:96 110 112 120 102 139 98 140 120 132 42 123 128 115 119 152a Find:


b Interpret the IQR.

3 The following set of data represents the sale price, in thousands of dollars, of 14vintage cars sold at auction.

89 46 76 41 12 52 76 97 547 59 67 76 78 30a For the 14 vintage cars, find:

i the lowest price paid ii the highest price paidiii the median price iv the lower quartile of the datav the upper quartile of the data vi the IQR

b Interpret the IQR for the price of the vintage cars.c If the price of the most expensive vintage car increased, what effect would this

have on Q1, Q2 and Q3? What effect would it have on the mean price?

18Example

Enrichment

4 The following two sets of data represent the number of boiled lollies found in 10jars purchased from two different confectionery stores, A and B.

Shop A: 25 26 24 24 28 26 27 25 26 28Shop B: 22 26 21 24 29 19 25 27 31 22a Find Q1, Q2 and Q3 for:

i Shop A ii Shop Bb The top 25% of the data is above which value for shop A?c The lowest 25% of the data is below which value for shop B?d Find the interquartile range (IQR) for the number of lollies in 10 jars from:

i Shop A ii Shop Be By looking at the given sets of data, why should you expect there to be a

significant difference between the IQR of Shop A and the IQR of Shop B?f Which shop offers greater consistency in the number of boiled lollies in each

jar it sells?

5 Investigate the consistency of weight of an item from a supermaket or other shop,for example the weight of a chocolate bar. Collect a number of samples and usestatistics to compare the data to the weight stated on the item packaging.

Th



If we know the quartiles for a set of data we can represent these as a box plot.

12.10 Box plots

Key ideas

A common graphical representation used to summarise data is the bbooxx pplloott.

There is one-quarter of the data in each of the four sections of the box plot. The length of each of

the sections represents the spread of the data in that section.

If an outlier has been identified, then the format for the box plot would be as follows:

minimum value maximum valueQ2Q1 Q3

outlier maximum valueQ2Q1 Q3

Example 19

ExplanationSolution

Consider the following set of data representing the 11 scores resulting from throwing andadding the scores of two dice.

7 10 7 12 8 9 6 6 5 3 8

a Find:i the minimum value ii the maximum value iii the medianiv the lower quartile v the upper quartile

b Draw a box plot to represent the data.

a Q1 Q2 Q3

4 5 6 6 7 7 8 8 9 10 12min. lower median upper max.

value quartile quartile value

i Min. value � 4ii Max. value �12iii Q2 � 7iv Q1 � 6v Q3 � 9

b Box plot: Throwing two dice.

Order the data.

Determine the minimum andmaximum value, median, lowerquartile and upper quartile.

Draw a horizontal axis.

Place the box plot above theaxis.3 4 5 6 7 8 9 10 11 12 13



Example 20

ExplanationSolution

The following set of data describes a daily native bird count in a small city park over a two-week period.

35 37 29 43 28 31 34 35 13 46 32 38 41 49

a Represent the data as a box plot which identifies any outliers.b On what percentage of days were the birds counted between

i 31 and 41? ii 13 and 49? iii 31 and 35? vi 31 and 49?

a 13 28 29 31 32 34 35 35 37 38 41 43 46 49min. � 13, max. � 49, Q1 � 31,Q2 � 35, Q3 � 41

13 is an outlier.

b i On 50% of the days between 31 and 41 birds were counted.

ii On 100% of the days between 13 and 49 birdswere counted.

iii On 25% of the days between 31 and 35 birds werecounted.

iv On 75% of the days between 31 and 49 birds werecounted.

Order the data.Determine the minimum andmaximum, median, lowerquartile and upper quartile.13 is an isolated value.

Draw the box plot.Use an appropriate scale.

Interpret the box plot, usingthe quartiles and range.

10 14 18 22 26 30 34 38 42 46 50

12JExercise

1 For each of the sets of data below:

i state the minimum value ii state the maximum valueiii find the median (Q2) iv find the lower quartile (Q1)v find the upper quartile (Q3) vi draw a box plot to represent the dataa 2 2 3 3 4 6 7 7 7 8 8 8 8 9 11 11 13 13 13b 43 21 65 45 34 42 40 28 56 50 10 43 70 37 61 54

88 19c 435 353 643 244 674 364 249 933 523 255 734d 0.5 0.7 0.1 0.2 0.9 0.5 1.0 0.6 0.3 0.4 0.8 1.1 1.2 0.8 1.3

0.4 0.5

2 The following set of data describes the number of cars parked in a street on 18 given days.

14 26 39 46 13 30 1 46 37 26 39 8 8 9 17 48 29 27

19Example

20Example

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a Represent the data as a box plot which identifies any outliers.b On what percentage of days were the number of cars parked on the street between

i 1 and 48? ii 13 and 39? iii 1 and 39? iv 39 and 48?

3 Draw box plots for the sets of data with the statistical measures given in the tablebelow.

4 A traffic count was conducted on a suburban street between the hours of 7.00 am and9.00 am. The results are as follows:

a Find:i the minimum value and the maximum valueii Q1, Q2 and Q3 iii the IQR

b Identify the outlier and when it occurs.c Give a practical reason that might explain why the outlier has occurred.d Draw a box plot for the data, marking the outlier with a cross.

Monday Tuesday Wednesday Thursday FridayWeek 1 83 112 71 107 109Week 2 96 99 83 111 101Week 3 12 108 96 94 115

Enrichment

5 Two data sets can be compared using comparative box plots on the same scale, asfollows.A

B

a What statistical measures do the above box plots have in common?b Which data set (A or B) has a wider range of values?c Find the IQR for:

i data set A ii data set Bd How would you describe the main difference between the two sets of data from

which the comparative box plots have been drawn?

1910

10 12 14 16 18 20 22

22

1817141312

Minimum value Q1 Q2 Q3 Maximum value Outlier

a 2 5 8 10 14 –b 7.3 8.1 8.5 9.6 12.3 –c 83 89 94 103 111 126d 53 61 73 78 86 32e 1.6 1.7 1.9 2.0 2.2 0.5

Th



Example: Consider this data. 6 7 8 0 1 6 9 5 5 5 3 4


Select STAT EDITand put your data inList 1 (L1).

Select STAT CALCmenu, Select 1-VarStats. Press 2nd L1

and .̧Note: You will need toscroll down using thearrow keys to observe allthe values available.

Press 2ndSTATPLOT andselect Plot 1.Select all items asshown.Note: The other box plotoption will allow outliersto be identified.

Press ZOOM andselect ZoomStat.Press TRACE andpress arrows to theleft and right to locateMinX, Q1, Q2, Q3 andMax.

Enter the data intolist1 using theStats/List Editorfound in the APPSmenu.

Use F4 Calc Select1-Var Stats andpress .̧For list type list 1(2nd A-LOCK).Then press ¸twice.

Select F2 PlotSetup followed byF1 Define. Select Boxplot.For list type list 1(2nd A-LOCK).Press ¸twice.

Press F5ZoomData. UsingF3 Trace, pressarrows to the leftand right to locateMinX, Q1, Q2, Q3

and MaxX.

Using technology to find statistical measures and draw box plots

Exercise1 Sketch box plots for the following set of data, and use TRACE to identify minX, Q1, Q2, Q3

and MaxX in each case.48 44 37 34 41 12 28 21 20 42 6 30 5 5 4023 17 15 10 27 28 21 27 3 33 30 8 4 16 34

2 Sketch a box plot, identifying the outliers, for each of these sets of data.40 33 38 62 23 35 24 26 35 37 32 34 32 22 24 37 29 2838 30


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W O R K I N G

StatisticsMathematically


The aim of this project is to gather data on 10 aspects of the students in your year level and make some conclusions about the average student in Year 9 and how this could affectdecisions made in your local council.

Choosing data categories

a As a class decide on six different data sets, on top of the four shown, you wish toinvestigate.1 the height of each student in the class by gender2 the eye colour of each student by gender3 favourite sport4 the number of people in each student’s family by gender5678910

b i Decide whether to collect this data from your class only or from the whole yearlevel.

ii Construct a survey and collect the data. If your data sets include both boys andgirls, collect the data separately.

Organising and representing the data

Each student should select four of these statistics and arrange them into a frequency tablewith one column for boys and one column for girls.

For example:

Eye colour No. of girls No. of boysBlue 6 7

Green 4 3Total 10 10

The four sets of data need to include:one set of continuous data (you will need to set up class intervals)one discrete set of datatwo sets of categorical data

Use column graphs, histograms, box plots, stem-and-leaf plots and dot plots to representyour data appropriately. You should try to represent boys and girls separately.

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Chapter 12 — Statistics

Analysing your data

a Using your graphs and determining themean, median and mode where possible,discuss what you have noticed from youranalysis with the rest of the class andcreate a model of Mr and Ms Average forYear 9 students.

b Draw a sketch of a person and label themwith your average statistics.

Conclusion and reflection

a Write a paragraph to explain what Mr and Ms Average might look like.b Discuss as a group if this project could be improved.

If you were to do this project again what other questions would you ask?

Would you modify your questionnaire/survey?

Would you have used different ways of representing your data?c Write a paragraph on how you think your local council could use this data when they

are planning for the future of their area.


Rev

iew


Chapter summary

Types of data

Data is classified as

Numerical Discrete (countable number of values)Types of data Continuous (not discrete, use class intervals)

Categorical Nominal (only classified by name)Ordinal (can be ordered in some way)

Column or bar graphs

These are used to represent categorical dataNote: The bars are drawn apart.

Frequency histograms and polygons

These are used to represent numerical data.Note: The bars are drawn together.

Dot plots

Dot plots give a picture of the data.We can identify clusters and outliers.

Back-to-back stem-and-leaf plots

These help compare twosets of data.The key showsthe place value of thestem and leaves.

Box plots

To draw a box plot determine the minimum value, Q1, Q2 (median), Q3 and the maximumvalue.

If an outlier has been identified, then theformat for the box plot would be as follows:

Freq

uenc

y

Categories

0

2468

101214

1 2 3 4 5Score

Fre

quen

cy

18

15

12

9

6

3

00 5 10 15 20 25

Score

Freq

uenc

y

clusters

outlieroutlier

Scores for the last 40 football gamesWinning scores Losing scores

7 4 5 6 68 0 0 3 3 4 4

6 6 3 3 9 0 1 2 2 6 69 9 8 5 4 10 0 4 5 6 6 9

5 5 0 0 115 5 0 0 12 1

10/9 means 109 points

The highest losing score is 121

points.

The lowest winning

score is 93. For both sides, the smallest

leaf is next to the stem.

Summary statistics

Mean: for ungrouped data for grouped dataMode: the most common dataMedian: the middle value of the data if it is arrangedin orderQ1, Q3, IQR: the quartiles and interquartile range

©xf>©f©x>n

maximum valueminimum value Q1 Q2 Q3

maximum valueoutlier Q1 Q2 Q3


Review


Multiple-choice questions

1 The heights of buildings in Melbourne would be classified as

A categorical, ordinal data B numerical, discrete data

C numerical, continuous data D categorical, nominal data

E categorical, discrete data2 Categorical data can be represented by

A a histogram B a dot plot C a frequency polygon

D a column graph E a stem-and-leaf plot

Questions 3–5 refer to the graph.3 The number of customers that rent fewer

than three videos were

A 27 B 35 C 14

D 2 E 84 The number of customers surveyed were

A 2 B 5 C 40

D 15 E 145 The total number of videos rented was

A 40 B 82 C 70 D 18 E 756 The mean of the data shown is

1 3 3 3 4 5 6 7

A 7 B 4 C 3 D 3.5 E 47 The median of the data shown is

4 4 5 6 7 7 7 8

A 7 B 6 C 6.5 D 7.7 E 48 The mode of the data shown is

1 1 2 3 3 4 5 6 7 7 7 8

A 7 B 6 C 4.5 D 4 E 5

Questions 9–10 refer to the box plot.

9 The interquartile range is

A 6–9 B 4–12 C 8 D 3 E 9–610 Which statement is false?

A the median is 7

B Q2 is 7

C 50% of values lie between 7 and 12

D 25% of values are less than 7

E 50% of values are between 6 and 9

3 4 5 6 7 8 9 10 11 12 13

002468

101214


Fre

quen

cy

Number of videos rented per customer

CUAT029_ch12[442-491].qxd 6/6/06 3:47 AM 9

Rev

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1 Consider these sets of data:a 29 26 38 28 28 29 33 34 31 25 27 28b 15 13 27 18 15 34 25 16 14 24 10c 0.1 0.4 1.2 1.3 0.2 0.4 0.5 1.6 2.1 0.7

Find: i the median (Q2)

ii the lower quartile (Q1)

iii the upper quartile (Q3)

iv the interquartile range (IQR)2 The data below gives the number of goals that

Christopher scored in his last 20 hockey games.

2 4 6 7 1 4 6 1 4 7 2

6 7 4 6 7 1 7 4 6

a Draw a dot plot for this data.

b On how many days did Christopher score fewer

than three goals?3 The data below shows the actual weight (in grams) of biscuits in 34 different 250 g packets

262 265 246 250 262 247 251 261 262 244 252 261

255 256 253 254 257 250 259 258 250 263 264 244

246 249 254 260 252 260 257 250 265 255

a Draw an ordered stem-and-leaf plot for this data.

b What percentage of packets have a weight equal to or more than 250 grams? (Round

to two decimal places.)4 Find the mean and mode of each of the following sets of data.

a

b

Short-answer questions

Score (x) Frequency (f )5 6

10 1215 820 1025 430 10

Score (x) Frequency (f )0–4 25–9 14

10–14 2815–19 1520–24 25

CUAT029_ch12[442-491].qxd 6/6/06 3:47 AM 0

Review


1 The owner of a movie theatre surveyed 50 customers regarding the number of moviesthey watched in a particular month. The results of the survey are shown below.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 3 3 3 3 3 3 3 3 4 4 4 4 5 5

a Organise the data into a frequency table.

b Draw a histogram to represent the data.

c Add a frequency polygon to the graph.

d How many customers watched more than three movies?

e What percentage of customers watched either one or two movies?

f What is the mean, median and mode for the data?2 Consider the following data representing the heights of some Bonsai plants in cm.

33 11 26 3 42 49 35 11 49 42

10 3 2 47 32 29 6 18 10 49

1 26 49 38 25 9 1 34 39 45a Organise the data into a stem-and-leaf plot, with the stem representing the tens

column and the leaves as the units column. You may need to do a second stem-and-leaf plot to order the leaves from lowest to highest.

b Transfer the data to a frequency table as shown.

c Draw a histogram for the data using the class intervals on the horizontal axis and thefrequencies on the vertical axis.

d Add a frequency polygon to your histogram.e Which class interval(s) of 10 units is/are the most common for the data provided?

Extended-response questions

Class interval Frequency0–

10–20–30–40–50

MC

TEST

D&D

TEST


Cyclic revision B

Chapter 7 Pythagoras’ theorem

Chapter 8 Linear relations

Chapter 9 Probability

Chapter 10 Trigonometry

Chapter 11 Quadratic relations

Chapter 12 Statistics

CUAT029-Cyclic Rev B.qxd 6/6/06 4:08 AM Page 492

Cyclic revisio

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Cyclic revision B 493

Pythagoras’ theorem

Multiple-choice questions1 For the right-angled triangle shown the length of the hypotenuse

is given by

A x2 � 152 � 202 B x2 � 152 � 202

C x2 � 202 � 152 D x2 � 152 � 202

E x � (15 � 20)2

2 For the right-angled triangle shown the value of x is given by

A B

C D

E

3 For the right-angled triangle shown

A B 9x2 � 2

C D x2 � 92 � x2

E

4 For the diagram shown the value of x and y are respectively

A 10.68 and 22.36 B 10.68 and 36.06

C 20, 36.06 D 20 and 22.36

E 20, 20

5 A Pythagorean triad that belongs to triad family (3n, 4n, 5n) is

A (6, 8, 12) B (9, 12, 15) C (15, 20, 30)

D (30, 40, 60) E (1, 2, 3)

Short-answer questions1 From point P a helicopter flies 40 km due north and then turns and travels another 60 km due

west to point Q. Find the direct distance between P and Q correct to two decimal places.

2 Two poles, 6.4 m and 4.2 m tall respectively, are 3.5 m apart. Find, correct to one decimal place, the length of wire needed to reach from the top of one pole to the other correct to two decimal places.

x2 �29

x2 �92

x2 �812

2(7 � 4)2

272 � 42242 � 72

272 � 42272 � 42

C H A P T E R

7

20

15x

7

4

x

9 x

x

30

12

16x

y

6.4 m4.2 m

3.5 m


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B3 For each of the following find the value of x correct to one decimal place.

a b

Extended-response question1 The map shows four key landmarks on an island.

Wombat Place (W ) is 2.3 km from Pirate’s Lookout (P)and 4.8 km from the shipwreck (S ). The lagoon (L) is3.2 km from the shipwreck.

a Find the distance, to two decimal places, to walk

from Pirate’s Lookout to:

i Wombat Place to the shipwreck and directly

back to Pirate’s Lookout

ii the lagoon via the shipwreck and directly back

to Pirate’s Lookout

b Which walk is longer and by how much?

c If the average walking speed is 5 km/h how long will each walk take?

Give all answers correct to one decimal place.

15 cm

5 cm

9 cm

x


C H A P T E R

8Linear relations

Multiple-choice questions1 The x- and y-intercepts for the graph of the relation 2y � 3x � 12 are respectively

A (0, 4), (6, 0) B (4, 0), (0, 6) C (�4, 0), (0, 6) D (0, 4), (0, 6) E (�3, 0), (0, 2)

2 The special lines x � �3 and y � 2 would intersect at

A (�3, 2) B (�3, 0) C (0, 2) D (3, 2) E (2, �3)

3 The gradient of a line passing through (�2, �2) and (1, a) is 2. The value of a is

A 6 B 1 C 3 D 4 E 2

4 A line has gradient and y-intercept (0, 3). The equation of the line is

A y � x � 3 B y � � x � 3 C y � 3 � � x D y � 3 � x E y � 3 � � x

5 The graph of the linear relation 2x � y � 4 has a gradient of

A 2 B �2 C D � E �412

12

12

12

12

12

12

�12

12 m3 m

10 mx

W

L

SP


Short-answer questions1 For the linear relation 2y � x � 6 complete the following.

a Make y the subject and write in the form y � mx � c.

b State the gradient and y-intercept.

c Sketch a graph.2 A straight line passes through the points (�1, 6) and (5, 3).

a Find the gradient of the line.

b Find the equation of the line.

c Find the x- and y-intercepts of the line.3 A cyclist travels at 20 km for half an hour, rests for one-quarter of an hour and then rides 8 km

in one-quarter of an hour.

a Find the cyclist’s speed in

i the first section ii the last section

b Find the cyclist’s average speed (correct to two decimal places):

i including the rest stop ii not including the rest stop

c At the cyclist’s average speed (including the rest stop), how far could the cyclist travel in

12 hours?

Extended-response question1 Janice has two landscaping jobs: (1) paving (approximately 6 hours’ work) and (2) fixing a

retaining wall (approximately 20 hours’ work).

She has received quotes from two landscapers and the details are:

1 Beautiful Backyards ($102 plus $40 per hour)

2 Landscape Plus ($52 per hour with no upfront fee)

a Write a rule for the cost ($C ) for t hours of work from:

i Beautiful Backyards ii Landscape Plus

b Sketch a graph for the cost of both landscapers on the same set of axes using

c Use your graph to estimate the number of hours (t) where the cost for both landscapers is

the same.

d If Janice is interested in keeping her costs as low as possible, which landscaper would you

recommend for each job? Explain.

0 � t � 20.

Cyclic revisio

n B


Probability

Multiple-choice questions1 A spinner is numbered 1 to 5. The chance that the spinner will result in a number less than 4 is

A 0.8 B 0.4 C 0.6 D 0.3 E 0.5

C H A P T E R

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B2 A biased coin is tossed 1000 times and the number of tails observed is 800. A good estimate

for the probability that the next toss will produce a head is

A 200 B 0.8 C 0.2 D 0 E 0.53 Two four-sided dice (numbered 1, 2, 3 and 4) are tossed and the total sum is recorded. The

probability that the sum is 8 is

A B C D E 0

4 A man randomly chooses a tie from two red and one blue and a shirt from two blue and onered. The probability that he will choose both a red tie and a red shirt is

A B C D E

5 If A is the set of odd numbers inside and B � {1, 2, 3} then is

A 2 B 4 C {1, 3} D {1, 2, 3} E {1, 2, 3, 5}

Short-answer questions1 A letter is chosen from the word ESSENTIAL. Find the following probabilities.

a Pr(N) b Pr(E or S) c Pr(not a vowel)

d Pr(letter belonging to the word NEST)2 A $5 bet is placed on a horse with odds of 9 : 1.

a State the chance of winning.

b Find the return if the horse wins.

c If the odds changed to 15 : 1, by how much does the chance of losing increase or decrease?3 A dot is selected from each of the sets A and B.

A = {�, •} and B = {�, �, •}a State the total number of outcomes. Draw a table showing all possible combinations to help.

b Find the probability that the outcome will:

i be (�, •) ii contain the same type of dot

iii contain one of each type of dot iv contain at least one black dot

v contain no more than one white dot

Extended-response question1 Three people randomly choose one of two muesli bars. One has nuts (N) and the other is free

of nuts (F).

a Show all possible outcomes of the selection of the three muesli bars, using a tree diagram,

and state the total number of outcomes.

b Find the probability that:

i all three people choose a muesli bar free of nuts

ii exactly one person chooses a muesli bar with nuts

iii at least two people choose a muesli bar with nuts

One of the three people is allergic to nuts.

c What is the chance that this particular person chooses a nut-free muesli bar?

d If the three people are allowed to swap muesli bars after the selection takes place, what is

the probability that the person with a nut allergy is able to have a nut-free muesli bar?

A ´ B�� 51, 2, 3, 4, 56,

59

49

12

29

13

12

14

18

116



Cyclic revisio

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C H A P T E R

10Trigonometry

Multiple-choice questions1

1 For the triangle shown

A B C

D E

2 Which of the following could be used to find the value of xin the triangle shown?

A 21 sin 35° B 21 cos 35° C

D E

3 The length of x in the given triangle is given by

A 12 sin 14° B 12 cos 14° C

D E

4 A 2.75-m slide has a ladder that reaches 1.43 m vertically above the ground. The angle the ladder makes with the ground to the nearest degree is

A 59° B 42° C 63° D 31° E 27°

5 To calculate the value of you need to evaluate:

A B C

D E

Short-answer questions1 Find the value of each pronumeral, correct to two decimal places.

a b c

78°12

x47°

4.5x

24°16

x

cos�1 a

56bsin�1

a65b

sin�1 a

56btan�1

a56btan�1

a65b

cos 14�

1212

sin 14o

12cos 14�

sin

cos 35�

21cos 35�

21

21cos 35�

cos �c

bcos �

bc

cos �ac

cos �ca

cos �a

b

ca

bθ

35°

21

x

14°x

12

2.75 m1.43 m

5

6θ


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B2 A roof of height 1.8 m has a span of 11.4 m as shown in the

diagram. Find the angle at the apex of the roof, correct tothe nearest degree.

3 From a hot air balloon the angle of depression to a markedlanding spot is 9°. If the balloon is 120 m above the ground find the horizontal distance (correct to the nearest metre) the balloon needs to travel to be directly above the landing spot.

Extended-response question1 A skateboard ramp is constructed as shown in the diagram.

a Calculate the total distance, d m, up the ramp.

b What is the angle of inclination () between

the ramp and the ground?

c i If a skateboarder rides from one corner of

the ramp diagonally to the other corner

what distance would he travel?

ii If the skateboarder travels at an average

speed of 12 km/h for how many seconds is he on the ramp?

Give all answers correct to two decimal places.


C H A P T E R

11Quadratic relations

Multiple-choice questions1 The curve shown in the diagram is the graph of

A y � (x � 2)2

B y � x2 � 3

C y � �(x � 2)2

D y � (x � 2)2

E y � �(x � 2)2

2 Which of the following are the coordinates of the turning point of the parabola whose equationis y � (x � 5)2 � 4?

A (�5, �4) B (0, 25) C (�5, 4) D (5, �4) E (5, 4)3 The curve shown in the diagram is the graph of

A y � (x � 2)2 � 1

B y � (x � 2)2 � 1

C y � (x � 2)2 � 1

D y � �(x � 2)2 � 1

E y � (x � 1)2 � 2

1.8 m

11.4 m

17.2 m

d m

d mθ 4.1 m

4.4 m

x

y

4

2

x–1–2–3

y


4 If x2 � x � 12 � 0 then x is equal to

A 3 or 4 B �4 or 3 C �3 or 4 D �3 or �4 E 0 or �125 The curve shown in the diagram is the graph of

A y � (x � 2) ( x � 3)

B y � (x � 2) ( x � 3)

C y � (x � 2) ( x � 3)

D y � (x � 2) ( x � 3)

E y � (x � 2) ( x � 3)

Short-answer questions1 Sketch the graphs of the following. You must show the y-intercept and the position of the

turning point.

a y � x2 � 2 b y � (x � 1)2 c y � (x � 2)2 � 1 d y � 3 � (x � 2)2

2 Solve each of the following for x.

a x (x � 6) � 0 b 2(x � 3)(x � 2) � 0 c 2x(x � 3) � 0

d x2 � 5x � 0 e x2 � 2x � 8 � 0 f 2x2 � 32 � 03 For each of the following quadratics:

i Find the x- and y-intercept. ii Find the axis of symmetry.

iii Find the coordinates of the turning point.

a y � (x � 4)(x � 2) b y � x(x � 6)

c y � x2 � x � 12 d y � x2 � 6x � 9

Extended-response question1 The flight path of a football kicked upwards from the ground is given by the equation

y � 20x � 2x2 where y is the height of the football at any time x seconds.

a What is the maximum height of the football?

b After how many seconds does it reach its maximum height?

c When does the football land on the ground?

d At what times was the football at a height of 32 m?

e Sketch a graph of the football’s flight path for 0 � x � 10.

Cyclic revisio

n B


C H A P T E R

12Statistics

Multiple-choice questions1 Which of the data described below is not discrete?

A the number of chairs in a classroom

B the number of people on a tram during peak hour

x

y

–2

–8

3


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BC the time taken to travel to work

D the number of pages in a book

E the number of tosses of a coin required before a ‘head’ appears2 Which of the data described below is not numerical?

A the pulse rate of each member of a swimming team

B the mass of 20 newborn babies

C the favourite colour of 20 people

D the height of 20 Year 9 students

E the volume of petrol used to fill a car3 Consider this set of data.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6

Which of the following are the mean value and the mode value, respectively, for this data?

A 3, 2 B 2, 2 C 3, 3 D 2, 3 E 2.5, 34 Which of the following is the lower quartile (Q1) for this set of data?

2 1 3 7 5 1 0 6 2 3 4

A 3 B 1 C 5 D 2 E 45 Which of the following is the interquartile range (IQR) for the data corresponding to the box

plot on the right?

A 23 B 4 C 7

D 22 E 11

Short-answer questions1 Fifty students were asked how

they like to spend their leisuretime at home. The results areshown in this table.

a Draw a bar chart to illustrate

the data.

b What percentage of the

students like to play a

computer game?2 The ages of 20 people queuing to see a movie are:

28, 24, 30, 18, 16, 20, 27, 32, 20, 35, 18, 18, 19, 20, 24, 32, 48, 36, 34, 38

a Draw an ordered stem-and-leaf plot to represent this data.

b Identify any outliers or clusters.3 Consider the scores shown in the table on the right.

a Draw a histogram and a frequency polygon to

represent this data.

b What is the modal class?

c What percentage of the scores are 30 or more?

d Determine the mean score.


Score Frequency

0� 10

10� 25

20� 20

30� 10

40�50 35

10 15 22 26 33

Leisure activity Number of students

Listening to music 12

Reading 8

Watching television 17

Watching a video 6

Playing a computer game 7


Extended-response questions1 In the last 40 one-day cricket matches a bowler took the following number of wickets per

game.

2 3 6 4 2 2 1 2 3 3

4 5 2 1 1 5 3 3 4 4

3 2 5 5 3 3 2 1 2 2

5 6 3 2 2 4 2 2 3 3

a Is this data continuous or discrete?

b Set up and complete a frequency table.

c Draw a column graph and a frequency polygon.

d For what percentage of matches did this bowler take 3 or more wickets?

e For what percentage of matches did this bowler take less than 2 wickets?

f What are the mode and the range of this data?

g Calculate the mean number of wickets taken per game by this bowler.

h How many wickets would the bowler need to take in the next game for the average per

game to increase by one wicket? Is this likely?2 The ages of 50 spectators passing through a turnstile at a cricket match were recorded as:

30 24 43 22 18 15 8 7 6 12

21 18 15 38 63 54 43 29 27 14

27 38 32 20 28 27 26 35 19 11

20 21 26 24 33 39 17 18 52 9

16 42 32 37 36 41 40 42 10 7

a Is this data continuous or discrete?

b Draw an ordered stem-and-leaf plot to represent this data.

c What is the range of the data?

d Set up and complete a grouped frequency table with intervals of 0–, 10–, 20–, etc.

e Draw a histogram and a frequency polygon of the data.

f What is the modal class?

g Calculate the mean age of the spectators.

h What percentage of the spectators are older than the mean age?

i What percentage of the spectators are aged within 5 years of the mean age?

Cyclic revisio

n B



Statistics - Maths and Science at Al Siraat -...

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