Maths A - Chapter 9

9

syllabussyllabusrrefefererenceenceStrand:Statistics and probability

Core topic:Data collection and presentation

In thisIn this chachapterpter9A Types of data9B Collecting data9C Organising and displaying

data using column and sector graphs

9D Graphical methods of misrepresenting data

9E Histograms and frequency polygons

9F Stem-and-leaf plots9G Five-number summaries

and boxplots

Collecting and entering data

MQ Maths A Yr 11 - 09 Page 325 Thursday, July 5, 2001 9:08 AM

326

M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Introduction

In Australia, we are fortunate to be ableto drink tap water. This is not the case inmany countries throughout the worldwhere the residents must consume bottledwater. If that were the case here, wewould be concerned with the price ofbottled water. Since we would requireongoing supplies, it would be wise toshop around for brands and sizes that suitour particular needs.

In this chapter we investigate:1. methods we could use to collect data2. how we could prepare and organise the

data3. ways in which we could display the

data4. drawing conclusions from the data col-

lected.

1

Distinguish between the terms

qualitative

and

quantitative

.

2

Consider the following temperatures (in degrees Celsius).

1, 3, 8, 6.5,

−

2, 25, 0,

−

1, 12

a

Arrange the data in ascending order.

b

Give the smallest, the largest and the middle temperatures.

3

At Central High School there are 117 students in Year 8 and 62 students in Year 12. InYears 9 and 10, there are 102 and 92 students respectively, while there are 77 studentsin Year 11.

a

Arrange this data set in a logical table format.

b

Draw a pie/sector graph to display the data.

c

Represent the data as a column/bar graph.

4

Express the following in the units indicated.

a

500 g for 98c (express in c/g).

b

$1.10 for 1.25 L (in $/L).

c

1.25 kg for $9.50 (in c/g).

5

Which is the better buy:

a 375-mL can of drink for $1, or a 1-L bottle of drink for $2.50?

MQ Maths A Yr 11 - 09 Page 326 Wednesday, July 4, 2001 5:49 PM

C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a

327

Types of data

Before we collect any

data

, we need to understand the various types of data we couldgather. The diagram below distinguishes the types.

Categorical data

are observations that fit some

qualitative

category. Data of thistype do not involve numbers or measurements. If there is no order associated withthe categories formed by the data, then they are termed

nominal

data

(for example,answers to questions about a student’s hair colour or method of transport used totravel to school). When the data categories are aligned to some qualitative scale,they are termed

ordinal data

. A response to a question on a scale (for example,

strongly disagree

to

strongly agree

) would constitute ordinal data (that is, someorder is implied).

Numerical data

, on the other hand, involve

quantitative

amounts.

Discrete

data

are responses, observations or records that can take only certain, set values. Examplesof discrete data would include, for example, the number of children in a family and thenumber of marks obtained in a test (even though you can be awarded half marks).

Continuous data

may take any value within the range of the data. Here, we often finddata arising as the result of taking measurements (for example, a person’s height, thedaily temperature).

Data

Categorical(qualitative)

Numerical(quantitative)

Nominal

Ordinal

Discrete

Continuous

State whether the following pieces of data are categorical or numerical.a The value of sales recorded at each

branch of a fast-food outletb The breeds of dog that appear at

a dog show

THINK WRITE

a The value of sales at each branch can be measured.

a The value of sales are numerical data.

b The breeds of dog at a show cannot be measured.

b The breeds of dog are categorical data.

1WORKEDExample


328


Types of data

1

State whether the data collected in each of the following situations would becategorical or numerical.

a

The number of matches in each box is counted for a large sample of boxes.

b

The sex of respondents to a questionnaire is recorded as either M or F.

c

A fisheries inspector records the lengths of 40 cod.

d

The occurrence of hot, warm, mild and cool weather for each day in January isrecorded.

e

The actual temperature for each day in January is recorded.

f

Cinema critics are asked to judge a film by awarding it a rating from one to five stars.

State whether each of the following records of numerical data is discrete or continuous.a The number of people in each car that

passes through a tollgateb The mass of a baby at birth

THINK WRITE

a The number of people in the car must be a whole number.

a

Give a written answer. The data are numerical and discrete.b A baby’s mass can be measured to

various degrees of accuracy.b

Give a written answer. The data are numerical and continuous.

1

21

2

2WORKEDExample

remember1. Data can be classified as either:

(a) categorical — the data are in categories, or(b) numerical — the data can be either measured or counted.

2. Numerical data can be either:(a) discrete — the data can take only certain values, usually whole numbers, or(b) continuous — the data can take any value depending on the degree of

accuracy.

remember

9AWORKEDExample

1


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 3292 State whether the numerical data gathered in each of the following situations are

discrete or continuous.a The heights of 60 tomato plants at a plant nurseryb The number of jelly beans in each of 50 packetsc The time taken for each student in a class of six-year-olds to tie his or her shoelacesd The petrol consumption rate of a large sample of carse The IQ (intelligence quotient) of each student in a class

3 For each of the following, state if the data are categorical or numerical. If numerical,state if the data are discrete or continuous.a The number of students in each class at your schoolb The teams people support at a football matchc The brands of peanut butter sold at a supermarketd The heights of people in your classe The interest rate charged by each bankf A person’s pulse rate

4 An opinion poll was conducted. A thousand people were given the statement‘Euthanasia should be legalised’. Each person was offered five responses: stronglyagree, agree, unsure, disagree and strongly disagree. Describe the data type in thisexample.

5 A teacher marks her students’ work with a grade A, B, C, D, or E. Describe the datatype used.

6 A teacher marks his students’ work using a mark out of 100. Describe the data typeused.

7

The number of people who are using a particular bus service are counted over a 2-weekperiod. The data formed by this survey would best be described as:A categorical data B numerical and discrete dataC numerical and continuous dataD quantitative data

8 The graph at right shows the number of days of each weather type for the Gold Coast in January. Describe the data in this example.

9 The graph at right shows a girl’s height each year for 10 years.Describe the data in this example.

WORKEDExample

2

mmultiple choiceultiple choice

0

Cool

WarmHot

Mild

2

4

6

8

10

12

14

Num

ber

of d

ays

in J

anua

ry

Weather

5 7 9 10 11 12 13 14 15Age

Hei

ght (

cm)

100

120

140

160

180

86


330 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Collecting dataData can be collected using a variety of techniques. Three common methods are:1. observation2. survey3. experiment

ObservationLet us return to consider collecting data on bottled water. We could obtain data by visiting outlets selling water and by observing the prices charged. Imagine that on a visit to three stores, a variety of brands, sizes and prices of bottled water were observed. The table below indicates the variationand costs of 1.5-L bottle of water at the three stores. The prices shown are shelf pricesfor one particular week.

Bottled water (1.5 L) Corner store Coles Woolworths

Natural spring water $1.88c

Mount FranklinAustralian spring water

$1.39 $1.35 $1.55

FrantelleSpring water

$1.89c $1.17 $1.99c

Rain FarmPure Australian rainwater

$1.14 $1.09 $1.19

Peats Ridge SpringsNatural spring water

$1.99c $1.20

First ChoiceSpring water

$1.09

EvianNatural spring water

$2.66 $2.67 $2.39

Brim-BrimNatural spring water

$2.22

TourquayNatural spring water

$1.19

SavingsStill spring water

$1.90c

ColesNatural spring water

$1.29

BellsPurified natural spring water

$1.19 $1.14

SchweppesCool Ridge still spring water

$1.70


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 331

Take note of:

1. the variety available at each venue

2. the variation in price between different brands (what could cause this?)

3. the brands that are common to the three stores

4. the variation in price for the same brand at the three venues.

Observations such as these would help you decide on a product suitable for your ownpersonal needs. However, the 1.5-litre bottle may not be appropriate for you. We shalldelve more deeply into this question with an investigation that also examines thevariety of sizes available.

FarmlandNatural spring water

$1.89c

SummitNatural spring water

$1.19

Fraser BlueNatural Australian spring water

$1.79

AquaqueenAustralian spring water

$1.59

Cottonwood ValleyNatural spring water

$1.39

Snowy MountainNatural spring water

$1.03

Collecting data by observing bottled water prices

In this activity, we attempt to determine the amount of bottled water that you or your family needs.

1 Visit your local suppliers of bottled water (as many of them as possible). Make a record of the brands, sizes and costs of all types available at each outlet.

2 Organise your data into a table format (slight modifications to the table shown previously would be appropriate). These records would be entered as discrete numerical data. Check to ensure you have not made any obvious recording errors.

3 Compare the price of bulk buying with that of purchasing several smaller bottles of equivalent capacity.

4 Now look for a solution to your water requirements. (Consider your needs for a weekly supply.) Take into account the following points.(a) Your bottled water supply must provide for all drinking and food

preparation needs.(b) The water has a long shelf life, so bulk purchases may be more economical

(if you have storage room and funds available).

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stigationinvestigatio

n



SurveysCollecting data by survey is the form that is most frequently used. The survey is admin-istered with the aid of a questionnaire. The degree of success of obtaining meaningfuland relevant data from a questionnaire depends largely on the care taken in designingthe questions. Methods used to collect data include:

1. personal interviews, where the interviewer usually asks prepared questions, thenrecords the respondents’ replies

2. telephone interviews, where the interview is conducted over the telephone

3. self-administered questionnaires, which are usually mailed to individuals whocomplete the questionnaire, then return it in a pre-paid envelope or hold it forcollection.

Reporting your resultsWrite a formal report of this investigation, detailing:1. Aim2. Method of obtaining data3. Table summarising data4. Your own personal weekly needs5. Purchases which would satisfy these needs — specify preferred brands, sizes

and the total cost. Explain your choice of brand and size.

More data collection by observationIt is possible to collect a variety of types of data by observation. The following are some suggestions you may wish to investigate.

1 Count the number of vehicles passing a particular point on a road during a given time period. (Road counters across roads typically record this information.)

2 Observe the number of people waiting at a department store for the doors to open the morning before a sale commences and on the morning on which it occurs.

3 Record the wildlife in a park.

4 Take note of the number of early morning joggers during the week compared with the numbers who jog during the weekend.

5 Record the variation in the price of petrol during the week.

There are many situations where data are obtained by observation. In some situations, such data provide evidence for development of a scheme, concept or physical resource. This form of data collection is commonly used for planning, marketing and the preparation of reports.

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QuestionnairesAs mentioned previously, it is important to take care in designing the questionnaire inorder to obtain relevant and reliable data. In formulating the questions, we must alsokeep in mind that we require the data so collected to be in a form which is not difficultto analyse.

The questions posed can be either open or closed.Open questions are those where the respondent has no guided boundaries within

which to answer. Questions that belong to this class include:‘Who is your favourite singer?’‘What is your favourite food?’The main difficulty with open questions is that they are often difficult to classify and

analyse.Closed questions are of the type where the respondent must answer within a

category. The question about food (above) could be rephrased as:‘Which of the following foods do you prefer most?

K meatK seafoodK poultryK vegetablesK fruit’

Analysis of these answers would be easier than trying to fit the open-ended answersinto a category.

It must be noted that options such as:K none of the above orK don’t know

are to be avoided, if possible, as they provide the respondent with a non-complianceexit.

Survey methodsThis investigation is best undertaken in a small group.

There are advantages and disadvantages of each of the survey methods mentioned previously. As a group, discuss the various methods, then copy and complete the table below with as many advantages and disadvantages as you can.

Retain the table for consultation later in designing your own questionnaire.

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n

Type of survey Advantages Disadvantages

Personal interview•

•

Telephone interview•

•

Self-administered questionnaire•

•



Twenty people answered the open question:‘What do you particularly dislike about the way the news is presented on TV?’.They provided the following responses.

1. The ads interrupt too often.2. Have to wait too long for the headline items.3. The violent scenes are too graphic.4. It was better when it was only half an hour instead of a full hour.5. There’s too much violence.6. The reporters are politically biased.7. The accident scenes are not sensitively handled.8. Some of the reports are too long.9. It’s all about politics.

10. It mixes up local, interstate and overseas news.11. I’d like it to be shorter.12. There are more advertisements than news.13. It seems to concentrate on murder and death.14. I find it far too long.15. The newsreaders don’t pronounce names correctly.16. It’s too informal.17. The newsreader is far too old.18. I don’t like ads interrupting the news.19. Some reports show too much blood and gore.20. The interviews by the reporters are too long.Classify the responses into categories to identify the main reasons given.

THINK WRITE

Look for 4 or 5 categories under which they could be classified.

Four main categories are apparent1. Length2. Violence3. Advertisements4. Newsreader

Classify the responses under these headings.

LengthResponses No. 4, 8, 11, 14, 20ViolenceResponses No. 3, 5, 7, 13, 19AdvertisementsResponses No. 1, 12, 18NewsreaderResponses No. 15, 16, 17

Identify those not classified. Those not classified are:No. 2. Too long for headline itemsNo. 6. Reporters politically biasedNo. 9. All about politicsNo. 10. Mixes up news

1

2

3

3WORKEDExample



Preparing a questionnaireIn preparing good questions, it is advisable to keep the following points in mind:

1. The questions should flow smoothly from one to the next.

2. Introductory remarks should be included outlining the aim and purpose of thequestionnaire, along with any necessary instructions.

3. Jargon, slang and abbreviations should be avoided.

4. Do not ask questions which are vague or ambiguous.

5. Avoid bias and emotional language.

6. Avoid double-barrelled questions.

7. Do not pose leading questions (that is, those that lead to an expected response).

8. Make sure your questions are capable of being answered by your respondents.

9. Avoid questions with double negatives.

10. At the conclusion, thank the respondent for answering.

Good and bad question writingAs an aid to good questionnaire writing, it is helpful to practise writing questions, then do a pilot test to determine whether the questions are clear.

1 In small groups, discuss each of the points in the list above to gain a comprehensive understanding of good question writing.

2 Choose a topic that interests you.

3 Illustrate each of the points mentioned above by designing questions that do not conform to the guidelines of ‘good question writing’.

4 Rewrite the questions in a clear format.

5 Conduct a pilot test of both sets of your questions on the members of your group.

THINK WRITE

Identify the main reasons. The main reasons for people disliking the TV news presentations seem to be centred around the length of the news and the violence portrayed.

Note: It is possible to classify these responses into different categories. No single way is correct. In practice, the responses from open-ended questions are generally classified several ways until the researcher is satisfied with the classifications.

4

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ExperimentsCollecting data by conducting experiments is commonly used in the sciences andrelated fields. In order to draw reliable conclusions from experiments, the data must beable to be reproduced. The conclusions from one experiment are frequently not suf-ficient evidence on which to base a report.

Survey of bottled waterWith an understanding of good question writing, we can now put these techniques into practice by conducting a classroom survey.

Let us consider the bottled-water situation we encountered earlier in the chapter.1. The aim of this survey is to discover:

(a) whether people prefer bottled water to tap water(b) if purchasing bottled water, their brand preference(c) how much bottled water they would drink in a week(d) what they really know about bottled water.

2. To encourage your respondents to take care in completing your survey, it should have variety and be interesting. All your questions should not be of the same style. Be creative in your question construction. Questions such as:‘Draw a mouth on each face to show how much you like the types of water’

create interest in your survey and encourage the respondents to answer with care. Also consider styles like ‘circle the answer’ and ‘tick the box’.

1 Design your questions to fulfil the aims of the survey. Decide on the variables to be recorded. Are they categorical, numerical, discrete or continuous?

2 Make copies of your questionnaire to be answered by other members of your class; alternatively, draw up a table to record the answers if you wish to use the interview technique; then conduct your survey.

3 Critically appraise your questionnaire’s effectiveness after its administration. Did your questionnaire achieve the aims of the survey? Did your respondents find it interesting? Did you feel that the data obtained were reliable? Should you disregard any data?

4 Retain these results for later when we will look at the variety of methods we can use to display data.

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Love it

Tap water

Nice OK Not nice Horrible

Bottled water


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 337Let us return to consider the bottled-water observation discussed earlier. Much vari-

ation is evident among the prices of the 1.5-litre bottles of water. An assumption wouldbe that the more expensive bottles contained better-quality water. An experiment couldbe devised to determine whether consumers can detect differences in the quality of thewater samples with a relatively simple taste test.

Bottled water taste testThe aim of this experiment is to discover whether people can distinguish, by taste,

1. between different samples of bottled water (Test 1)

2. tap water from bottled water (Test 2).

This experiment would be best performed as a whole class activity.

1 Prepare three jugs: one containing tap water, one with a cheap variety of bottled water and the third with an expensive variety of bottled water. Provide small cups for each jug.Because the samples may look different (for example, tap water may be slightly coloured), this experiment would be best conducted with the participants blindfolded.

2 For Test 1, blindfolded students will each be given a sip of a cheap variety and a sip of an expensive variety of the bottled water. The test is to see whether the taster can identify which is the cheaper variety and which the dearer.

3 Test 2 is conducted similarly, using bottled water and tap water.

4 Prepare a result sheet (as below).

a Record the taste tests.

b Are any conclusions obvious at his stage?

c Identify any problems with the tests.

5 Retain the results of this experiment for use later, when we will consider ways of displaying data.

We must not forget that the results of this taste test could not be published as conclusive evidence of the ability of consumers generally to be able (or unable) to determine the quality of water. The results obtained by your class would need to be confirmed by similar results from tests performed on many other groups.

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Student Test 1 (� or ✗) Test 2 (� or ✗)



Gathering data from the World Wide Web (www)

The Web is an excellent resource for data collection. We will spend some time now investigating web sites that provide interesting and relevant data for use later on.

1 A census is conducted on the Australian population every five years. The Australian Bureau of Statistics (ABS) has published the results of the 1996 census with data available for viewing on their web site. It takes some time for the data from a census to be collated and analysed, so the results of the next census (2001) may not be available until some years afterwards. Browse the site, noting the data available. Take a note of the web address for future reference.

2 The Bureau of Meteorology publishes climatic data for numerous towns in Queensland. Locate its web site and investigate the range of data displayed. Take particular note of the data recorded for the city or town closest to where you live. Record the site address for future use.

3 What is a ‘gallup poll’? This famous poll is named after its founder, the American statistician, George Gallup, who was born in 1901. Another well-known poll is the Morgan Gallup Poll which publishes statistics on a variety of topical issues throughout the world. Search the Web for information on gallup polls. Record your findings in the form of a poster and present the results of your search to the class.

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remember1. Some important methods of collecting data are observations, surveys and

experiments.2. All data collection requires careful planning with regard to the variables

recorded in order to maintain the quality of the data collected.3. A survey may be conducted personally, via the telephone or it may be self-

administered. Each of these methods has advantages and disadvantages.4. Care must be exercised in constructing a questionnaire so that the responses

collected are of good quality.5. The questions may be of an open or closed format.6. Open questions must eventually be placed in categories before proceeding to

the stage of presentation.7. The data obtained from experiments must be capable of being reproduced

before generalisations can be made.8. In all cases of data collection, the quality and reliability of the data must be

examined before processing.

remember



Collecting data

1 Explain what you understand by the terms ‘open’ and ‘closed’ questions. Give anexample of an open question. Rewrite your question in a closed format.

2 Twenty students were asked their opinions about the cause of congestion at the school’sfront gate. Analyse their responses below, suggest categories into which they could beclassified and identify the most commonly stated reasons for the congestion.

1. The cars shouldn’t come up the front driveway.2. The front entrance is too small.3. There should be another entrance.4. The buses are the problem.5. The students get in the way of the cars.6. Bike riders should have a separate entrance.7. The Senior school and the Junior school should start and finish at different times.8. The cars block the gate.9. The bike riders don’t know the road rules.

10. The buses should stop further down the road.11. Too many students.12. Parents don’t care where they park.13. The front gates should be wider.14. Bike riders should go out the back gate.15. Kids block the cars.16. Kids just sit around talking there.17. There should be a traffic control officer there to direct the traffic.18. Students should not just sit around there.19. The buses all arrive at the same time.20. The road is too narrow.

3 Thirty students were asked: ‘Identify one thing in your maths course which you par-ticularly don’t like’. Classify the responses below into appropriate categories, thenidentify the main reasons.

1. It’s too hard.2. There’s too much homework.3. I can’t understand the teacher.4. It’s boring.5. The boys are too distracting.6. The teacher doesn’t like girls.7. It’s too much work.8. We get homework every night.9. I can’t understand it.

10. The boys show off.11. The boys always get better marks.12. I can’t concentrate for that length of time.13. We do something new every lesson.14. I don’t like working in groups.15. The teacher expects too much.16. Our teacher is too strict.

9B

WORKEDExample

3



17. The teacher doesn’t help us enough with our problems.18. There’s too much work to cover.19. We’re expected to do assignments.20. I don’t like doing presentations to the class.21. Our class is too big.22. The work is not interesting.23. I just don’t understand maths.24. I don’t like the teacher.25. The teacher expects too much of us.26. There’s too much work to cover.27. We’re expected to remember too much.28. It won’t help me later in life.29. The teacher picks on me because I don’t understand the work.30. The course is not relevant.

4 Identify the areas of concern in the following questions, then rewrite each so that themeaning is clear and understandable.(a) How much do you earn?(b) Do you exercise regularly?(c) Is the GST in Australia less than the VAT in England?(d) Do you generally support the causes of murderous terrorists who threaten the lives

of peace-loving people?(e) Do you support the Prime Minister’s policy on wildlife preservation?(f) What is your height in inches?(g) Did you buy your sneakers for comfort and quality?(h) You don’t agree with charging more for skim milk (where they’ve taken out the

cream) than for full cream milk, do you?(i) Do you agree that we should do more for our ‘diggers’ who risked their lives

during the war so that we could be free?

5 Write the following open questions in closed format.(a) What is your age?(b) How much pocket money do you get each week?(c) How do you travel to school?(d) What type of destination do you prefer for a holiday?

Data preparationHaving considered so far the collection of data by observation, survey and experimentalmethods, we must now reflect on the techniques at our disposal for treating these data.However, before we rush into doing calculations, compiling tables and drawing graphs,we must first carefully examine the data for anomalies.

We must make a decision with regard to non-compliant responses from a respon-dent. Sometimes a decision is made just to disregard those non-compliant responsesand to count the remainder of the compliant responses of the survey from the respon-dent. At other times, the whole survey from that respondent is disregarded. Alterna-tively, we could cope with the dilemma (after the survey has been administered) byproviding another category in the question to absorb all those responses which do notslot into the categories given.


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 341Data must also be checked for recording errors, which commonly occur. Recording

errors should not be included as part of the data as they will distort calculations. Wherepossible, when errors are detected, every effort should be made to correct the data(make new observations, contact the respondent etc.). If for any reason the errorscannot be rectified, that record should be excluded.

In the business world, databases are often created to allow responses for each ques-tion only within defined constraints (a database may not allow a number entry where aword was expected, for instance). This reduces the chance of recording invalid data, butwould not stop you entering your age as 66 instead of 16, so careful scrutiny andchecking of data are essential before the results are displayed and analysed.

Organising and displaying dataOrganising dataOnce data have been collected and checked for errors they need to be put into an organ-ised form. This involves tallying the responses to a questionnaire, accurately recordingyour observations or tabulating the results of your research.

This task is made easier if the questionnaire is designed with ease of tabulation inmind. Usually the results are first organised into a table and the number of responses ineach category recorded. This is often done with tally marks and using the gatepostmethod.

A survey is conducted among 24 students who were asked to name their favourite spectator sport. Their responses are recorded below.AFL Cricket Cricket Soccer Rugby League Cricket Tennis Cricket AFL Rugby League AFL AFL Rugby Union Soccer NetballBasketball Basketball Netball AFL CricketCricket AFL Rugby League CricketPut these results into a table.

THINK WRITE

Draw a table and beside each sport put a tally mark for each response. Every fifth tally mark becomes a gatepost.

Sport Tally Frequency

AFL |||| | 6

Basketball || 2

Cricket |||| || 7

Netball || 2

Rugby League ||| 3

Rugby Union | 1

Soccer || 2

Tennis | 1

4WORKEDExample



For simplicity, numerical data may be tabulated in groups.

Displaying dataThe most common way of displaying data is by using a graph. Different graphs havedifferent purposes. We will now look briefly at column graphs and sector graphs, thenlook at histograms, stem plots and boxplots.

Column graphs

A column graph (or bar graph) is used when we wish to show a quantity. Categoriesare written on the horizontal axis and frequencies on the vertical axis.

A Year 11 class was surveyed on their weekly income. The responses are shown below.$75 $115 $60 $54 $88 $0 $98 $102$56 $45 $83 $71 $40 $37 $87 $117$43 $79 $58 $89 $70 $105 $99 $55Complete the table below.

Income Tally Frequency

0–20

21–40

41–60

61–80

81–100

101–120

THINK WRITE

Count the number of responses within each category and put a tally mark in the column.

Income Tally Frequency

0–20 | 1

21–40 || 2

41–60 |||| || 7

61–80 |||| 4

81–100 |||| | 6

101–120 |||| 4

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343

Sector graphs

A

sector graph

(circle graph, or piegraph) is used when we want thegraph to display a comparison ofquantities. An angle is drawn at thecentre of the circle that is the samefraction of 360° as the fraction ofpeople making each response.

The table below shows the results of the survey on favourite sports.

Show this information in a column graph.

Sport Frequency

AFL 6

Basketball 2

Cricket 7

Netball 2

Rugby League 3

Rugby Union 1

Soccer 2

Tennis 1

THINK WRITE

Draw the horizontal axis showing each sport.

Draw a vertical axis to show frequencies up to 7.Draw the columns all the same width with gaps between.Use a ruler.Label the axes.Give the graph a title.

1

AFL

Baske

tball

Cricke

t

Netball

Rugby

Lea

gue

Rugby

Unio

n

Socce

r

Tennis

Sport

Favourite sports of 24 students

Freq

uenc

y

01234567

23

456

6WORKEDExample

MQ Maths A Yr 11 - 09 Page 343 Thursday, July 5, 2001 10:54 AM


These graphs can also be drawn using a spreadsheet and the charting tool.In our next investigation we shall explore how to enter data into a spreadsheet and

how to display the results in a variety of graphical forms.

For the table in worked example 6, draw a sector graph.

THINK WRITE

Calculate each angle as a fraction of 360°.

AFL = × 360° Basketball = × 360°

= 90° = 30°Cricket = × 360° Netball = × 360°

= 105° = 30°Rugby League = × 360°

= 45°Rugby Union = × 360°

= 15°Soccer = × 360° Tennis = × 360°

= 30° = 15°Draw the graph.Label each sector or provide a legend.

1 624------ 2

24------

724------ 2

24------

324------

124------

224------ 1

24------

23 AFL

BasketballCricketNetballRugby LeagueRugby UnionSoccerTennis

Sport

7WORKEDExample



Organising and displaying data using column and sector graphs

1 A class of students was asked to identify the make of car their family owned. Theirresponses are shown below.

Holden Ford Nissan Mazda Toyota Holden Ford Holden Ford Mitsubishi Toyota ToyotaNissan Holden Holden Ford Toyota MazdaMazda Toyota Ford Holden Holden FordMitsubishi Toyota Holden Ford Ford Toyota

Put these results into a table.

2 The results of a spelling test done by 30 students are shown below.6 7 6 8 4 6 6 7 5 95 7 8 10 5 9 7 7 7 64 7 8 8 7 8 6 5 9 7


3 The marks scored on a Maths exam, out of 100, by 25 Year 11 students are shownbelow.87 44 95 66 78 69 66 92 7854 60 66 69 66 77 79 66 7171 83 74 81 69 70 57

Copy and complete the table below.

Mark Tally Frequency

40–49

50–59

60–69

70–79

80–89

90–99

remember1. When data are collected they are usually first organised into table form.2. Data can be easily counted using a tally column and the gatepost method.3. Sometimes numerical data are better organised into categories.4. A column graph is drawn when we want to display quantities.5. A sector graph is drawn when we want to compare quantities.

remember

9CWORKEDExample

4

EXCEL Spreadsheet

Frequencytally tables

EXCEL Spreadsheet

Frequencytally tables

DIY

WORKEDExample

5



4 The data below show the number of customers that entered a shop each day in acertain month.114 195 175 163 180 120 204 199178 216 200 147 168 173 102 150169 185 173 164 130 119 158 163 141 155 132 143 190 179 200

Choose suitable groupings to tabulate these data.

5 Draw a column graph to display the data from question 1.

6 Draw a sector graph to display the data from question 1.




10 Draw a sector graph to compare the number of people in each category from question 3.

For questions 1–4, state if the data are quantitative or qualitative. If they are quantitative,also state whether they are continuous or discrete.

1 Customers in a video shop vote for their favourite movie.

2 Customers in a video shop have records kept on the number of movies they hire eachyear.

3 The video shop keeps records of the number of times each movie has been hired.

4 The video shop keeps records of the length of each movie.

The bar chart at right shows the marital status of respondents to a survey.

5 How many people responded to the survey?

6 What was the most common marital status?

7 How many people were married?

8 How many respondents were either divorced or separated?

9 How many people had been married at some time?

10 Draw a pie/sector graph of the data.

EXCEL

Spreadsheet

Column graphs

WORKEDExample

6

EXCEL

Spreadsheet

Column graphs DIY

WORKEDExample

7

SkillSH

EET 9.1

WorkS

HEET 9.1

1Fr

eque

ncy

Nev

er

mar

ried

Mar

ried

Div

orce

d

Sepa

rate

d

Wid

owed

5

10

15



Spreadsheets — Displaying numerical data

Worked examples 6 and 7 explained the calculations needed to display manually the same numerical data as a column graph and as a sector (pie) graph. In this activity, we look at entering the data into a spreadsheet, then exploring some of the graphical tools available. The instructions provided refer to the Excel spreadsheet. If you are using a different spreadsheet, your teacher will give you the equivalent commands.

1 Enter the heading ‘Sport’ and the categories of sport in Column A as shown above.

2 Enter the heading ‘Frequency’ and the relative frequencies in Column B as shown.

• Highlight the sporting classifications and the frequencies, enter the Chart Wizard and select the Column graph option. Follow the instructions, remembering to label the axes and title the graph, to produce the column graph shown above.

• Select the data again and follow through the Chart Wizard to produce the pie graph shown.

• Print out a copy of your table and graphs.

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Spreadsheets — Displaying categorical coded data

The previous investigation dealt with entering and graphing numerical data. Frequently, data are collected in categories that are each given a code. Examine the shopping survey shown below. It represents a section of the ‘Australian Shoppers Survey’ distributed by PMP Data Based Marketing to households in Australia.

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1. How many litres of soy milk do you consume per week?1–2 litres 1 3–4 litres 2 4 litres + 3

2. Which of the following beverages would you consider having home delivered?Juice 1 Water 2 Soft Drink 3

3. How often do you have your hair coloured at a salon?More than once per 6 weeks 1 Every 3 months 3 Once a year 5Once every 6 weeks 2 Every 6 months 4 Never 6

4. How often do you purchase shampoo?Once a week 1 Once a month 3Once a fortnight 2 Once every 2 mths 4

5. What type of washing machine do you own/use?Top Loader 1 Front Loader 3

6. What brands of laundry detergent do you regularly use?Once a week 01 Dynamo 05 Radiant 09Cold Power 02 Dynamomatic 06 Surf 10Cold Powermatic 03 Omo 07 Other 11Drive 04 Omomatic

7. Which of the following is your main toy store?Big W 01 Mr Toys 05 Toys ‘R’ Us 09dstore 02 Myer 06 toyspot.com 10Grace Bros. 03 Target 07 Toyworld 11K-mart 04 Toy Kingdom

8. How many dogs do you have in your home?None 1 One 2 Two 3 3 or more 4

9. How many cats do you have in your home?None 1 One 2 Two 3 3 or more 4

10. Do you treat your pets for any of the following?Fleas/Ticks 1 Worms 2

11. Do you or your partner smoke? Yes 1 No 2

12. Please sign that you are over 18 and a smoker:

Your signature_______________________________________________________________

Partner’s signature ___________________________________________________________

13. Have you purchased or would you consider purchasing any of the followingby mail or telephone? Have Bought Considering

You Ptnr You PtnrBooks 01 23 12 34Computer Equipment 02 24 13 35Computer Software 03 25 14 36Cosmetics 04 26 15 37Craft Products 05 27 16 38Fashion Wear 06 28 17 39Kitchen Products 07 29 18 40Music (Tapes, CDs, Records) 08 30 19 41Videos 09 31 20 42Vitamins/Health Supplements 10 32 21 43Wine 11 33 22 44

14. Have you/your partner bought or searched for goods to buy over the internet?You 1 Your partner 2

Thinking about buying beauty care products which are new to the market, please indicate whether you agree or disagree with the following statements:

15. In general, I am among the first to buy new beauty care products when they appear on the market.

Strongly Agree 1 Somewhat Disagree 4Somewhat Agree 2 Strongly Disagree 5Neither Agree or Disagree 3

16. I enjoy taking chances in buying new beauty care products.Strongly Agree 1 Somewhat Disagree 4Somewhat Agree 2 Strongly Disagree 5Neither Agree or Disagree 3

2. SHOPPING



Notice that:1. all the questions are of the closed type2. the response categories for the questions each have a code associated with them

(this enables them to be entered into a computer database or spreadsheet for analysis)

3. the ordinal data in questions 15 and 16 are coded.

Let us take a small number of responses to question 15 and see how the data might be treated.

1 Open a spreadsheet, head Column A with ‘Respondent’ and enter the respondents’ names as shown below.

2 Head Column B as ‘Code’, then enter the coded responses shown beneath the heading.

3 Columns A and B represent the raw data. We will collate these records on the spreadsheet to the right of these columns.

4 Head Column D with the word ‘Code’, then enter the coded categories 1 to 5 beneath.

5 In Column E, enter the meanings of the coded categories beside their respective code (this just makes the spreadsheet and graphs more meaningful).

6 Head Column F with the word ‘Number’. In this column we are going to count the number of 1’s, 2’s etc. which occur in all the responses in Column B. The formula that enables us to do this is the COUNTIF command. Its format is =COUNTIF(range,criteria).

7 To count the number of 1’s in the range B3 to B14, the formula would be=COUNTIF(B3:B14,1). Enter this formula in Cell F3.



Methods of misrepresenting dataMany people have reasons for misrepresenting data: politicians may wish to magnifythe progress achieved during their term, or business people may wish to accentuatetheir reported profits. There are numerous ways of misrepresenting data. In this section,only graphical methods of misrepresentation are considered.

Vertical axis and horizontal axisIt is a truism that the steeper the graph the better the growth appears. A ‘rule of thumb’for statisticians is that for the sake of appearances, the vertical axis should be two-thirds to three-quarters the length of the horizontal axis. This rule was established inorder to have some comparability between graphs.

The following figure illustrates how distorted the graph appears when the verticalaxis is disproportionately large.

8 To count the number of 2’s, in Cell F4 enter the formula =COUNTIF(B3:B14,2).

9 Complete Cells F5, F6 and F7 with similar formulas to count the number of responses coded 3, 4 and 5 in Column B.

10 Use the Chart Wizard with Columns E and F to display graphically the responses.

11 Print out a copy of your spreadsheet and graph/s.

Spreadsheets — Displaying data from the Web

In previous web searches you have noted web sites with interesting and relevant data. Select one of these sites and obtain the necessary data.

1 Enter the data into a spreadsheet. Remember to include meaningful headings to enhance the quality of your display.

2 Present your data in both table and graphical form (don’t forget to label axes and include a title).

3 Print out a copy of your presentation and arrange it in a poster format.

4 Examine your graph/s and write a summary of the data.

5 Present your findings to the class.

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351

Changing the scale on the vertical axis

The following table gives the holdings of ROPE Corporation during 2001.

QuarterHoldings in $’000 000

J–MA–JJ–SO–D

200200201202

Secondaryroads

1000

2000

3000

4000

5000

6000

7000

8000

9000

10 000

11 000

12 000

13 000

Sealed

Paved

Formed

Unconstructed

Statehighways

Urban andsub arterials

Main roads

Developmentalroads

Declared roads as at 30 June 1979

Len

gth

of r

oad

(km

)

Source: Dept of Mapping & Surveying (1980),Queensland resources atlas, 2nd rev. ed.(Courtesy Dept of Lands)


352


Here is one way of representing this data:

But it is not very spectacular, is it? Now look at the following graph.

The shareholders would be happier with this one.

Omitting certain values

If one chose to ignore the second quarter’s value, which shows no increase, then the graph would look even better.

Foreshortening the vertical axis

Look at the figures below. Notice in graph (a) that the numbers from 0 to 4000 havebeen omitted. In graph (b) these numbers have been inserted. The rate of growth of theQueensland Police Force looks far less spectacular in graph (b) than in graph (a).

Foreshortening the vertical axis is a very common procedure. It does have theadvantage of giving extra detail but it can give the wrong impression about growthrates.

100

200

300

Hol

ding

s ($

’000

000

)

Quarter

x x x x

J-M A-J J-S O-D

200.5

201

201.5

Hol

ding

s ($

’000

000

)

Quarter

x x

x

x

J-M A-J J-S O-D200

202

200.5

201

201.5H

oldi

ngs

($’0

00 0

00)

Quarter

x

x

x

J-M J-S O-D200

202

4500

5000

5500

Qld

pol

ice

stre

ngth

794000

1977 81 83 85 87

x x x

xx x

x xx

x

x

1000

2000

3000

Qld

pol

ice

stre

ngth

791977 81 83 85 87

xx

xx x x x x

(a) (b)

4000

5000 x x

x

Source: Qld Year Book, 1989, p.53 and The Australian Bureau of Statistics.

YearYear



353

Visual impression

In this graph, height is the property that gives the true relation, yet the impression of amuch greater increase is given by the volume of each money bag.

A non-linear scale on an axis or on both axes

Consider the following two graphs.

Both of these graphs show the same numerical information. But graph (a) has alinear scale on the vertical axis and graph (b) does not. Graph (a) emphasises the ever-increasing rate of growth of pollutants while graph (b) suggests a slower, linear growth.

400

300

200

100

$ m

1990 1995 2000

S SYear

Net value of production

S

100

200

300

400

500

95 96 97 98 99

Part

icle

s / u

nit a

rea

Year(a)

100

200

300

95 96 97 98 99

Part

icle

s / u

nit a

rea

Year

500

(b)

The following data give wages and profits for a certain company. All figures are in millionsof dollars.

Continued over page

Year 1985 1990 1995 2000

Wages% increase in wagesProfits% increase in profits

625

120

9501·550

13442·566

2054

5100

8WORKEDExample


354


Now consider the graphs:

Consider the following questions.a Do the graphs accurately reflect the data?b Which graph would you rather have published if you were:

ii an employer dealing with employees requesting pay increases?ii an employee negotiating with an employer for a pay increase?

THINK WRITE

a Look at the scales on both axes. All scales are linear.

a Graphs do represent data accurately. How-ever, quite a different picture of wage and profit increases is painted by graphing with different units on the y-axis.

Look at the units on both axes. Graph (a) has y-axis in $ while graph (b) has y-axis in %.

b ii Compare wage increase with profit increases.

i The employer wants high profits.

b ii The employer would prefer graph (a) because he/she could argue that employees’ wages were increasing at a greater rate than profits.

ii Consider again the increases in wages and profits.The employee doesn’t like to see profits increasing at a much greater rate than wages.

ii The employee would choose graph (b), arguing that profits were increasing at a great rate while wage increases clearly lagged behind.

4

8

12

16

20

1985 1990 1995 2000

Wag

es a

nd p

rofi

ts (

$m)

Year

2

6

10

14

18Wages

Profits

(a)

25

50

75

100

1985 1990 1995 2000

Wag

es a

nd p

rofi

ts (

% in

crea

se)

Year

Profits

Wages

(b)

1

2

1

2

1

2

rememberTo determine whether data in graphical form have been misrepresented, check that:1. scale on both axes is linear2. scales on vertical and horizontal axes have not been lengthened or shortened to

give a biased impression3. certain values have not been omitted in the graph4. picture graphs are drawn to represent a height and not a volume

remember



355

Graphical methods of misrepresenting data

1

This graph shows the dollars spent on health care for 1990, 1994, and 1998. Draw another bar graph that minimises the fall in health care funds.

2

Examine this graph of employment growth.

Why is this graph misleading?

3

Examine this graph.

a

Redraw this graph with the vertical axis showing road fatalities starting at 0.

b

Does the decrease in road fatalities appear to be as significant as the graphsuggests?

9DWORKEDExample

8

101

102

103

104

1990 1994 1998

Hea

lth c

are

cost

($

m)

100

105

0

1

2

3

1947 1954 1961 1966

Growth of total employment, 1947–1981

1971 1976 1981

4

5

6

Tota

l em

ploy

men

t(m

illio

ns)

400

450

500

550

600

650

1987 1984 1981 1978 1975 1972 1969

Road fatalities, Queensland

Source: Qld Year Book, 1989, p. 205 and the Australian Bureau of Statistics.


356


4

This graph shows the student-to-teacher ratio in Queensland for the years 1979 to 1987.Apart from the fact that the 0 on the vertical scale has not been shown, this graph ismisleading.

In 1981 what was the student-to-teacher ratio at:

a

non-government schools?

b

government schools?

How can you explain this when most classes in the city government schools are 25 ormore? Explain fully.

5

You run a company that is listed on the Stock Exchange. During 2002 you have givensubstantial rises in salary to all your staff. However, profits have not been as spectacularas in 2001. The following table gives the figures for the mean salary and profits foreach quarter. Draw 2 graphs, one showing profits, the other showing salaries, that willshow you in the best possible light to your shareholders.

6

You are a manufacturer and your plant is discharging heavy metals into a waterway.Your own chemists do tests every 3 months and the following table gives theresults for a period of 2 years. Draw a graph which will show your company in thebest light.

1st quarter 2nd quarter 3rd quarter 4th quarter

Profits$’000 000Salaries$’000 000

6

4

5·9

5·9

6

6

6·5

7·5

2000 2001

DateConcentration(parts per million)

Jan.

7

Apr.

9

July

18

Oct.

25

Jan.

30

Apr.

40

July

49

Oct.

57

15

17

19

21

1987 1985 1983 1981

Student to teacher ratio, Queensland

1979

Rat

e

Non-government

Government(a)

(a) Break in continuity of seriesSource: Qld Year Book, 1989, p. 125 and The Australian Bureau of Statistics.



357

7

This pie graph shows the break-up of workers compensation costs incurred byemployers other than the government.

a

What fraction of the total costs are weekly compensation payouts and statutorylump sum claims?

b

What angle should be at the centre of this sector?

c

What angle is at the centre of this sector?

d

Why has this distortion of angle occurred? Discuss how this might be used to mis-lead the reader.

8

This graph shows how the $27 that a buyer pays for a CD is distributed among thedepartments involved in its production and marketing.

You are required to find out whether or not the graph is misleading, to explain fullyyour reasoning, and to support any statements that you make. Also,

a

comment on the shape of the graph and how it could be obtained.

b

Does your visual impression of the graph support the figures?

The community is constantly bombarded with graphical representations of data. Wemust be aware that these displays are carefully constructed so that a cursory glanceconveys the impression intended by the presenter. On closer inspection, a differentimpression is often revealed. It is wise to look at the detail in a graph and not to relysimply on the overall visual impact.

Break-up of non-government workers compensation costs

Common law claims $143.5m Common law fees

and outlays $19m

Total $202.8m

Source: Courier-Mail, 21 September 1991.

Weekly compensationpayouts and statutory lump

sum claims $40.3m

Where your $27 goes

Mechanicalroyalties $1.57

Record companyprofit $1.54

Advertising $1.94 Sales tax

$3.27Production $3.40

Other recordingcosts 65c

Distribution 56c

Record shop $7.40

Record companyadministration costs $1.54

Royalties and costs to artist $3.86

Record companysales process $1.27



Misleading reports1 Look in any newspaper, magazine or government publication and select three

different types of statistical presentation. Discuss the reasons why each particular type of presentation was chosen and whether other forms of presentation might have been just as good.

2 Look at the annual report of a company and discuss critically the forms of statistical presentation used.

3 Collect three examples of misleading graphs. For each graph explain how it is misleading. How do such misleading graphs benefit the people who produced the graphs?

4 Work in groups to prepare misleading company reports with graphs of wages, profits and percentage increases. Present a report to the class. Other students must spot the flaws.

5 Work in groups to design a suitable advertisement to explain why someone should become a member of your club.a Draw a graph to support your claims.b Present your graph and argument to the class.c While other groups are presenting, list any information they have left out.d Find other advertisements that use this technique to give certain viewpoints.e List some additional information you think should have been given.

Spreadsheets creating misleading graphs

Let us return to worked example 8 and use a spreadsheet to draw graph (a).

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vestigation

investigation



Other forms of graphical displayWe shall now consider three other forms of graphical display of data: histograms, stemplots and boxplots.

Displaying data using frequency histogramsA frequency histogram is similar to a column graph with the following essentialfeatures.1. Gaps are never left between the columns, except for a half-unit space before the first

column.2. If the chart is coloured or shaded then it is done all in one colour. (The columns are

essentially all representing different levels of the same thing.)3. Frequency is always plotted on the vertical axis.4. For ungrouped data the horizontal scale is marked so that the data labels appear

under the centre of each column. For grouped data the horizontal scale is marked sothat the class centre of each class appears under the centre of the column.

1 Enter the data as indicated in the spreadsheet above.

2 Graph the data using the Chart Wizard. You should obtain a graph similar to Graph 1.

3 Copy and paste the graph twice within the spreadsheet.

4 Graph 2 gives the impression that the wages are a great deal higher than the profits. This effect was obtained by reducing the length of the horizontal axis. Experiment with shortening the horizontal length and lengthening the vertical axis.

5 In Graph 3 we get the impression that the wages and profits are not very different. This effect was obtained by lengthening the horizontal axis and shortening the vertical axis. Experiment with various combinations.

6 Print out your three graphs and examine their differences.

Note that all three graphs have been drawn from the same data using valid scales. A cursory glance leaves us with three different impressions. Clearly, it is important to look carefully at the scales on the axes of graphs.

Another method which could be used to change the shape of a graph is to change the scale of the axes.

7 Right click on the axis value, enter the Format axis option, click on the Scale tab, then experiment with changing the scale values on both axes.

Techniques such as these are used to create different visual impressions of the same data.

8 Use the data in the table of worked example 8 to create a spreadsheet, then produce two graphs depicting the percentage increase in both wages and profits over the years giving the impression that:a the profits of the company have not grown at the expense of wage increases

(the percentage increase in wages is similar to the percentage increase in profits)

b the company appears to be exploiting its employees (the percentage increase in profits is greater than that for wages).



A frequency polygon is a line graph that can be drawn by joining the centres of the tops of each column of the histogram. The polygon starts and finishes on the horizontal axis a half column width space from the group boundary of the first and last groups.

The figure at right shows the frequency polygon drawn on top of the histogram for the previous worked example.

It is common practice to draw the histogram and the polygon on the same set of axes.

The table below shows the number of people living in each house in a street.

Show this information in a frequency histogram.

No. of people Frequency1 12 43 104 155 8

THINK WRITE

Draw a set of axes with the number of people living in a house on the horizontal axis and frequency on the vertical axis.Draw the graph, leaving half a column width space before the first column.

1

102468

10121416

2 3 4 5Number of people in a house

Freq

uenc

y

2

9WORKEDExample

102468

10121416

2 3 4 5Number of people in a house

Freq

uenc

y

The frequency table below shows a class set of marks on an exam. Draw a frequency histogram and polygon on the same set of axes.

Mark Class centre Frequency51–60 55.5 361–70 65.5 571–80 75.5 1281–90 85.5 791–100 95.5 3

10WORKEDExample



A graphics calculator can also be used to draw histograms. The instructions givenbelow are for the Texas Instruments TI 83 graphics calculator. If you have another var-iety, your teacher will show you the equivalent commands.

THINK WRITE

Draw a set of axes with the exam mark on the horizontal axis and frequency on the vertical axis. Show the class centres for the exam marks.Draw the columns, leaving a half-column-width space before the first column.Draw a line graph to the centre of each column.Make sure the line graph begins and ends on the horizontal axis.

1

55.502468

1012

65.575.5

85.595.5

Exam mark

Freq

uenc

y

2

34

The marks out of 20 received by 30 students for a book-review assignment are given in the frequency table below.

Display these data on a histogram using a graphics calculator.

Mark 12 13 14 15 16 17 18 19 20

Frequency 2 7 6 5 4 2 3 0 1

THINK DISPLAY

Enter the data.(a) Clear any previous equations.(b) Press and clear any functions.

(c) Press , select 1:Edit and press .(d) Enter the marks in L1 and the frequency in L2.

Set up the calculator for graphing.(a) Press [STAT PLOT] and select 1:Plot1.

Press .(b) Select On and press .(c) Select the type of graph required. The

histogram is the third along on the top row.(d) At Xlist type in L1.(e) At Freq type in L2.(f) Press and highlight 9:Zoom Stat;

press .(g) If not all of the histogram is shown, press

and reset the x- and y-range and step values.

1

Y=STAT ENTER

2

2ndENTER

ENTER

ZOOMENTER

WINDOW

11WORKEDExample



Histograms and frequency polygons

This exercise can be completed manually, or with the aid of a graphics calculator.

1 A survey is done on young drivers taking the written test for their licence. The number of mistakes each makes is recorded and the results are shown in the frequency distribution table at right.Show this information in a frequency histogram.

2 Students in a class were asked the number of children in their families. The results are shown in the frequency distribution table at right.Show this information in a frequency histogram and polygon.

3 The table below shows the age in years of the members of a surf club.Show this information in a frequency polygon.

remember1. Numerical data can be graphed using histograms and polygons.2. When drawing histograms always put frequency on the vertical axis and never

leave gaps between columns.3. If the histogram is illustrating ungrouped data, the data labels on the horizontal

axis are placed under the centre of each column.4. If the histogram is illustrating grouped data, the data labels on the horizontal

axis (that is, the class centres) are placed under the centre of each column.5. A frequency polygon is a line graph, which can be drawn by joining the centres

of the tops of each column of the histogram.

remember

9E

GCpro

gram

UV statistics

WORKEDExample

9

No. of mistakes (score)

No. of drivers (frequency)

0 5

1 8

2 11

3 4

4 3

5 1

EXCEL

Spreadsheet

Histograms and frequency polygons

No. of children in a family Frequency

0 3

2 5

3 8

4 4

5 2

6 1

Age No. of members

18 3

19 5

20 8

21 13

22 15

23 10

24 8

25 5


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 3634 The label on a box of matches states that the average contents of a box is 50 matches.

Quality control surveyed 50 boxes for the number of matches and the results areshown below.

48 50 50 51 50 49 53 52 48 51

50 50 51 49 48 53 52 50 49 49

49 50 50 51 53 52 54 47 50 49

48 49 47 53 49 52 50 51 50 50

50 48 47 50 51 49 50 49 52 51

a Put this information into a frequency table.b Show the results in a frequency histogram and polygon.

5 The table below shows the length of 71 fish caught in a competition.

Show this information in a frequency histogram and polygon.

6 Sixty people were involved in a psychology experiment. The following frequencytable shows the times taken for the 60 people to complete a puzzle for the experiment.

a Copy the frequency table and complete the class centre column.b Show the information in a frequency histogram and polygon.

Length of fish (mm) Class centre Frequency

300–309 304.5 9

310–319 314.5 15

320–329 324.5 20

330–339 334.5 12

340–349 344.5 8

350–359 354.5 7

Time taken (seconds) Class centre Frequency

6 to almost 8 1

8 to almost 10 4

10 to almost 12 15

12 to almost 14 18

14 to almost 16 12

16 to almost 18 8

18 to almost 20 2

WORKEDExample

10



Stem-and-leaf plotsAs an alternative to a frequency table, a stem-and-leaf plot may be used to group andsummarise data.

A stem is made using the first part of each piece of data. The second part of eachpiece of data forms the leaves. Consider the case below.

The following data show the mass (in kg) of 20 possums trapped, weighed thenreleased by a wildlife researcher.

1.8 0.9 0.7 1.4 1.6 2.1 2.7 2.2 1.8 2.32.3 1.5 1.1 2.2 3.0 2.5 2.7 3.2 1.9 1.7

The stem is made from the whole number part of the mass and the leaves are thedecimal part. The first piece of data was 1.8 kg. The stem of this number could be con-sidered to be 1 and the leaf 0.8. The second piece of data was 0.9. It has a stem of 0 anda leaf of 0.9.

To compose the stem-and-leaf plot, rule a vertical column of stems then enter the leaf of each piece of data in a neat row beside the appropriate stem. The first row of the stem-and-leaf plot records all data from 0.0 to 0.9. The second row records data from 1.0 to 1.9 etc.

Attach a key to the plot to show the reader the meaning of each entry.It is convention to assemble the data in order of size, so this stem-and-leaf plot

should be written in such a way that the numbers in each row of ‘leafs’ are in ascendingorder.

Key: 0 | 7 = 0.7 kg

When preparing a stem-and-leaf plot, it is important to try to keep the numbers inneat vertical columns because a neat plot gives the reader an idea of the distribution ofscores. The plot itself looks a bit like a histogram turned on its side.

Stem0123

Leaf7 91 4 5 6 7 8 8 91 2 2 3 3 5 7 70 2



It is also useful to be able to represent data with a class size of 5. This could be donefor the previous stem-and-leaf plot by choosing stems 0*, 1, 1*, 2, 2*, 3 where the classwith stem 1 contains all the data from 1.0 to 1.4 and stem 1* contains the data from 1.5to 1.9 etc. If stems are split in this way it is a good idea to include two entries in thekey. The stem-and-leaf plot for the ‘possum’ data would appear as follows:

Key: 1 | 1 = 1.1 kg 1* | 5 = 1.5

A stem-and-leaf plot has the following advantages over a frequency distribution table.1. The plot itself gives a graphical representation of the spread of data. (It is

rather like a histogram turned on its side.)2. All the original data are retained, so there is no loss of accuracy when

calculating statistics such as the mean and standard deviation. In a grouped frequency distribution table some generalisations are made when these values are calculated.

Stem0*1*1*2*2*3*

Leaf7 91 45 6 7 8 8 91 2 2 3 35 7 70 2

The information below shows the mass, in kilograms, of twenty 16-year-old boys.65 45 56 57 58 54 61 72 70 6961 58 49 52 64 71 66 65 66 60

Show this information in a stem-and-leaf plot.

THINK WRITE

Make the ‘tens’ the stem and the ‘units’ the leaves.Write a key. Key: 5 | 6 = 56 kgComplete the plot.

Note: Complete the plot with the leaves in each row in ascending order.

1

23 Stem

4567

Leaf5 96 7 8 4 8 25 1 9 1 4 6 5 6 02 0 1

Stem4567

Leaf5 92 4 6 7 8 80 1 1 4 5 5 6 6 90 1 2

12WORKEDExample



Since all the original data are recorded on the stem-and-leaf plot and are convenientlyarranged in order of size, the plot can be used to locate the upper and lower quartilesand the median.Notes1. the median is the middle score or the average of the two middle scores2. the lower quartile is the median of the lower half of the data3. the upper quartile is the median of the upper half of the data.

Using the ‘possum’ mass data as an example:

Key: 0 | 7 = 0.7 kg

Stem0123

Leaf7 91 4 5 6 7 8 8 91 2 2 3 3 5 7 70 2

The following data give the length of gestation in days for 24 mothers. Prepare a stem-and-leaf plot of the data using a class size of 5.

280 287 285 276 266 292288 273 295 279 284 271292 288 279 281 270 278281 292 268 282 275 281

THINK WRITE

A group size of 5 is required. The smallest piece of data is 266 and the largest is 295 so make the stems: 26*, 27, 27*, 28, 28*, 29, 29*.The key should give a clear indication of the meaning of each entry.Enter the data piece by piece. Enter the leaves in pencil at first so that they can be rearranged into order of size. Check that 24 pieces of data have been entered.Now arrange the leaves in order of size. Key: 26* | 6 = 266 days

27 | 0 = 270 days

1

2

3

Stem26*27*27*28*28*29*29*

Leaf6 80 1 35 6 8 9 90 1 1 1 2 45 7 8 82 2 25

13WORKEDExample


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 367There were 20 records so the median is the

average of the 10th and 11th scores. Counting each score as it appeared in the stem-and-leaf plot we can see that the 10th score is the number 1.9 and the 11th score is the number 2.1.

Median =

= 2.0 kgThe median divides the data into halves.The lower quartile is the median of the lower half which has ten scores in it.

So the position of the lower quartile is given by the average of the 5th and 6th scores.The 5th score is the number 1.5. The 6th score is the number 1.6.

The lower quartile =

= 1.55 kgThe upper quartile is the median of the upper half which also has ten scores in it. The 5th score in this half is the number 2.3. The 6th score is the number 2.5.

The upper quartile =

= 2.4 kgThe interquartile range is the difference between the upper and lower quartiles.

1.9 2.1+2

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

1.5 1.6+2

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

2.3 2.5+2

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

Find the interquartile range of the data presented in the following stem-and-leaf plot.Key: 15 | 7 = 157 kg

Stem151617181920

Leaf4 8 81 3 3 6 80 0 1 4 7 9 9 91 2 3 3 5 7 8 8 92 7 80 2

THINK WRITE

There are 30 scores and so the median will be the average of the 15th and 16th scores.

Median =

Median = 179 kg

There are 15 scores in each half and so the lower and upper quartiles will be the 8th score in each half.

The lower quartile = 168 kgThe upper quartile = 188 kg

The interquartile range is the difference between the upper and lower quartiles.

Note: Remember to provide appropriate units for the answer.

Interquartile range = upper quartile − lower quartile

= 188 − 168= 20 kg

1179 179+

2------------------------

2

3

14WORKEDExample



Stem-and-leaf plots

1 The data below give the number of errors made each week by 20 machine operators.Prepare a stem-and-leaf diagram of the data using a stem of 0, 1, 2, etc.

6 15 20 25 28 18 32 43 52 2717 26 38 31 26 29 32 46 13 20

2 The data below give the time taken (in minutes) for each of 40 runners on a 10 km funrun. Prepare a stem-and-leaf diagram for the data using a class size of 10 minutes.

36 42 52 38 47 59 72 68 57 8266 75 45 42 55 38 42 46 48 3942 58 40 41 47 53 68 43 39 4871 42 50 46 40 52 37 54 48 52

3 The typing speed (in words per minute) of 30 word processors is recorded below.Prepare a stem-and-leaf diagram of the data using a class size of 5.

96 102 92 96 95 102 95 115 110 10888 86 107 111 107 108 103 121 107 96

124 95 98 102 108 112 120 99 121 130

4 Twenty transistors are tested by applying increasing voltage until they are destroyed.The maximum voltage that each could withstand is recorded below. Prepare a stem-and-leaf plot of the data using a class size of 0.5.

14.8 15.2 13.8 14.0 14.8 15.7 15.5 15.6 14.7 14.314.6 15.2 15.9 15.1 14.3 14.6 13.9 14.7 14.5 14.2

5 The stem-and-leaf plot at right gives the exact mass of 24 packets of biscuits. Find the interquartile range of the data.

rememberWhen presenting stem-and-leaf plots, observe the following points.1. Always include a key to assist in the interpretation of the plot.2. Choose a suitable class size. A class size of 5 is possible by using *notation on

class stems.3. After initially recording each score, rearrange the leaves so that they appear in

ascending order.

remember

9FWORKEDExample

12

WORKEDExample

13

WORKEDExample

14Key: 248 | 4 = 248.4 g

Stem248249250251252253

Leaf4 7 82 3 6 60 0 1 1 6 9 91 5 5 5 6 71 5 80


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 3696 The time taken (in seconds) for a test vehicle to

accelerate from 0 to 100 km/h is recorded during a test of 24 trials. The results are represented by the stem-and-leaf plot at right.a Find the median of the data.b Find the upper and lower quartiles of the data.c Find the interquartile range of the data.

Questions 7 to 10 refer to the stem-and-leaf plot below.Key: 12 | 1 = 1210Key: 12* | 5 = 1250

7

The class size used in the stem-and-leaf plot is:

8

The number of scores that have been recorded is:

9

The median of the data is:

10

The interquartile range of the data is:

11 The maximum hand spans (in cm) of 20 male concert pianists are recorded as follows.

23.6 20.2 22.8 21.4 25.1 24.8 23.2 21.6 20.7 23.622.8 24.6 21.8 22.8 23.1 24.6 21.7 24.7 22.2 23.0

a Complete a stem-and-leaf plot to represent the data.b Find the median of the data.c Find the upper and lower quartiles of the data.d Find the interquartile range of the data.

Stem12*12*13*13*14*14*

Leaf1 2 45 7 7 9 90 1 1 2 3 4 45 6 6 7 9 90 2 3 46 7

A 1 B 10 C 33 D 50

A 27 B 33 C 1210 D 1410

A 13.4 B 14 C 1335 D 1340

A 14 B 100 C 1290 D 1390

Key: 7 | 2 = 7.2 sKey: 7* | 6 = 7.6 s

Stem7*7*8*8*9*9*

Leaf2 4 45 5 7 90 0 1 2 4 4 45 5 6 8 92 2 35 7







12 The heights (in cm) of a sample of 30 plants are recorded as follows.

93 88 94 99 91 85 126 107 110 11198 96 117 101 97 92 101 132 103 82

114 84 96 103 108 115 90 110 126 85

a Complete a stem-and-leaf plot to represent the data.b Find the median of the data.c Find the upper and lower quartiles of the data.d Find the interquartile range of the data.

Boxplots (box-and-whisker plots)In drawing a boxplot, five values are extracted from the data set. These values sum-marise the data and hence attract the name of a five-number summary.

This five-number summary consists of:• lower extreme — the lowest score in the data set• lower quartile — the middle score of the lower half of the data set• median — the middle score • upper quartile — the middle score of the upper half of the data set• upper extreme — the highest score in the data set.

Once a five-number summary has been developed, it can be graphed using a box-and-whisker plot, a powerful way to display the spread of the data.

The box-and-whisker plot consists of a central divided box with attached whiskers.The box spans the interquartile range, the vertical line inside the box marks the medianand the whiskers indicate the range. Each section of the boxplot represents one quarterof the scores of the data set.

For the set of scores below, develop a five-number summary.

12 15 46 9 36 85 73 29 64 50

THINK WRITE

Re-write the list in ascending order. 9 12 15 29 36 46 50 64 73 85Write the lowest score. Lower extreme = 9Calculate the lower quartile. Lower quartile = 15

Calculate the median. Median =

= 41Calculate the upper quartile. Upper quartile = 64Write the upper extreme. Upper extreme = 85

Five-number summary = 9, 15, 41, 64, 85

123

436 46+

2------------------

56

15WORKEDExample

Lowerextreme

Upperextreme

Lowerquartile

Upperquartile

Median

of scores

1–4 of scores

1–4

of scores

1–4 of scores

1–4


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 371Box-and-whisker plots are always drawn to scale. This can be drawn with the five-

number summary attached as labels:

or with a scale presented alongside the box-and-whisker plot. (This representation ispreferable.)

4 2815 21 23

0 5 10 15 20 25 30Scale

The box-and-whisker plot drawn at rightshows the marks achieved by students on their end of year exam.a State the median.b Find the interquartile range.c What was the highest mark in the class?

THINK WRITE

a The mark in the box shows the median (72).

a Median = 72 marks

b The lower end of the box shows the lower quartile (63).

b Lower quartile = 63 marks

The upper end of the box shows the upper quartile (77).

Upper quartile = 77 marks

Subtract the lower quartile from the upper quartile.

Interquartile range = 77 − 63 = 14 marks

c The top end of the whisker gives the top mark (92).

c Top mark = 92 marks

0 10 20 30 40 50 60Marks achieved

70 80 90 100

1

2

3

16WORKEDExample

After analysing the speed (in km/h) of motorists through a particular intersection, the following five-number summary was developed.

The lowest score is 82 km/h.The lower quartile is 84 km/h.The median is 89 km/h.The upper quartile is 95 km/h.The highest score is 114 km/h.

Show this information in a box-and-whisker plot.Continued over page

17WORKEDExample



A graphics calculator can also be used to display data in the form of a boxplot.

THINK WRITE

Draw a scale from 70 to 120 using1 cm = 10 km/h.

Draw the box from 84 to 95.

Mark the median at 89.

Draw the whiskers to 82 and 114.

1

70 80 90Speed in km/h

100 110 120

2

3

4

Use a graphics calculator to draw a boxplot displaying the data below.

3 6 4 8 17 12 9 7 13 13 5 9 7 2 1 7 5 4 2

THINK DISPLAYClear any previous functions.(a) Press .(b) Press to clear any functions.

Enter data.(a) Press [STAT PLOT].(b) Highlight 4:Plots Off.(c) Press .(d) Enter the list in L1 (press , select

1:Edit... and press to access the screen).

Draw the boxplot.(a) Press [STAT PLOT] then select

1:Plot1.(b) Press .(c) Select Plot1, then On, then select the

5th icon along which indicates a boxplot; then at Xlist: enter L1 (use [L1]), and at Freq: enter 1.

(d) Press and select 9:ZoomStat.(e) Press and the boxplot will

appear.(f) Use the trace key to locate the minimum,

maximum and quartile values.

1Y=

CLEAR

1 47

9 17

22nd

ENTERSTAT

ENTER

32nd

ENTER

2nd

ZOOMENTER

18WORKEDExample



Exploring boxplotsBoxplots provide a powerful tool for comparing sets of data. It is essential, therefore, to understand and interpret correctly the shape of a boxplot.

Situation 1Consider the following situation.Ten randomly chosen students from Class X and Class Y each sit for a test where the highest possible mark is 10. The results of the ten students from the two classes are:Class X: 1 2 3 4 5 6 7 8 9 10Class Y: 1 2 2 3 3 4 4 5 9 10Note that the range of the marks is the same in both class samples (1 to 10). In fact, the two highest and two lowest marks are the same in both cases.

Determining the five-number summaries for both class samples shows:Class X 1, 3, 5.5, 8, 10Class Y 1, 2, 3.5, 5, 10You should confirm these figures.If a boxplot was drawn for each set of data and the two plots were shown on the same scale, we would observe the following:

In interpreting the differences between the two plots, we must recall the following.1. The boxplot is divided into four sections and each section represents one

quarter of the scores.2. The median represents a score in the middle of the data.3. A long ‘box’ or a long ‘whisker’ does not tell us that there are more scores in

that section. There is still one quarter of the scores in each whisker and one half of the scores in the whole box. An increase in length indicates that the scores are more spread out.

From the boxplots, answer the following questions:

1 The scores in the lower whisker of the Class X sample range from 1 to 3. What is the range of scores for the lower whisker of the Class Y sample? How many scores lie in each of these whiskers?

2 What is the range of scores in the upper whisker of the Class Y plot? How many scores lie in this area? Why is the upper whisker longer than the lower whisker?

3 It is true to say that three-quarters of the Class Y student sample performed more poorly than half the student sample in Class X. Explain how the graph shows this.

4 It is also true to say that more than half the student sample from Class X performed as well as the top quarter of the student sample from Class Y. Explain how the graph indicates this.

inve


n

1 2 3 4 5 6 7 8 9 10

Class Y

Class X



Five-number summaries and boxplots

A graphics calculator can be used for many of the following.

1 Write a five-number summary for the data set below.

15 17 16 8 25 18 20 15 17 14

2 For each of the data sets below, write a five-number summary.

a 23 45 92 80 84 83 43 83b 2 6 4 2 5 7 1c 60 75 29 38 69 63 45 20 29 93 8 29 93

3 From the five-number summary 6, 11, 13, 16, 32 find:a the medianb the interquartile rangec the range.

4 From the five-number summary 101, 119, 122, 125, 128 find:a the medianb the interquartile rangec the range.

Situation 2Ten randomly chosen students from Class Z sat for the same test with the following results.Class Z: 1 2 5 5 6 7 8 8 9 10

5 Determine the five-number summary for the Class Z sample.

6 Draw parallel boxplots of the three class samples, using the same horizontal scale.

7 Write a report comparing the performances of the three classes on the test. You must refer to specific data from the boxplots to support your conclusions.

remember1. A five-number summary is a summary set for a distribution.2. The five numbers used in a five-number summary are the lower extreme, lower

quartile, median, upper quartile and upper extreme.3. A box-and-whisker plot can be used to graph a five-number summary.4. The box is used to show the interquartile range and the median is marked with

a line in the box. The whiskers then extend to show the range of the data set.

remember

9G

WORKEDExample

15

EXCEL

Spreadsheet

Interquartile range

WORKEDExample

16


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 3755 A five-number summary is given below.

Lower extreme = 39.2 Upper quartile = 52.3Lower quartile = 46.5 Upper extreme = 57.8Median = 49.0

Draw a box-and-whisker plot of the data.

6 The box-and-whisker plot at right shows the distribution of final points scored by afootball team over a season’s roster.a What was the team’s greatest points score?b What was the team’s smallest points score?c What was the team’s median points score?d What was the range of points scored?e What was the interquartile range of points scored?

7 The box-and-whisker plot at right shows the distribution of data formed by counting the number of honey bears in each of a large sample of packs.In any pack, what was:a the largest number of honey bears?b the smallest number of honey bears?c the median number of honey bears?d the range of numbers of honey bears?e the interquartile range of honey bears?

Questions 8, 9 and 10 refer to the box-and-whisker plot drawn below.

8

The median of the data is:A 20 B 23 C 35 D 31

WORKEDExample

17

EXCEL Spreadsheet

Boxplots

50 70 90 110 130 150 Points

30 35 40 45 50 55 60 Scale

5 10 15 20 25 30 Scale




9

The interquartile range of the data is:

10

Which of the following is not true of the data represented by the box-and-whiskerplot?A One-quarter of the scores is between 5 and 20.B One-half of the scores is between 20 and 25.C The lowest quarter of the data is spread over a wide range.D Most of the data are contained between the scores of 5 and 20.

11 The number of sales made each day by a salesperson is recorded over a fortnight:

25, 31, 28, 43, 37, 43, 22, 45, 48, 33

a Write a five-number summary of the data.b Draw a box-and-whisker plot of the data.

12 The data below show monthly rainfall in millimetres.

a Provide a five-number summary of the data.b Draw a box-and-whisker plot of the data.

A 23 B 26 C 5 D 20 to 25

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

10 12 21 23 39 22 15 11 22 37 45 30

Drawing conclusions from the bottled-water investigations

Having considered ways of collecting, organising and displaying data, it is now appropriate that you should draw together all your results from your investigations on bottled water and form them into a report. You have collected data through:1. observation2. survey and3. taste testing.

1 Consider each activity separately. For each:a define the AIM of the investigationb outline the PROCEDURE undertakenc present a table of your RESULTS (preferably on a spreadsheet)d represent your data in an appropriate GRAPHICAL FORM (preferably

using a computer or graphics calculator)e summarise and draw some CONCLUSIONS from your investigation.

2 Present your whole investigation as a report to the class.



GCpro

gram

UV statistics

WorkS

HEET 9.2

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n



Classification of data• Data can be classified as being categorical or numerical.• Categorical data are data that are non-numerical. For example, a survey of car types

is not numerical.• Numerical data are data that can be either counted or measured. For example a

survey of the daily temperature is numerical.• Numerical data can be either discrete or continuous. • Discrete data can take only certain values such as whole numbers.• Continuous data can take any value within a certain range.

Collecting data• Data can be collected by observation, survey or experiment.• Observations can be made on categorical or numerical data.• Surveys can be conducted through personal interview, telephone interview, or self-

administered questionnaire.• Questionnaires require careful construction, particularly with question construction

and classification.• Experiments must be capable of being reproduced for conclusions to be drawn.• In all cases of collecting data, the quality of the data must be investigated before

any processing occurs.

Organising and displaying data• Data are usually organised by first collating the records in the form of a table using

grouped or ungrouped data.• Data are generally displayed graphically. Types of graphs include column, pie,

histogram, stem plots or boxplots.

Misrepresenting data graphically• View graphical presentations carefully to detect signs of bias in impressions.• Check that scales on both axes are linear and not lengthened or shortened.• Check that picture graphs are not represented as volumes.

Histograms and frequency polygons• Numerical data are best displayed by a frequency histogram and polygon.• A frequency histogram is a column graph that is drawn with a 0.5-unit (half

column) space before the first column and no other spaces between the columns.• A frequency polygon is drawn as a line graph from the corner of the axes to the

centre of each column.

Stem-and-leaf plot• A stem-and-leaf plot is like a histogram on its side.• The first part of the data forms the stem.• The last part of the data forms the leaves.• The data in the leaves must be in ascending order.

Boxplots (box-and-whisker plots)• A five-number summary of a data set is the lower extreme, lower quartile, median,

upper quartile and upper extreme.• A five-number summary can be graphed using a box-and-whisker plot.• A box-and-whisker plot shows the spread of a data set on a scale.

summary



1 State whether each of the following data types is categorical or numerical.a The television program that people watch at 7:00 pmb The number of pets in each householdc The amount of water consumed by athletes in a marathon rund The average distance that students live from schoole The mode of transport used between home and school

2 For each of the numerical data types below, determine if the data are discrete or continuous.a The dress sizes of Year 11 girlsb The volume of backyard swimming poolsc The amount of water used in householdsd The number of viewers of a particular television programe The amount of time Year 11 students spent studying

3 Identify the areas of concern with the following questions, then rewrite each so that the meaning is clear and understandable.a Do you think that the school holidays are too long at Christmas time and that they

should be spread evenly over the year?b Have you never said that you don’t like chocolate?c Do you support the building of a fence around the duck pond to protect the poor

defenceless ducklings from the savage dogs in the area?d Did you know that the BSSSS administer the QCS test?e You support the police in their endeavour to reduce the road toll, don’t you?

4 A survey is conducted on the number of people living in each household in a street. The results are shown below.

1 4 5 2 2 3 4 6 1 2 56 4 4 6 3 2 3 5 1 3 43 3 4 2 2


5 A group of Year 11 students was asked to state the number of CDs that they had purchased in the last year. The results are shown below.

12 1 13 20 5 22 35 12 17 209 5 11 0 14 25 3 8 10 9

12 6 18 7 10 9 6 23 14 19

Put the results into a table using the categories 0–4, 5–9, 10–14 etc.

6 Draw a column and a sector graph to represent the results to question 4.

7 Draw a column and a sector graph to represent the results to question 5.

CHAPTERreview

9A

9A

9B

9C

9C

9C9C


C h a p t e r 9 C o l l e c t i n g a n d e n t e r i n g d a t a 3798 This table shows the number of students in each year level from Years 8 to 12.

Draw two separate graphs to illustrate the following:a The principal of the school claims a high retention rate of students in Years 11 and 12

(that is, most of the students from Year 10 continue on to complete Years 11 and 12).b The parents claim that the retention rate of students in Years 11 and 12 is low (that is, a

large number of students leave at the end of Year 10).

9 The table below shows the number of sales made each day over a month in a car yard.

Show this information in a frequency histogram and polygon.

10 The frequency table below shows the crowds at football matches for a team over a season.

11 Display the following scores in a stem-and-leaf plot.

45 21 38 46 42 41 42 49 35 29 24 2836 21 38 45 44 40 29 28 35 35 33 3840 41 48 39 34 38 45 28 23 29 30 40

12 Use the stem-and-leaf plot drawn in the previous question to find:a the range b the median c the interquartile range.

Year No. of students

8 200

9 189

10 175

11 133

12 124

Number of sales Frequency

0 2

1 5

2 12

3 6

4 2

5 0

6 1

Class Class centre Frequency

5000–9999 1

10 000–14 999 5

15 000–19 999 9

20 000–24 999 3

25 000–29 999 2

30 000–34 999 2

9D

9E

a Copy the frequency table and complete the class centre column.

b Show the information in a frequency histogram and polygon.

9E

9F

9F



13 For the data set below, give a five-number summary.

24 53 91 57 29 69 29 15 84 6

14 For the box-and-whisker plot drawn at right:a state the medianb calculate the rangec calculate the interquartile range.

15 The number of babies born each day at a hospital over a year is tabulated and the five-number summary is given below.

Lower extreme = 1 Upper quartile = 16Lower quartile = 8 Upper extreme = 18Median = 14

Show this information in a box-and-whisker plot.

The stem-and-leaf plot below refers to questions 16 to 20. It represents the number of typing errors recorded by a class of students in 1 page of typing.

16 How many students are in the class?

17 What is the median number of errors?

18 State the value of the lower quartile.

19 Determine the interquartile range.

20 Draw a boxplot of the data.

The box-and-whisker plots at rightrefer to questions 21 to 25. They show the sales of two different brands of washing powder at a supermarket each day.

21 State the range for Brand A.

22 State the interquartile range for Brand A.

23 State the range for Brand B.

24 State the interquartile range for Brand B.

25 Describe the spread of the sales for each brand of washing powder.

Key: 1 | 2 = 12 1* | 5 = 15Stem

0*0*1*1*2*

Leaf0 1 46 7 8 90 0 1 1 2 3 3 45 6 8 93

9G

9G0 5 10 15 20 25 30 35 40 45 50 55 60

9G

10 15 20 25 30 350 5 40 Scale

Brand A

Brand B

45 50

testtest

CHAPTERyyourselfourself

testyyourselfourself

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Technology

Maths A - Chapter 9