chapter_5

Collecting and Organising DataCollecting and Organising DataChapter 5Chapter 5This chapter deals with the display of statistical information.

At the end of this chapter you should be able to:

Sy l labus re ference DS4.1

recognise data as quantitative or categorical draw frequency distribution tables, histograms and polygons draw and use dot plots draw and use stem-and-leaf plots use the terms cluster and outlier when describing data.

116 Collecting and Organising Data (Chapter 5) Syllabus reference DS4.1

Diagnostic TestDiagnostic Test

An example of a categorical variable is:A height B number of carsC shoe size D eye colour

The amount of rainfall each month is a:A categorical variableB discrete numerical variableC continuous numerical variableD discrete categorical variable

Organise this data into a frequency distribution table and use it to answer questions 3 and 4.

Number of lbw decisions per innings:

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

The number of innings with 3 lbwdecisions is:A 3 B 4 C 5 D 6

The two scores with the same frequencyare:A 0 and 1 B 2 and 4C 0 and 4 D 2 and 3

Which score would be placed in the2125 class?A 20 B 23 C 26 D 29

Which looks the most like a line graph?A frequency polygonB frequency histogramC sector graph D column graph

Using the code C cola, W water, J orangejuice, D cordial and O other, which set ofdata was used to draw this dot plot?

A CCWCC CWOCW WDCCJB WOCCW JDCWC CCJWCC WOCCC WWCCJ WJCCWD CCCCC WJCCW WJDOC

The number with a stem of 2 and a leaf of8 is:A 10 B 82 C 28 D 8

This stem-and-leaf plot is used to answer questions 9 and 10.

The highest and lowest scores are:A 0 and 9 B 11 and 14C 10 and 41 D 110 and 141

The outlier score is:A 110 B 0 C 9 D 129

A stem-and-leaf plot is preferred over afrequency distribution table using classeswhen:A there are more than 50 scoresB all scores need to be shownC all scores dont need to be shownD the scores are large numbers

Which display would not show that ascore occurred three times?A dot plotB stem-and-leaf plotC frequency distribution tableD frequency distribution table with

classes

1

2

3

4

5

6

7

Stem Leaf

11 0

12 4 4 5 5 6 8 9 9 9

13 0 0 3 3 3 4 5 6 9

14 0 0 0 0 1 1 1

8

9

10

11

12

117Collecting and Organising Data (Chapter 5) Syllabus reference DS4.1

The Diagnostic Test questions refer to the sections of text listed in the table below.

Statistics is an important branch of mathematics. We use statistics to collect and display information.

Some examples include the top 10 pop songs, average monthly temperatures and the number of people usingparticular products.

The building blocks of statistics are the items of data collected. When this data is organised in some way that is useful, it has become information. When numerical data is organised in a meaningful way, the data is statistics. Statistics may be used to help people make informed decisions.

This chapter examines some of these aspects.

A. VARIABLES

The word variable is used to describe a number of things in many areas of mathematics, including algebra, graphing and statistics.

In statistics a variable is something that has different values for different people or items.

Examples of variables include: age, weight, height, method of travelling to school, eye colour, left or righthandedness, marks in a test.

So, when collecting data we are interested in a particular property or characteristic of a group of people orobjects. The particular characteristic that we are interested in is the variable.

A nominal or categorical variable describes categories that are the names of things. Examples include method of transport to school, eye colour, and gender.

A numerical or quantitative variable is one that has a numerical value.

Further, a numerical variable can be discrete or continuous.

A discrete variable takes exact numerical amounts, and is usually a result of counting. Examples include the number of children in a family, marks in a test, and shoe size.

Question 1, 2 3, 4 5 6 7 8, 9 1012

Section A B C D E F G


A continuous variable has values that are usually the result of measuring. They are within a certain range. One example is weight that is measured to the nearest kilogram. If someone weighs 42 kg to the nearest kilogram, their exact weight could be anywhere between 41.5 kg and 42.5 kg. Other examples include height, and distance from home.

Exercise 5A

Indicate whether the following are nominalor numerical variables.a ageb weightc method of travel to schoold eye coloure the time spent talking on the telephonef the number of people in your family g the speed of a carh hair colour i favourite TV showj type of pet k temperature during the dayl number of aces served by a tennis player m number of goals scored by a soccer teamn the mass of grapes on a vine o the number of grapes on a vinep the length of a line

Give three examples of:a nominal variablesb numeric variables that are different from question 1.

Indicate whether the following numerical variables are continuous or discrete variables.a age b heightc the number of people in your family d the speed of a care the number of words in a book f the amount of money in your pocketg the time taken to run 100 metres h the length of your hairi the length of time spent talking on the telephonej the number of male students in your school

Example 1

Classify the following variables as nominal, discrete numerical or continuousnumerical.a The number obtained when a die is rolled.b The brand of runners worn by students in the class.c The weights of the students in the class.

a The values are obtained by counting the number of dots on the uppermost face ofthe die. The values are 1, 2, 3, 4, 5 or 6. It is discrete numerical.

b The variable describes the brand of runners. It is a nominal variable.c This is numerical data obtained by measuring. The results would be determined by

the accuracy of the measurement. It is continuous numerical.

1

Nominal nameNumerical number

2

3


Give three examples of:a continuous variablesb discrete variables that are not mentioned in question 3.

For each of the following investigations, classify the variable as nominal (categorical), discrete or continuous numerical.a the number of goals scored each week by a netball teamb the height of the members of a football teamc the most popular radio stationd the number of children in an Australian familye the number of loaves of bread bought each week by a familyf the pets owned by a class of studentsg the quality rating of different brands of washing machinesh the number of leaves on the stems of plantsi the amount of sunshine in a dayj the number of people who die from cancer each yeark the amount of rainfall in each month of the yearl the countries of origin of immigrantsm the most popular colours of carsn the number of pets owned by the students in a classo the gender of school principalsp the time spent doing homeworkq the sports played by students in high schoolr the stopping distances of carss the number of cars passing through an

intersectiont the marks scored in a class testu the items sold at the school canteenv the number of matches in a boxw the pulse rates of a group of athletesx the reasons people use taxisy the fuel consumption of different cars

B. FREQUENCY DISTRIBUTION TABLES

Once data has been collected, the next step isto organise the data before it is analysed anddisplayed. One method of doing this is to usea frequency distribution table.

A frequency distribution table is a table thatdisplays the frequency (the number of timeseach piece of data occurs) for each of thecategories of the data.

Tally marks are often used to help recordthe data in the table.

4

5

They are called gate post tally

marks because of how they look.


Exercise 5B

a Three coins were tossed simultaneously and the number of heads each time was recorded,using tally marks, in the table below. Copy and complete the table.

Number of heads Tally Frequency

0

1

2

3

Total

Example 1

A census is taken of year 8 Red. The method by which the students in the class travelled to school on a particular day is recorded below using the code: walk (W), cycle (C), bus (B), train (T) and car (M).

WCBWC BBBWB BBCBT CMCBT MMTMM MWCCB

Rearrange this information into a frequency distribution table using a tallycolumn.

Note: Every fifth tally mark is placed through the four preceding tally marks. (This makes

counting easier.) The frequency is the total of the tally marks, that is, the number of times a

particular mode of travel is used. Always check that the total of the frequency column is the same as the number

of observations recorded.

Method of travel Tally Frequency

Walk 4

Cycle 7

Bus 10

Car 6

Train 3

Total 30

1


b The number of days absent in a week for the students in year 8 are shown in this tableusing tally marks. Copy and complete the table.

The maximum daily temperatures in the month of February (in C) were:42 41 40 42 43 42 41 43 39 4141 43 41 40 40 42 40 43 39 4039 40 38 39 41 40 42 37

Copy and complete this table for the data given using tally marks, and answer the questions.

a On how many days in February was the maximum temperaturerecorded?

b On how many days did the highest maximum temperature occur?c How many days had a maximum temperature of 40C or more?

The type of vehicle passing a certain point during a 10-minute period was recorded.Copy and complete this table using tally marks.

Days absent Tally Frequency

0

1

2

3

4

5

Temperature (C) Tally Frequency

37

38

39

40

41

42

43

Vehicle Tally Frequency

Car

Motor cycle

Bus

Truck

2

3


Code: car (C), bus (B), truck (T) and motorcycle (M)

CCMCC TBCCC TTMMC CTBCCCCTMC CCTCM CCMTT CCCMCCCCCT CMBCC CCCTM TCCCC

a How many cars passed this point?b What was the total number of trucks?c How many more trucks than buses were there?

Copy and complete this table for the data given using tally marks.

Month of birth of the students in a year 8 class:Jan March Dec MayJune July Jan DecOct Apr Jan JulySept Aug July OctJuly May Nov MarchJuly Oct May JulyMay Aug Sept DecDec Sept

a How many students are in the class?b What was the most common month for birthdays?c Which month had no birthdays?d How many students were born in winter?

The colours of cars passing the front of a school in a 30-minute period are recorded belowusing the code: white (W), blue (B), grey (G), red (R), other (O).

BWROW BRGWO BGRWW GBRWO GBRWGBRRGW BRGOW BWGRB WWBRG RRWWW

a Rearrange this information into a frequency distribution table using a tally column.b How many cars passed the front of the school in this time period?c What was the most popular colour in this survey?d Calculate the percentage of each colour.

The eye colours of a group of students are recorded below using the code: blue (B), brown(b), green (G), grey (g).

BBgBb bGbBg bBgbG bbBgB BBbBG BbgbBa Organise this data into a frequency distribution table using a tally column.b How many students were observed?c What is the most common eye colour in this sample?

Month Tally Frequency

January

February

December

4

5

6


The marks out of 10 for a mental arithmetic test were:9 10 4 9 7 9 3 2 9 5 8 7 1 73 6 3 10 5 9 0 8 7 4 6 1 6 57 7 5 3 3 5 8 2 6 7 9 8

a Rearrange the data into a frequency distribution table using a tally column.b How many students sat for the test? c How many scored full marks?d How many scored zero? e What percentage of students scored 5 or more?

C. GROUPED DATA

Sometimes the observed data consists of observations over a large range of values. When this is the case, it is often desirable to group the data into classes and determine the frequency of each class. Otherwise the frequency distribution table would have too many rows.

Note: The frequency distribution table would have 25 rows if classes were not used.

7

Example 1

Measurements of the maximum temperature on Mondays were taken over a year with the results below. Measurements were taken in C.14 29 32 21 19 23 14 20 25 31 24 16 2624 21 20 15 26 23 30 16 19 27 22 23 2616 24 35 25 27 14 21 20 27 21 14 32 2319 26 29 20 15 22 25 31 19 23 14 25 19a Draw a frequency distribution table with groups of 1115, 1620, 2126, and so on.b What is the size of each class?c How many days had temperatures of 26C or more?d How many days had temperatures of 20C or less?e How many days had a temperature of 27C?

a

b The class size is 5 as there are 5 numbers in each class.c 10 + 5 = 15 days.d 12 + 7 = 19 days.e Cannot answer from the table. To answer this question the raw data must be used.

By looking through the raw data the answer is 3 days.

Class Tally Frequency

1115 7

1620 12

2125 18

2630 10

3135 5

Total 52

Class sizesmust be the

same.


Exerise 5C

The scores for students in a mathematics exam out of 100 were:69 90 61 58 94 68 77 80 64 87 59 81 73 5672 66 85 60 62 75 78 75 87 73 81 70 73 5885 73 76 92 82 86 70 75 82 71 79 80

a Using classes 5059, 6069, 7079, 8089, 9099, tabulate the data using columns forthe score, tally and frequency.

b What is the size of each class?c How many students are in the 7079 class?d How many students scored less than 80?e How many students scored 70 or better?

A group of young footballers was invited to participate in a long kicking competition.The following results in metres were obtained.

55 30 51 49 58 31 32 47 41 36 40 57 45 4849 69 47 43 35 49 41 45 45 46 49 53 41 4030 51 35 37 23 33 68 49 49 31 61 48 57 4737 48 56 38 40 29 45 65 45 45 41 40 39 3636 39 30 39 35 42 58 37 47 43 43 41 53 4242 55 48 59 45 46 57 49 48 66 40 42 43 3746 68 42 41 37 31 39 49 36 23 47 53

a Copy and complete this table using the groupings given.b What is the size of each class?c How many footballers kicked less than 44 metres?d How many footballers were able to kick at least 51 metres?e Which distance group has the most footballers in it?

A plant inspector takes a random sample of two-week-old seedlings from a nursery andmeasures their height to the nearest mm. The results are:

333 423 349 425 370 404 370 323 351 351 438 398 390398 326 311 368 360 411 389 381 378 377 408 359 357374 301 350 361 329 370 375 373 321 340 358 400 448366 380 348 385 371 308 440 351 390 379 340 402 366380 342 369 368 300 370 330 424

Distance (metres) Tally Frequency

2329

3036

3743

4450

5157

5864

6571

1

2

3


a Copy and complete this table.b What is the size of each class?c How many seedlings were measured?d How many seedlings were between 375 and 399 mm in height?e How many seedlings were between 349 and 400 mm tall?f How many seedlings were 400 mm high or more?g How many seedlings were less than 400 mm high?h What fraction of seedlings were less than 400 mm in height?i If the total number of seedlings in the nursery is 1462, estimate the number that might

be less than 400 mm tall.

D. HISTOGRAMS AND POLYGONS

Numerical data may be displayed using a frequency histogram or frequency polygon.

Height (mm) Tally Frequency

300324

325349

350374

375399

400424

425449

A histogram is a column graph in which the numerical values of the variable are placed on thehorizontal axis and the frequency of the numerical variable on the vertical axis.

A frequency polygon is a line graph with the first and last points joined to the horizontal axis toform a polygon.

Example 1

The maximum temperature on each dayin September was recorded and theresults summarised in a frequencytable, as shown.

Draw:a a frequency histogramb a frequency polygon for the distribution

Temperature (C) Frequency

17181920212223

1247664


Exercise 5D

Draw a frequency histogram for each of the following distributions.

a

b

Draw a frequency polygon for each of the following distributions.a

a

Both axes are labelled. The histogramhas a heading. The area of each columnrepresents the frequency of each score.Hence the total area of the histogramrepresents the total number of scores.

b The frequency polygon can be drawnfrom the histogram by: joining the midpoints of the

columns or separately, by plotting

points.Note: As the area under the polygonis equal to the area of the histogram,then the first and last points shouldbe joined to the points on thehorizontal axis where the nextscore would be found.

The columns are centred about the scores theyrepresent. They are the same

width and are joined.

1

Number of snacks 0 1 2 3 4 5 6

Frequency 5 7 9 7 6 4 2

Score 10 11 12 13 14

Frequency 7 11 17 0 5

2

Number of goals Frequency

01234

89631


b

On the same diagram, draw a frequencyhistogram and a frequency polygon forthe data given.

From this frequency histogram draw a table of values.

A sample survey of the weeklyallowances of 800 studentsfrom a certain school wastaken by randomly choosing 38 students and asking eachof them the question:

How much pocket money do youreceive per week?

The results are given in thetable opposite.a Copy and complete the table.b Use the table to complete this frequency

histogram.c Draw a frequency polygon for this

information.d What fraction of the sample receive

an allowance between $2 and $3.99?e Why cant you say exactly how many

students receive an allowance of $3.00?

Salary Frequency

15 00020 00025 00030 00035 00040 000

867301

3 Mark 14 15 16 17 18 19 20

Frequency 4 6 9 0 2 3 1

4

5Size of allowance Tally Frequency

$2 to $3.99

$4 to $5.99

$6 to $7.99

$8 to $9.99

$10 to $11.99

Total


A survey of the heights of 14-year-old boys was taken and thefollowing results were obtained.

Measurements were made to thenearest centimetre.a Copy and complete this table.b Draw a frequency histogram

representing this information.c Draw a frequency polygon

representing this information.d Draw a picture graph

representing this information.e Which type(s) of graph(s) best show the information? Why?

Consider the following set of data.55 30 51 49 58 31 32 47 41 36 40 57 45 4849 69 48 47 43 35 49 41 45 45 46 49 53 4140 30 51 35 37 43 23 33 68 49 49 31 61 4857 47 37 48 56 38 40 29 45 65 45 45 41 4039 36 36 39 30 39 35 42 58 37 47 43 43 4153 42 42 55 48 59 45 46 57 49 48 66 40 4243 37 46 68 42 41 37 31 39 49 36 23 47 53

a Sort these scores into a frequency distribution table using:i classes 2329, 3036, 3743, 4450, 5157, etc.ii classes 2127, 2834, 3541, 4248, etc.

b Draw a frequency histogram and polygon to represent each frequency distribution table.c Compare the two histograms.

E. DOT PLOTSA dot plot is able to convey information more simply and clearly than a column graph. It is especially suitable when there are a large number of categories to be displayed. If there are more than five categories, then a dot plot is usually preferred over a column graph. A dot plot usually displays the categories in order of size, except when the categories are numerical and then they are in correct numerical order.

6Height (cm) Tally Frequency

less than 140

140149

150159

160169

170179

180189

190 and over

Total

7

Example 1

Fifty professional people weresurveyed and asked the question:

In which country was your carmanufactured?

The results are shown in the tableopposite.

Represent this information using a dot plot.

Country Frequency

AustraliaJapanEnglandGermanyOther

1715387

Total 50


Exercise 5E

The following figures represent cheese imports (in tonnes) into Australia.

Draw a dot plot representing this information.

The table below represents the salesof six different washing powdersin a corner store over a one-weeksurvey period.Draw a dot plot representingthis information.

This table represents the salesof eight different drink types atthe school canteen.a Draw a dot plot representing

this information. Rememberdot plots usually displaycategories in order of size.

b Use the dot plot to find thethree most popular drinktypes.

Country NZ UK Denmark Canada Sweden Spain

Tonnes 83 56 38 20 15 8

1

2 Brand Number of packets sold

BrightoKleen-itWunder-lifeHandy-mateDirt-ridSuper-clean

12167

1086

Total 59

3 Flavour Number

OrangeLemonMineral waterColaDiet colaOrange juiceLemonadePlain mineral water

2834117532182518


The number of video recorders sold by an electricalstore over a one-month period is shown in thetable opposite.a Draw a dot plot illustrating this

information.b Which is the third most popular

brand?

Year 8 were asked which colour ice-block was their favourite. The results are listed using theabbreviations R for Red, B for Blue, G for Green, W for White, O for Orange and Y for Yellow.Draw a dot plot showing this information.

RBGOO RROYG WRRBB GBRWO RWYRRGWROY RORYG GGBBW OORBO GBBBY

Use the dot plot opposite to complete the table.

Year 8 were surveyed about their favouritevegetable. The results for 8 Red and 8 Blueare shown in the back-to-back dot plot.

a Which is the most popular vegetable among8 Blue students?

b Which vegetable is the most popular amongYear 8 students? Explain.

c Redraw the dot plot combining theinformation from both classes.

4Brand Number sold

SonyRank ArenaPhillipsJVCNationalOther

304085702860

Total 313

5

6

Colour Frequency

RedBlueGreenBrownOrangePinkNone

Total

7


F. STEM-AND-LEAF PLOTS

Stem-and-leaf plots are another way of displaying information. A stem-and-leaf plot is usually used when there are a large number of scores that are spread out. If there are more than 15 different values for scores, a stem-and-leaf plot is useful.

Each score is broken into two parts. The leaf is the units digit of the score and the stem contains all other digits. This means a stem may have more than one digit but a leaf has only one.

Here is a table showing how some numbers are divided into stems and leaves.

Example 1

Complete this table showing scores, stems and leaves.

Note: A leaf has only one digit but a stem may have more than one digit.

Score Stem Leaf

28

153

91

8

1 9

2 8

18 6

204 9

0 6

Score Stem Leaf

28 2 8

153 15 3

91 9 1

8 0 8

19 1 9

28 2 8

186 18 6

2049 204 9

6 0 6


Exercise 5F

Copy and complete this table.

Score Stem Leaf

a 39

b 47

c 125

d 8 3

e 11 4

f 9 3

g 0 4

h 350

i 5

j 1384

1

Example 2

The results in a mathematics class test out of 70 are given below.

43 45 46 22 65 65 23 53 45 26 46 61 51 57 55

55 66 57 42 41 63 70 57 65 48 23 67 62 70 46

In this stem-and-leaf plot, the tens digit forms the stem and the units digit forms the leaf. This means that for the value 45, the stem is the 4 and the leaf is the 5.

The leaves are now put into ascending numerical order, from smallest to largest, to make working easier.

Stem Leaf

2 2 3 6 3

3

4 3 5 6 5 6 2 1 8 6

5 3 1 7 5 5 7 7

6 5 5 1 6 3 5 7 2

7 0 0

Stem Leaf

2 2 3 3 6

3

4 1 2 3 5 5 6 6 6 8

5 1 3 5 5 7 7 7

6 1 2 3 5 5 5 6 7

7 0 0

This row represents the numbers 22, 23, 26 and 23.

There were no scores in the 30 range of numbers.


As no student scored marks in the 30s, the leaf is left blank. The stem-and-leaf plot looks like a histogram or column graph on its side, with the length of the stem corresponding to the height of the histogram or column graphs column.

With a stem-and-leaf plot:

The first three scores have been placed in the stem-and-leafplot. Copy the table and add the remaining 17 scores.

34 49 41 57 38 59 33 31 61

68 55 39 51 53 63 61 58 33

49 60

Copy and complete this stem-and-leaf plot.135 148 157 169 160 111 123129 138 125 128 155 133 129158 114 119 133 144 122

Draw a stem-and-leaf plot using stems of 3, 4, 5 and 6 for these 20 scores.39 45 52 68 37 44 59 62 66 4058 35 49 62 41 59 32 52 47 48

Draw a stem-and-leaf plot with stems of 4, 5, 6, 7, 8 and 9 for these 32 scores.49 68 74 89 49 96 83 98 98 92 48 9547 93 48 79 78 93 67 96 91 42 66 7445 71 77 44 65 78 70 41

Draw a stem-and-leaf plot with stems of 11, 12, 13 and 14 for these 40 scores.138 146 126 140 146 134 119 136 142 146144 132 140 113 124 144 139 117 130 130142 125 140 133 112 118 146 118 143 136146 143 146 122 113 131 112 112 124 140

Draw stem-and-leaf plots for these sets of scores.a 27, 49, 40, 40, 13, 21, 30, 30, 45, 49, 38, 29, 27, 38, 38, 32, 47, 44, 36, 45b 109, 145, 111, 111, 130, 137, 127, 112, 135, 116, 127, 121, 124, 106, 122, 120, 136,

129, 110, 141, 140, 141, 133, 134, 131, 141, 111, 135, 138, 138c 216, 225, 225, 203, 206, 223, 221, 190, 190, 193, 208, 203, 193, 225, 224, 208, 205,

225, 194, 223

all of the data is used and displayed

the largest and smallest measurements can be found

the clustering of data can be more easily seen

the length of the leaf column indicates the number of scores belonging to that stem.

2Stem Leaf

3 4

4 9 1

5

6

3 Stem Leaf

11

12

13

14

15

16

4

5

6

7


a Draw a stem-and-leaf plot for the 50 scores below.44, 59, 59, 107, 65, 43, 85, 60, 64, 86, 64, 46, 62,57, 71, 43, 62, 84, 84, 63, 79, 70, 46, 81, 104, 44,73, 105, 106, 65, 97, 108, 107, 45, 103, 59, 40, 41, 52,101, 107, 102, 105, 75, 46, 105, 105, 102, 70, 43

b Draw a grouped frequency distribution table with classes 4049, 5059 etc.c Construct a frequency distribution histogram and polygon.d What can be seen on the stem-and-leaf plot that cannot be seen on the histogram or in

the frequency distribution table?

A back-to-back stem-and-leaf plot has been drawn comparing the marks of 8L and 8R in amathematics test.

a In 8L what was the:i highest score? ii second highest score?

b The highest score in 8L is called an outlier. An outlier scoreis a score that is much smaller or larger than the rest of thescores. What is the outlier score in 8R? Give a reason.

c The teachers say that the classes are of equal ability.Comment on this statement.

d In 8R the scores are clustered around the stems2 and 3. Describe the cluster of the scores in 8L.

A back-to-back stem-and-leaf plot has been drawncomparing the marks of 8L and 8R in an English test.

a Describe the test scores in 8L, using the terms cluster and outlier.b Are the scores in 8R clustered? Explain.

8L 8R

Leaf Stem Leaf

0 2

9 9 9 8 8 5 5 5 4 4 2 1 9

9 9 8 7 6 5 5 5 1 0 0 0 0 2 0 0 2 2 3 3 5 6 6 7 9

9 3 0 0 1 2 2 4 4 4 5 8 8 9

8L 8R

Leaf Stem Leaf

3 2 0 7 9 9

3 0 1 1 3 5 7

9 8 5 4 4 4 4 3 2 1 1 1 1 0 0 4 2 4 4 4 8 8 9

4 3 3 3 2 2 1 1 1 5 0 1 6 6 6

6 2 3 4

8

9

Cluster is astatistical term for grouped.

10


c In 8R, 20 is the lowest score. Is it an outlier? Explain.d Compare the results of the two classes.

G. DISPLAYING DATA

When we choose a method to display data there are a number of factors to consider. Some types of displays are better than others in certain situations. This section considers different types of displays.

Exercise 5G

Twenty households in a street were surveyed on the number of people living in each house.These are the results:

4 2 3 5 4 4 2 3 1 54 4 3 7 2 3 4 4 2 4

a Draw a stem-and-leaf plot displaying this data.b Draw a dot plot displaying this information.c What was the greatest number of people living in one house?d What was the most common number of people living in a house?e Which display was easiest to use? Explain.f Write a question that uses one of the displays to get the answer 1.

The local under-14 soccer team has listed the number of goals scored in each of their15 games.

0 1 2 2 0 0 1 1 08 0 3 1 1 1

a Display this data in a dot plot.b Draw a stem-and-leaf plot.c Put the information into a frequency distribution table and draw a histogram.d Are there any outlier scores? Which display(s) show(s) this best? Explain.e List the advantages of the histogram over the stem-and-leaf plot for this data.f Write a question for this information that has the answer 3.

The scores in the year 8 final mathematics exam out of 100 were:79 70 71 68 74 78 87 90 74 97 69 9183 66 95 70 72 85 88 85 97 83 91 63

a Display this data in a stem-and-leaf plot.b Greg suggested drawing a dot plot. Explain why this is not a good idea.c Display this data in a frequency distribution table using the classes 6069, 7079, 8089,

9099.d Is it possible to use the frequency distribution table to find if any student scored 75?

Explain.e Give a reason why this data should be displayed using a stem-and-leaf plot.f Write a question for this data that has the answer 85.

1

2

3


This table of information shows eye colour of year 8 students.

Carmine drew a line graph showingthis information.a Does it make sense to join each

point on the graph?b What is between each eye

colour? Explain.c Display this information in

a dot plot.d What type of data should not be

displayed using a line graph?

List the advantages and disadvantages of:a stem-and-leaf plotsb dot plotsc frequency distribution tablesd frequency distribution tables using classese line graphs

Eye colour Number

blue 3

brown 10

green 2

hazel 6

grey 4

4

5

Investigation 1WM: Applying strategies

Spreadsheet exerciseHere are some charts to use with a spreadsheet package.

Enter the information into a spreadsheet.

Favourite sport Number

Rugby 4

Netball 11

Tennis 8

Hockey 6

Cross-country 3


H. PRESENTING DATAGraphs are used to display data and information in a way that is visually attractive. Although there is some loss of detail compared with a table, it is often easier to see trends and relationships. Often both a table and a graph are used to convey the maximum amount of information.

Sometimes graphs are purposely drawn to display the information in a biased way. We will discuss this inSection H.

In Section C you drew frequency histograms and polygons. Remember that histograms and polygons areonly drawn for numerical (quantitative) data. Categorical (nominal) data are represented in column graphsand dot plots.

This section examines column and line graphs as well as sector and divided bar graphs, revising the section from year 7.

Sector graphsIn a sector graph (sometimes called a pie chart), each category is represented by a sector of a circle. Sector graphs were covered in year 7.

The area of each sector is proportional to the size of each category, so each sector angle is proportional to the size of each category.

This is usually called a column graph but thespreadsheet labels it as a bar graph.

Other graphs and different versions of these graphs are available through the spreadsheet. Type the information given in the beginning of the investigation and experiment with different graphs.


Exercise 5H

The table shows the quantities of meats sold by a butcher.a Write down the fraction of each type of meat sold.b For a sector graph, calculate the size of the sector angle

for each category of meat.c Draw a sector graph to illustrate this information.

The table shows the percentage of theworkforce in each of the industrycategories given.a For a sector graph, calculate the

size of the sector angle for eachcategory.

b Draw a sector graph to illustratethis information.

Example 1

In a survey, 50 salespeople were asked thecountry of manufacture of their car. Theresults are given in the table.a Write down the fraction of cars built

in each country.b For a sector graph, calculate the size

of the sector angle for each country.c Draw a sector graph to illustrate

this information.

a The fraction of cars manufactured in:

Australia = Japan = Korea =

Germany = Other =

b Hence the sector angle for: c

Australia = 360 Japan = 360

122 137

Korea = 360 Germany = 360

43 22

Other = 360

36

1750------

1950------

650------

325------=

350------

550------

110------=

1750------

1950------

650------

350------

550------

Country Frequency

Australia 17

Japan 19

Korea 6

Germany 3

Other 5

1 Meat Sales (kg)

Beef 160

Lamb 120

Chicken 90

Pork 80

2 Industry % of workforce

Agriculture 5

Manufacturing 26

Construction 12

Hospitality 35

Finance 16

Public administration 6


Divided bar graphsDivided bar graphs are also useful for illustrating how a whole is divided into parts. They have all the advantages of sector graphs, apart from perhaps being less attractive visually. The calculations and measurements required are easier.

Example 2

In a survey, 50 salespeople were asked the country of manufacture of their car. The results are given in the table. (This is the same data used in Example 1.)

a Calculate the percentage of cars manufactured in each country.b Hence, draw a divided bar graph to illustrate this data.

a

b Step 1: Draw a rectangle bar of convenient length, that is, one that can easily bedivided into percentages. (100 mm is a convenient length.)

Step 2: Divide the bar into parts in the proportion of the percentages foundabove, that is, length of part representing:

Australia = 34% of 100 mm = 34 mmJapan = 38% of 100 mm = 38 mmKorea = 12% of 100 mm = 12 mmGermany = 6% of 100 mm = 6 mmOther = 10% of 100 mm = 10 mm

Hence, the divided bar graph looks like this.

Country Australia Japan Korea Germany Other

Frequency 17 19 6 3 5

Country Frequency Percentage of total

Australia 17 100% = 34%

Japan 19 100% = 38%

Korea 6 100% = 12%

Germany 3 100% = 6%

Other 5 100% = 10%

Total 50

1750------

1950------

650------

350------

550------


The table shows the quantities of meats sold by a butcher.a Add up the sales to find the total number of kilograms sold.b Calculate the percentage of each type of meat sold.c Draw a divided bar graph of length 100 mm to illustrate this

information.

The table shows the percentage of the workforce in eachof the industry categories given. Draw a divided bargraph of length 100 mm to illustrate this information.Use the percentages given to calculate the lengths.

Column graphs

3Meat Sales (kg)

Beef 160

Lamb 120

Chicken 90

Pork 80

4

Industry % of workforce

Agriculture 5

Manufacturing 26

Construction 12

Hospitality 35

Finance 16

Public administration 6

Example 3

The number of each brand of dishwasher soldby an electrical store in one week is shown inthe table.

Draw a vertical column graph to illustratethis information.

Note: The rectangular columns are

the same width and are evenlyspaced.

The vertical axis shows the scale. The number of dishwashers is

shown by the height of eachcolumn.

Brand Frequency

Washwell 10

Fabwash 8

Greatwash 4

Nova 7

PACT 2

Note thecolumns are not joined, unlike a

histogram.


The table shows the averagepercentage of protein incertain foods.

Draw a vertical column graphto illustrate this information.Use percentage on thevertical scale.

The sales of a real estatecompany for the monthsgiven are shown in thetable.

Draw a column graph toillustrate this data. Usesales in millions on thevertical axis, usingincrements of 1 unit.

Bar graphsWhen the rectangles are drawn horizontally, the graph is usually called a bar graph.

5Food % Protein

Beef 70

Rice 80

Eggs 95

Fish 70

Milk 80

6 Month Sales ($million)

JanFeb 6.4

MarApr 7.0

MayJun 2.7

JulAug 3.6

SeptOct 2.9

NovDec 5.1

Example 4

The average price for a house inthe capital cities of Australia is shown.

Illustrate this information on ahorizontal bar graph.

City Av. price ($000s)

Sydney 290

Melbourne 240

Canberra 210

Brisbane 210

Perth 190

Adelaide 170

Hobart 140


This table shows the number of injuries in each of theeight main sports played in Australia.

Illustrate this information on a horizontal bar graph.

Line graphsLine graphs are useful for showing upward and downward trends in a quantity.

They are also useful for finding in between values when the quantity on the horizontal axis is continuous.

Line graphs may only be drawn for quantitative (numerical) continuous data, otherwise the in between values do not exist.

7Sport

Number ofinjuries/year

AFL 270

Soccer 160

Cricket 99

Rugby league 88

Rugby union 58

Netball 40

Hockey 32

Indoor cricket 30

Basketball 23

Example 5

This table shows the variation inprice of Minaus shares for oneweek in 2003. The value wastaken at the close of tradingat 4 p.m.a Draw a line graph to show this information.b Approximate the share value at midday on Tuesday.

a The graph is drawn by plotting thepoints corresponding to theinformation given in thetable and then joiningthese points.

b About 46 cents.

Day Mon Tue Wed Thu Fri

Value (cents) 45 48 55 52 46

Note the lines arenot joined to the

horizontal axis as they were in a polygon.


The monthly sales figures for a computer firm are recorded at the end of the month.a Draw a line graph for this information.b Estimate the sales figures halfway through May.

The weight of a baby at various ages is shown below.a Illustrate this information on a line graph.b Estimate the babys weight at 10 months.

Scatter graphsScatter graphs compare two quantities. They are usually the result of entering information into a spreadsheet. They are used to find trends, if they exist.

Comment on any relationship between the variables in these scatter diagrams.a b c d

8

Month J F M A M J J A S O N D

Sales ($000s) 54 36 30 30 28 34 25 26 31 38 44 56

9

Age (months) 0 3 6 9 12 15 18 21 24

Weight (kg) 3.2 5.1 7.0 8.8 10.0 10.7 11.2 11.8 12.5

Example 6

Describe any relationship between the two variables. Each point represents both results.a b c

a From the graph, as mathematics marks increase so do science marks.b There seems to be no correlation between height and mathematics marks.c From the graph, as the number of cigarettes per day increases, the level of

fitness decreases.

10


I. MISLEADING GRAPHS

Example 1

The table shows the profits of a company over the five year period 1998 to 2002. Below are five ways of presenting this information graphically. What features, if any, are misleading?

Graph B has exaggerated the increase in profit by not starting the scale on the vertical axis at zero and by enlarging this scale.Graph C has the opposite effectdiminishing the rate of increase by enlarging thehorizontal scale.Graph D, by using a smaller scale on the horizontal axis, gives a different impression again.Graph E has an irregular scale on the vertical axis.Graph A, C and D are fair, although each gives a different impression. Graphs B and E are misleading.

Year 1998 1999 2000 2001 2002

Profit ($millions) 12.3 12.9 13.2 13.8 14.6


The main causes of graphs being misleading are:

Exercise 5I

Describe the misleading or poor features of the following graphs.a b

Example 2

Here are three ways of presenting theinformation in the table graphically.Comment on any misleadingfeatures about the graphs.

The column graph A correctly shows that the sales of brand X are double those of brand Y (as does the table). The width of each column is the same and the height of thesecond column is twice that of the first.In graph B, both dimensions, height and width, of the picture of a TV have been doubled creating the impression that the sales of brand Y are (2 2 =) 4 times those of brand X.In graph C, the three dimensions have been doubled creating the impression that the sales of brand Y are (2 2 2 =) 8 times those of brand X.

Brand No. of TVs sold

XY

40008000

the scale on the vertical axis does not start at zero

the scale on the vertical axis is irregular

the scale on the vertical axis is missing

the use of area or volume to create a false impression.

1


c d

e f

g

a Which graph gives the impression of rapidly increasing sales?b Have sales in fact rapidly increased over this 7-year period?c According to graph A, the sales for 2000 appear to be

double those of 1999. Is this true?

a Explain how this graph is misleading.b Redraw the line graph without the bias.

2 Graph

Year Year

A

2000

2100

2200

2300

2400

2500

2600

97 98 99 00 01 02 03

Sale

s o

f N

u-C

ho

cch

oco

late

s

Sale

s o

f N

u-C

ho

cch

oco

late

s

Graph B

0

500

1000

1500

2000

2500

3000

97 98 99 00 01 02 03

3


non-calculator

As this chapter uses very little calculator, there will not be a non-calculator section.

Language in Mathematics

The last two letters of some words from the glossary are given. What are the words?a ________am b ________on c ________ot d ________le e ________erf ________ph g ________ta h ________cy i ________ve j ________or

Complete the following passage by filling in the blanks with words or numbers from the list.continuous discrete numerical outlierscores small together

a A distribution has clustered ________ when they are grouped ________.b An ________ is a score that is very large or ________ when compared to the other scores.c Quantitative data is also called ________ data. d The height of students is ________ data.e Line graphs should not be drawn for ________ data.

Glossarybar graph categorical variable clustercolumn graph continuous variable discrete variabledot plot frequency distribution table grouped datahistogram line graph nominal variablenumerical variable outlier polygonquantitative variable scale scatter diagramsector graph stem-and-leaf plot variable

1

2


An example of a continuous numerical variable is:A height B number of cars C shoe size D eye colour

The country of origin is a:A categorical variable B discrete numerical variableC continuous numerical variable D numerical variable

This information refers to questions 3 and 4. Organise this data into a frequency distribution table.

Number of children in a family2 1 3 2 3 0 4 6 2 1 3 24 0 2 1 2 5 0 2 0 3 3 10 3 1 4 2 1 2 4 3 3 0 45 2 2 4

The number of families with two children is:A 9 B 10 C 11 D 12

The two scores with the same frequency are:A 2 and 3 B 0 and 3 C 0 and 4 D 2 and 4

A possible score in the 3035 class is:A 4 B 25 C 30 D 39

Which looks the most like a column graph?A frequency polygon B frequency histogramC sector graph D line graph

Which is the correct dot plot for the data below?

Favourite apple

Apple type Number

Jonathon (J) 3

Delicious (D) 7

Crofton (C) 4

Granny Smith (G) 5

Roma (R) 2

CHECK YOUR SKILLSCHECK YOUR SKILLS

1

2

3

4

5

6

7


A B

C D

When written into a stem-and-leaf plot, the number 113 would have:A a stem of 1 and leaf of 13 B a stem of 11 and leaf of 3C a stem of 3 and leaf of 11 D a stem of 13 and leaf of 1

This stem-and-leaf plot is used to answer questions 9 and 10.

The highest and lowest scores are:A 0 and 9 B 2 and 5C 23 and 59 D 9 and 3

The outlier score is:A 32 B 59C 9 D there is none

The display that does not allow easy comparison of actual scores is:A dot plot B stem-and-leaf plotC frequency distribution table D frequency distribution table with classes

Which display cannot be used with categorical data?A dot plot B stem-and-leaf plotC frequency distribution table D column graph

If you have any difficulty with these questions, refer to the examples and questions in the sections listed in the table.

8

Stem Leaf

2 3 3 4 5 6 6 9

3 0 1 2 2 2 2 3 4 5 8 8

4 0 0 1 1 1 2

5 9

9

10

11

12

Question 1, 2 3, 4 5 6 7 8, 9 1012

Section A B C D E F G

150 Collecting and Organising Data (Chapter 5) DS4.1 Syllabus reference

Which of these are nominal variables?a eye colour b weight c temperature

a Construct a frequency distribution table for this information.Winning margins in a series of soccer matches:2 1 3 0 1 2 0 4 2 1 3 50 4 3 3 4 1 2 2 2 0 3 10 0 2 1 3 4 5 3 2 1 0 25 0 4 3 2 2 3 0 1 3 4 01 0 2 0 1 2 3 2 1 4 0 2

b Draw a frequency histogram and polygon for this information.c i How many soccer matches were played?

ii How many winning margins of 2 were there?iii What does a winning margin of 0 mean?

a Draw a frequency distribution table to show the following information:8 9 5 10 7 6 6 7 5 1 8 62 8 6 2 6 4 9 4 7 4 9 26 5 9 10 10 1

b How many numbers were 8 or more?

Draw a stem-and-leaf plot for the following information. Use stems of 2, 3, 4, 5, 6.46 45 28 38 61 50 49 42 32 4250 45 51 31 48 45 32 45 44 53

a Draw a column graph for the datain this table.

b Draw a dot plot showing thisinformation.

The price of CHP shares over a fortnight varied as shown.

Draw a line graph to show this variation in price.

REVIEW SET 5AREVIEW SET 5A

1

2

3

4

5 Drink Frequency

Soft drink 14

Still water 10

Juice 6

Tea/coffee 8

Other 3

6

Day M T W T F M T W T F

Price ($) 8.25 8.27 8.30 8.00 8.06 8.06 8.14 7.99 8.08 8.04


Which of these are numerical variables?a eye colour b weight c number of grapes on a vine

The mathematics results for five students are givenin the table.Draw a column graph representing this information.

a Copy and complete the grouped frequencydistribution table for the data:15 19 22 22 20 24 15 2126 21 25 17 27 25 22 2116 27 24 20 17 20 28 2324 27 19 21 16 11

b How many were 24 or less?

a Draw a stem-and-leaf plot for the information in question 3.b Which shows the information more clearly?

From the frequency distribution table in question 3, draw:a a sector graph b a frequency histogram c a frequency polygon

Which of these are discrete numerical variables?a hair colour b weight c number of peas in a pod

a Construct a frequency distribution table for this information.Marks out of 10 for a mental arithmetic test:9 10 6 0 8 7 7 8 6 2 9 7 0 75 10 5 8 5 10 3 7 6 10 9 3 5 39 9 8 8 5 4 6 5 6 10 9 7

REVIEW SET 5BREVIEW SET 5B

1

2 Student Mark

Renee 75

Jim 65

William 89

Harry 94

Yousef 62

3 Group Tally Frequency

1014

1519

2024

2529

4

5

REVIEW SET 5CREVIEW SET 5C

1

2

152 Collecting and Organising Data (Chapter 5) DS4.1 Syllabus reference

b Draw a frequency histogram and polygon for this information.c i How many students sat for the test?

ii How many scored full marks?iii How many scored zero?iv How many scored one?v If a score of 5 or better was considered a pass, how many passed?

The number of milligrams of cholesterol per100 g of various foods is shown in the table.Draw a horizontal bar graph 100 mm longto illustrate this data.

Month of birth of the students in a year 8 class:Jan March Dec May June July Jan MarchMay Aug Oct Apr Jan July Sept AugJuly Oct Sept Dec July May Nov MarchJuly Oct May July July Sept

a Use a tally column to put this data into a frequency distribution table.b How many students are in the class?c What was the most common month for birthdays?d Which month had no birthdays?e How many students were born in winter?f Organise this data into a dot plot.

The number of jellybeans in 40 packets were counted, and the data appears below.28 32 29 32 33 29 31 32 27 28 27 30 26 3127 28 29 31 32 28 31 30 29 30 27 32 29 1831 29 28 29 27 31 28 32 33 32 31 27

a Draw a stem-and-leaf plot for this information using the stems 1, 2, 3.b What comment can you make about the packet containing 18 jellybeans?c What statistical term is given to the score of 18?

3 Food Cholesterol (mg/100 g)

Lobster 70

Beef 80

Chicken 100

Duck 105

Prawns 145

4

5


Which of these are continuous numerical variables?a number of words on a page b length of a line c number of grapes on a vine

The table shows the percentage of the TV audience gained by each of the majorchannels, both for Sydney and nationally.Draw a sector graph to illustrate the setof results:a for Sydney b nationally

The colours of cars passing a school were recorded using the code White (W), Blue (B),Silver (S), Red (R) and Other (O).

WBSSR RWBWW OWWBO SOBWR ROSBBWBWSR BWWSS SWWRW RWBWW WWOWW

a Rearrange this information into a frequency distribution table using a tally column.b What was the most popular colour?c Instead of using the code the students could have drawn a dot plot. Draw a dot plot for the

information.

A survey was conducted on the number of matches contained in a box. The results obtainedfrom the first 60 boxes were as follows:

47 52 49 51 50 47 50 49 48 51 49 5048 49 50 52 20 52 48 50 51 49 50 5125 52 50 50 48 50 51 50 49 51 49 4849 50 48 50 51 47 50 48 50 51 48 4950 48 50 50 51 52 50 50 51 50 49 50

a Draw a stem-and-leaf plot using the stems 2, 3, 4, 5.b Are there any clusters? If so, where are they located?c Are there any outlier scores? Explain how these scores may have occurred.

REVIEW SET 5DREVIEW SET 5D

1

2 Channel TV audience (%)

Sydney Nationally

9 36.0 33.6

7 26.4 29.1

10 21.3 20.4

ABC 13.2 13.8

SBS 3.1 3.2

3

4

Chapter 5: Collecting and Organising DataDiagnostic TestA. VariablesExercise 5AB. Frequency distribution tablesExercise 5BC. Grouped dataExerise 5CD. Histograms and polygonsExercise 5DE. Dot plotsExercise 5EF. Stem-and-leaf plotsExercise 5FG. Displaying dataExercise 5GInvestigation 1: Spreadsheet exerciseH. Presenting dataSector graphsExercise 5HDivided bar graphsColumn graphsBar graphsLine graphsScatter graphsI. Misleading graphsExercise 5INon-calculatorLanguage in MathematicsGlossaryCheck your SkillsReview Set 5AReview Set 5BReview Set 5CReview Set 5DHomework

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