chapter 10

Data

Exploringdata

10

Statistics is an important tool for governments, businesses, community groups and individuals. The information that is collected, displayed and analysed provides a basis for decisions on a variety of matters such as the location of schools and hospitals, the environment, tourism, welfare and even sport.

Information obtained from a census or through market research (surveys, telephone polls and so on) can lead to advertising campaigns for health issues, road safety and new products.

However, before any action is taken, the statistical information must be carefully studied to avoid misinterpretation of the data and to check its relevance and reliability.

10_NC_Maths_9_Stages_5.2/5.3 Page 372 Friday, February 6, 2004 2:23 PM

■ recognise data as quantitative (either discrete or continuous) or categorical

■ organise data into a frequency distribution table and draw frequency histograms and polygons

■ draw and use dot plots and stem-and-leaf plots using the terms ‘cluster’ and ‘outlier’ when describing data

■ find measures of location (the mean, mode and median) and the range for sets of data

■ construct a cumulative frequency table, histogram and polygon for ungrouped data

■ use a cumulative frequency table or graph to find the median and interquartile range

■ construct a frequency table, histogram and polygon for grouped data■ find the mean, median class and modal class for grouped data■ determine the upper and lower quartiles for a set of scores and calculate

the interquartile range■ construct a box-and-whisker plot using the median, the upper and lower

quartiles and the extreme values (the ‘five-point summary’).

■ quantitative data Also called numerical data, this is data that can be counted or measured.

■ categorical data This is data that is represented as a category (such as ‘hair colour’ or ‘make of car’) rather than a number.

■ measures of location The mean, median and mode, which give an indication of the central value or ‘average’ of a set of data.

■ measures of spread A measure that indicates how much a set of data is spread out or dispersed.

■ bias Any unwanted influence on a sample that stops it from being representative of the population.

Nicole scored 75%, 83% and 68% in her last three maths tests. What mark must she score in her next test to achieve an average score of 80%?

In this chapter you will:

Wordbank

Think!

EXP LOR ING DATA 373

CHAPTER 10


374

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

1 The chart below shows the tides at Fort Denison over a three-day period.

a What was the highest tide over this three-day period and when did it occur?b What was the lowest tide and when did it occur?c When was the high tide just over 1 m?d At what times was the low tide 0.5 m?e What is the time difference between low tides and high tides?f What was the tide at:

i Thursday, 6:00am? ii Saturday, 1:00pm?

2 Sector graphs (sometimes called pie charts) are often used to show information as part of the whole population.

The following sector graph shows the results of a survey asking students at a local high school to name their favourite sport.a What is the most popular sport?b Which sport is least popular?c Estimate the percentage of students whose favourite sport is:

i rugby league ii basketball iii netball.d What extra information needs to be placed on this sector

graph to make it more useful?

3 This unusual graph shows the sales made by a bookstore in one year.a What percentage of book sales were

children’s books?b Apart from ‘Other’, which category makes

up the smallest percentage of sales?c If book sales amounted to $1 200 000 in one

year, calculate how much was spent on:i educational books ii general books.

d If the store increased its sales by 6% in the following year, how much would be spent on general books in that year?

4 The stopping distance for a car braking suddenly is found by adding the reaction (or thinking) time to the braking distance.

am 3:00 6:00 9:00 noon 3:00 6:00 9:00pm am 3:00 6:00 9:00 noon 3:00 6:00 9:00pm am 3:00 6:00 9:00 noon 3:00 6:00 9:00pm

1:00am 7:14am 1:45pm 8:11pm 2:07am 8:09am 2:49pm 9:32pm 3:23am 9:10am 3:56pm 10:50pm1.4 m 0.4 m 1.5 m 0.5 m 1.3 m 0.5 m 1.5 m 0.5 m 1.2 m 0.5 m 1.6 m 0.5 m

Thursday Friday Saturdaym

2

1

0

Rugby League

Netball

Basketball

AFL

Cric

ket

Soccer

RugbyUnion

Favourite sports

General40.5%

Educational31.2%

Other5.8%

Sport10.5%

Children’sbooks12%

Reactiontime

Brakingdistance

Stopping distance

Start up

Worksheet 10-01

Brainstarters 10

10_NC_Maths_9_Stages_5.2/5.3 74 Friday, February 6, 2004 2:23 PM

EXP LOR ING DATA

375

CHAPTER 10

The graph on the right shows the stopping distance for cars travelling at various speeds.

a What is the distance travelled during the reaction time at:

i 30 km/h?ii 60 km/h?

b Use the graph to estimate the reaction time at 120 km/h.

c What is the braking distance of a car travelling at 80 km/h?

d Dana is driving past a school at a speed of 40 km/h. She needs to stop suddenly. Use the graph to estimate her stopping distance. Explain how you obtained your answer.

5 This divided bar graph (also called a composite bar graph) shows the favourite ice cream flavours of a group of Year 9 students.

a By measurement, find the percentage of students who preferred:i caramel ii chocolate iii strawberry.

b If 80 students were surveyed, how many preferred vanilla ice cream?

6 a Find the range of the following scores: 6 3 0 7 1 5 5 6 8

b Find the mode of each of these sets of scores:i 3 4 2 3 5 4 3 6 0ii 8 10 11 7 6 9iii 4 4 4 4 4 4iv 3 6 5 4 5 8 6 5 6

c Find the average, or mean, of each of the following sets of scores:i 6 9 7 6ii 6 −3 −2 0 −1

d Find the median (middle score) of each of the following sets of scores:i 3 5 7 8 10 12 12 15 17ii 21 24 24 26 30 33 37 40

7 Students were surveyed on the number of DVDs they purchased in the last six months. The results were as follows:

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

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

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

30

50

60

80

100

10 20 30 40 50 60 70 80 90Distance (m)

Sp

ee

d (

km/h

)

Stopping distance

Caramel Chocolate Strawberry

Vanilla

Other

Favourite ice cream flavours


376

N EW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

a Copy the frequency table and use tally marks to organise the data in it.

b How many students were surveyed?c What was the most common number

of DVDs purchased?d What percentage of students

purchased:i 3 DVDs?ii more than 3 DVDs?iii not more than 2 DVDs?

Number of DVDs

Tally Frequency

0

1

�

6

Using Chart Wizard in ExcelWith Excel it is possible to draw a great variety of graphs (or ‘charts’). Before the graph is produced, you must enter the data into the cells of a spreadsheet. We shall use the data from the following example.

The number of cars sold by staff each month at a car dealership is recorded below.

Step 1: Open Excel.

Step 2: Enter the data in the spreadsheet, as shown.

Step 3: Use your mouse to select all the information, numbers and headings.

Step 4: Open Chart Wizard , then select the type of chart you want.

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

Cars Sold 61 70 43 80 88 75 60 86 80 72 55 70

Staff 12 15 10 13 18 14 11 16 17 13 10 12

B C D E F G H I J K L M N

5

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

7 Cars sold 61 70 43 80 88 75 60 86 80 72 55 70

8 Staff 12 15 10 13 18 14 11 16 17 13 10 12

Using technology

Spreadsheet 10-01

Using Chart Wizard in Excel


EXP LOR ING DATA 377 CHAPTER 10

Types of dataData refers to information that is collected and organised so that conclusions can be drawn and decisions made. Data that is collected is either categorical or quantitative (numerical).

Categorical data is information that can be put into different categories.

Step 5: Follow the Next few steps until you are ready to Finish.

Step 6: Remember to label your axes.

Sales performance

Month

Nu

mb

er

Population factsThe population of Australia has gradually been increasing. Population figures from 1901 to 2001 are given below.

Aboriginal people were included in the population figures for the first time in 1961. The population exceeded 18 million for the first time in March 1995.

On 4 April 2005, the resident population was projected to be 19 831 163. This projection was based on the estimated population at 30 September 2002 and assumed growth of:• one birth every 2 minutes and 5 seconds• one death every 3 minutes and 48 seconds• an overall total population increase of one person every 2 minutes and 42 seconds.

(Source: Australian Bureau of Statistics)

The world population was projected to be 6 284 586 182.(Source: US Census Bureau)

What are the current figures for the population of Australia and the world?

Year Population Year Population

1901 3 825 000 1961 10 643 000

1911 4 574 000 1971 13 198 400

1921 5 511 000 1981 15 054 100

1931 6 553 000 1991 17 414 300

1941 7 144 000 2001 19 603 500

1951 8 528 000

Just for the record


378 NEW CENTURY MATHS 9 : S TAGES 5 .2/5 .3

Examples of categorical data include:• hair colour (black, blonde, brown, grey, etc.)• eye colour (brown, blue, green, hazel, etc.)• nationality (Australian, Chinese, Greek, New Zealander, Vietnamese, etc.)• gender (male or female)

Quantitative (or numerical) data is information that is represented by numbers. This data can be discrete or continuous.

Discrete data is obtained through counting. This type of data has ‘gaps’ or ‘jumps’. For example, the number of girls in a family may be 0, 1, 2, 3 etc, but not 1.5, 2.78, 3.2. ‘In-between’ values are not possible.

Examples of discrete data include:• the number of girls in a family• the number of people in a room• the number of goals scored in a football or netball game• the daily attendance figures at your school.

Continuous data is obtained through measuring. The values for this type of data are on a smooth continuous scale, which means there are no gaps or ‘jumps’ between values.

An example of continuous data is the temperature during the day. Between any two values, say, 11°C and 12°C, it is possible to have ‘in-between’ values such as 12.5°C, 12.3°C, 12.28°C, etc.

Other examples of continuous data include:• the heights of students in your school• the weights of students in your class• the lengths of telephone calls• noise levels at a concert.

1 State whether each of the following types of data is quantitative (numerical) or categorical.a the number of students in maths classesb the time it takes to roast a chickenc the masses of babies at birthd brands of toothpastee a hospital patient’s blood pressuref the make of a carg the number of videos or DVDs watched during a weekh the time taken to download programs from the Interneti the country of birth of people living in Australiaj the distance from Sydney to Brisbanek the number of employees in a factoryl favourite TV shows.

2 State whether the following quantitative (numerical) data are discrete or continuous.a the amount of rainfallb the time taken to run 100 mc the number of heats for the 100 m at a swimming carnivald the number of cars passing a busy intersectione the number of traffic lights between Sydney and Melbournef the ages of students at a school

Exercise 10-01

Worksheet 10-02

Statistical data match-up



Collecting dataWhen the marketing department of a large company wishes to launch a new product, it collects information about consumer tastes. From this data it makes decisions about what type of products it will manufacture, the amount it expects to sell and so on. This process consists of three steps:

i collecting dataii organising and displaying the dataiii analysing and interpreting the data, and making predictions and decisions using the data.

Defining important termsThe following terms are often used when collecting data.• Population refers to all of the items under consideration. For example, if data on the best-

selling CD in NSW is required, the population is all CDs on sale in stores or shops in NSW.• A census is a survey of the entire population.• A sample is a part of the population that has been selected in order to find information about

the whole population.• An unbiased sample is one in which there are no factors that may influence the sample and

so prevent it from being a true indicator or representative of the population.• A random sample is one in which each member of the population has the same chance of

being selected as any other.• A survey is the collection of information for a specific purpose.

Methods of data collectionData is collected in many ways and by many different groups of people such as businesses, government departments and individuals.

g the shoe sizes of students in Year 9h the cost of ice creamsi the average contents of a box of matchesj the number of words per minute at which a person typesk the number of road fatalities over a weekendl the amount of time spent completing this exercise.

Working mathematicallyReasoning and reflecting: Collecting data — which method?Work in groups of three or four to complete this activity.

1 Write the meaning of each of the following (you may need a dictionary to help you).a population b census c sampled unbiased e random sample f survey

2 A market research company uses a telephone survey to determine the most popular news program on TV. List some of the advantages and disadvantages of using this type of survey to collect data.



3 In order to determine what type of music should be played on the school radio station, 20 students were given a questionnaire.a List some of the advantages of using a questionnaire to obtain the data.b List some of the disadvantages of using this method.c What other methods could be used to find out the type of music that should be played?

4 Describe how you could collect data about each of the following.a the most popular fast food eaten by students at your schoolb the average daily amount of money spent by students at the school canteenc the most popular TV series watched by students at your schoold the most popular Australian band among students at your schoole the most common brand of toothpaste used by students attending your schoolf the average number of girls in a classroom in NSW

5 You need to determine how many students in your school have part-time jobs. Which of the following methods would be most suitable? Give reasons.A You ask 15 friends in your class.B You randomly select 15 students from each year level at your school.C You ask all the students at your school to fill in questionnaires and return them to you.

6 When collecting data, it is important to have an unbiased sample. Write, in your own words, what is meant by an ‘unbiased sample’.

7 What bias could occur in surveys that use the following methods or places to select people to be surveyed?a a supermarketb a shopping mall in a Sydney suburbc houses with odd numbersd a telephone surveye a football match

8 List the possible areas of bias in each of the following.a a survey about the amount of time spent on homework, taken at a football gameb a survey about violence in sport, taken at a football gamec the average number of cars per household, taken at a car show

Working mathematicallyApplying strategies and communicating: Year 9 student surveyThe survey below is designed to collect data about Year 9 students. Your CD-ROM has a copy of the survey for you to print and use in class. You can add or change questions if you want to, but remember that the data is only useful if good questions are asked in a controlled way.

Instructions: Please answer every question.

Write the answer, or circle the one best answer for each question.

The survey normally takes about 15 minutes.



Steps in conducting the surveyStep 1: Finalise the questions and print the survey sheets.Step 2: Decide how many students you want to respond to the survey.Step 3: Decide how you will select the sample of students.Step 4: Decide whether the survey will be handed to the selected students or an interviewer

will record responses.Step 5: Decide whether you will interpret the questions/responses or not.Step 6: Ask the selected students to complete the survey.Step 7: Collect all the sheets and label the first sheet 1, the second sheet 2 and so on.Step 8: Decide on the code you will use for each response, then code the responses and

enter the results in a table, either on a spreadsheet or in your workbook.Step 9: Decide what to do if a question is not answered or the answer is a mistake.

Questions Data code

1 What is your gender? Male Female

2 What is your age (as of last birthday)?__________ years

3 How tall are you? __________ cm

4 How many people are in your household?__________ people

5 What is your position in your household (1 = oldest)? __________

6 How many pets do you have?__________ pets

7 How far from school do you live (to nearest 0.1 km)?__________ km

8 How did you travel to school today?Car Bus Walk Train

9 How long did the journey take?__________ minutes

10 On a scale of 0–10, rate your enjoyment of school (10 = most). ________

11 Which subject do you like best?Maths Science English Art PE History

12 Which subject do you study most?Maths Science English Art PE History

13 How much home study do you do each week?__________ hours

14 How many hours do you study maths each week?__________ hours

15 On a scale of 0–10, rate your enjoyment of maths (10 = most). __________

16 What was your last mark in a maths test?__________ %

17 How much competitive sport do you play out of school?__________ hours

18 How many competitive sports do you play out of school?__________ sports

19 How many hours each week do you spend on getting fit?__________ hours

20 Which sport do you prefer to watch?Football Soccer Netball Tennis Surfing

Spreadsheet 10-02

Year 9 student survey

Spreadsheet 10-03

Survey dummy data



Frequency histograms and polygonsA histogram is a graph that uses the height of a column to show the frequency of each score. The scores are always placed on the horizontal axis, and the frequency on the vertical axis.

Another type of graph is the frequency polygon. This is a line graph obtained by joining the midpoints of the tops of the columns of a histogram.

Features of a histogram Features of a polygon

Australia’s first censusThe first census held in Australia took place in November 1828. Before this, statistical information was collected by means of musters on the same day each year.

Worried about the accuracy of the figures, Governor Darling and the Legislative Council passed an Act requiring a census to be held at regular intervals. Each citizen needed to disclose his or her name, age, marital status, number of children, the name of the ship on which they arrived, the amount of land owned (cleared and uncleared) and the number of cattle and other stock owned. Collectors were employed to gather this information by personal interview.

The census showed that the total non-Aboriginal population in November 1828 was 36 598, and nearly 30% of that population lived in Sydney.

When was the most recent census taken in Australia?

Just for the recordWorksheet 10-03

Australian statistics

• Each column must be the same width.• No gaps are allowed between columns.• A space (equal to half the column width)

is left between the vertical (frequency) axis and the first column.

• The label for each score is placed under the centre of its column, on the horizontal axis.

• The midpoints at the tops of the histogram columns are joined by a line.

Example 1

45 students were asked the question: How many hours of homework did you do last week? The results were as follows:3 8 7 7 8 4 3 6 6 9 2 7 7 8 57 8 4 9 5 2 8 7 5 8 3 8 2 8 65 7 8 8 7 5 6 9 4 4 6 2 5 4 8a Arrange the information in a frequency table.b Draw a frequency histogram and frequency polygon.



Solutiona This information has been organised in a frequency table (sometimes called a frequency

distribution ) as shown below.

b

Number of hours, x

Tally Frequency,f

2 |||| 4 The left-hand column, which in this example is headed ‘Number of hours’, is sometimes referred to as the score column and the symbol that is used is x. The letter f is often used for frequency.

3 ||| 3

4 |||| 5

5 |||| | 6

6 |||| 5

7 |||| ||| 8

8 |||| |||| | 11

9 ||| 3

Total 45

Frequency histogram and polygon

Number of hours spent on homework

1

0

2

3

4

5

6

7

8

9

10

11

21 3 4 5 6 7 8 9

Fre

qu

en

cy

The frequency polygon can also be drawn without showing the histogram.

Frequency polygon

Number of hours spent on homework

1

0

2

3

4

5

6

7

8

9

10

11

21 3 4 5 6 7 8 9

Fre

qu

en

cy

1 A class of 30 students was given a test of 10 questions and the results are as follows:

7 8 6 6 4 7 8 10 7 9 7 5 6 5 97 5 6 8 4 9 5 4 6 7 7 8 3 8 7a Arrange this information into a frequency distribution table.b Draw a histogram and frequency polygon of this data.

2 The results of a survey conducted to determine the number of children in the families of students at a Sydney high school are shown below.

1 6 7 4 3 2 2 3 5 9 3 3 3 3 1 7 3 7 2 32 5 4 4 1 1 2 6 7 8 4 2 6 5 2 4 5 2 1 3

Exercise 10-02Example 1

Worksheet10-04

Student survey form



a Arrange the data into a frequency distribution table.b Construct a histogram and frequency polygon for the data.c What percentage of families have two children?d What percentage of families have four or more children?

3 The numbers of goals scored by a football team in 26 matches were as follows:

3 2 5 1 4 0 3 3 4 5 2 4 44 3 3 4 2 1 0 1 2 5 4 3 3a Arrange the data in a frequency distribution table.b Draw a frequency histogram with a frequency polygon for this data.c In how many matches were:

i two goals scored? ii more than three goals scored?

4 The numbers of letters in the surnames (last names) of 55 students were recorded. The results are shown in this frequency histogram.

a How many surnames had six letters? b Which result occurred eight times?c How many surnames had seven or more letters?d How many had less than five letters?e Copy and complete the frequency distribution table for the histogram.

5 Four coins were tossed 50 times and the number of heads was recorded after each toss. The results are shown in the frequency histogram below.

2

4

6

8

10

12

14

3 4 5 6 7 more than7

Frequency histogram

Number of letters in surname

Fre

qu

en

cy

Number of letters

Frequency

3

4

5

6

7

More than 7

3

6

9

12

15

18

21

24

00

1 2 3 4

Frequency histogram

Number of heads

Fre

qu

en

cy

Number of heads

Frequency

0

1

2

3

4

SkillBuilder 15-06–15-10

Histograms



a How many times were three heads thrown?b Which result occurred 12 times?c What was the most common result?d Copy and complete the frequency distribution table for the histogram.

6 The frequency polygon below shows the number of calls per minute made to a school over a 30-minute period.a On how many occasions were no calls

received?b How many calls were received altogether

over the 30-minute period?c How many times were three or more calls

received per minute?d At what time of the day do you think this

30-minute period occurred? Why?

1

2

3

4

5

6

7

8

9

10

00

1 2 3 4 5 6

Frequency polygon

Number of calls per minute

Fre

qu

en

cy

Drawing histograms on a graphics calculatorFor ungrouped data (such as that given in the table on the right about the marks of a group of students), we follow the steps below.

Using a TI-82 or TI-83 graphics calculator

Step 1: Open

Step 2: Select 1 : Edit.

Step 3: Enter the data for both columns.

Test Score Frequency

4 1

5 2

6 1

7 3

8 5

9 2

Total 14 STAT

Using technology



Step 4: Select PLOT 1: Plot 1 … Off. Change to On, and select the histogram icon.

Step 5: Select 9: ZoomStat.

Step 6: Select and move the cursor across the histogram.

Using a Casio CFX-9850GB PLUS graphics calculator

Step 1: Select STAT from the MAIN MENU.

Step 2: Enter the data for both columns.

Step 3: Select GPH1, then select SET and set the following options.

Step 4: Select (GPH1).

Step 5: Set the Start to 4 and the pitch to 1.

Step 6: Select (DRAW).

Step 7: Select (Trace) to move from one box to the next. The calculator displays the value (x) and its frequency (f).

2nd Y =

ZOOM

TRACE

F1 F6

F1

F6

SHIFT F1



Dot plotsAn easy way of displaying a small set of scores is to draw a number line with dots or crosses drawn above the numbers to represent the scores.

Stem-and-leaf plotsA stem-and-leaf plot is another way of displaying data.

Example 2

Construct a dot plot for the following scores, which are marks obtained in a test by 22 students.

11 8 12 12 15 13 10 12 13 15 16

22 12 11 11 10 13 16 12 15 12 11

Solution

An outlier is a value that is separated from the main body of the data. In this example, 22 is an outlier. We can also say the scores are clustered or bunched around the marks 11 and 12.

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22Marks obtained by students

Example 3

The data collected shows the pulse rates of 30 people in a health centre. Arrange the data in a stem-and-leaf plot.

67 86 53 67 68 71 79 64 60 55

64 66 52 73 67 61 87 82 58 62

61 69 73 91 40 85 77 68 59 70

SolutionThis data can be arranged in an unordered stem-and-leaf plot as follows:

Stem Leaf

4 0

5 3 5 2 8 9

6 7 7 8 4 0 4 6 7 1 2 1 9 8

7 1 9 3 3 7 0

8 6 7 2 5

9 1

The stem is formed by the tens digit of the scores.

The leaves are formed by the units digits of the scores.

The data shown by this row is 86, 87, 82, 85.



Back-to-back stem-and-leaf plotsWhen two related sets of data are compared, we can use back-to-back stem-and-leaf plots.

However, the stem-and-leaf plot is usually arranged in order as shown:

Stem Leaf

4 0

5 2 3 5 8 9

6 0 1 1 2 4 4 6 7 7 7 8 8 9

7 0 1 3 3 7 9

8 2 5 6 7

9 1

The outlier for this set of data is 40. The scores are also clustered in the 60s.

Example 4

Two different brands of batteries (Brand A and Brand B) are to be compared by testing 15 batteries of each brand until they fail. The lengths of the battery lives are recorded to the nearest hour as shown:

Brand A: 22 28 17 34 26 25 30 24 15 26 27 16 36 31 20

Brand B: 12 37 32 39 18 34 36 35 31 31 42 37 32 35 33

SolutionThe two sets of data can be displayed in an ordered back-to-back stem-and-leaf plot as shown below.

Which brand of battery do you think is better? Why?

Brand A Brand B

7 6 5 1 2 8

8 7 6 6 5 4 2 0 2

6 4 1 0 3 1 1 2 2 3 4 5 5 6 7 7 9

4 2

1 Draw a dot plot for each of the following sets of data.

a 15°C 22°C 18°C 20°C 19°C 21°C 20°C 19°C 22°C 27°C21°C 17°C 21°C 19°C 15°C 21°C 22°C 18°C 21°C 20°C

b 3 2 4 1 1 2 3 5 3 9 2 11 4 2 3

2 The following data represents the number of goals per game a team has scored in a season.

34 42 37 35 20 48 38 35 36 27 38 37 12 41 37

Would you display the data in a dot plot? Why?




3 The following set of scores represents the number of hours spent doing homework during one week by 20 students.

3 2 0 1 5 15 8 6 7 6 3 5 4 5 6 7 9 2 5 7

a Draw a dot plot to display the data.b Identify scores that could be considered outliers.c Around which of the scores is the data clustered?

4 The numbers of points per game scored by a basketball team during a season are as follows:

58 64 32 82 77 57 68 72 40 88 79 59 65 44

72 81 77 49 73 67 82 71 57 56 48 79 84 62

a Construct an ordered stem-and-leaf plot to display the data.b What was the highest number of points scored during a game?c In how many games were more than 60 points scored?d What percentage of games had fewer than 50 points scored?e Are there any: i outliers? ii clusters?

5 When 30 students ran 100 m, their times (in seconds) were as follows:

13.5 12.1 13.1 11.8 14.6 15.7 16.1 11.9 12.6 12.5

15.0 17.6 18.7 15.4 14.9 14.5 14.3 15.1 15.7 17.6

12.1 17.2 15.7 14.9 15.0 18.1 16.3 15.4 14.4 13.7

a Construct an ordered stem-and-leaf plot to display this data.b How many students ran the 100 m in less than 13.5 seconds?c What percentage of students took between 14.0 and 15.0 seconds to run the 100 m?d Are there any: i outliers? ii clusters?

6 The dot plot below shows the numbers of goals scored by a hockey team during the season.

a How many games were played?b What was the most frequent number of goals scored by the team?c In how many games were more than two goals scored?d In what percentage of games were more than two goals scored?e Which of the scores can be considered an outlier? Give reasons.

7 The following data represents the masses in kilograms of 20 teenagers.

63 55 68 71 53 62 77 54 55 47

57 55 53 53 54 64 56 52 60 51

a Construct an ordered stem-and-leaf plot to show the data.b What percentage of teenagers had a mass of more than 50 kg?c What percentage of teenagers had a mass of less than 50 kg?d Are there any clusters?

0 1 2 3 4 5 6 7 8 9 10Number of goals scored

Example 3

SkillBuilder 16-01–16-04Stem-and-leaf

plots



8 This ordered stem-and-leaf plot shows the marks obtained by 30 students in a history exam.a One of the marks in the 50s is missing. Decide what the

missing mark could be.b The entry for the lowest mark is missing. If the range of the

marks is 58, find the lowest mark.

9 This back-to-back, ordered stem-and-leaf plot shows the goals scored by two netball teams during a season.a How many times did each

team score more than 30 goals?

b Find the highest number of goals scored by each team.

c Which team performed better? Give reasons.

10 The number of hamburgers sold by Sandy and Greg during a two-hour period over 14 days was as follows:Sandy: 37 23 33 35 17 42 37 54 45 38 27 35 40 25Greg: 35 28 29 28 42 37 54 45 47 40 25 34 36 36a Construct a back-to-back, ordered stem-and-leaf plot of this data.b Who sold the highest number of hamburgers?c On how many days did Sandy sell more hamburgers than Greg?

11 The following sets of scores are the results of a science exam completed by classes 9 Al and 9 Au.9 Al: 37 48 63 55 77 64 72 55 81 69 74 70 60 54 49 40 78

68 61 53 72 71 64 62 60 73 679 Au: 56 60 62 65 38 46 74 80 83 67 68 54 50 82 73 73 64

60 68 72 71 79 70 67a Construct an ordered back-to-back stem-and-leaf plot.b Which class had:

i the highest mark? ii the lowest mark?c Were there any outliers?d Was there any clustering?e Which class performed better? Give reasons.

Stem Leaf

2

3 0 2 3 4

4 1 5 7 8

5 2 4 6 9 9

6 0 2 3 3 6 8 8 7

7 2 5 6 7 7

8 2 5

Rockets Blues

7 5 4 3 1 7

8 7 6 5 4 2 2 0 5 6 8

7 6 4 3 3 4 5 7 7 8 9

5 4 0 4 2 3 7 8

2 5 1 3 7

Example 4



Analysing dataVarious types of graphs and tables have been used to organise and display data. However, a graph or table may not always provide enough information, so we use certain measures that will give additional information about the data.

RangeThe first measure is the range, which is a measure of spread or dispersion.

Mean, median and mode: measures of locationWe now look at three other measures that are used in statistics: the mean, median and mode.

These measures give an indication of a central value, or average, around which a set of scores usually tends to cluster, and so the mean, median and mode are called measures of location.

When analysing data using the mean, mode and median, it is important to use the most appropriate measure of location. We need to consider the following points.• The mean is appropriate when there are no extreme values.• The median is more appropriate when there are extreme values.• The mode is appropriate when the most common characteristic or result of data is required.

Range = highest score − lowest score

Mean = sum of scoresnumber of scores-----------------------------------------

The median is (when scores are arranged in order, from lowest to highest):• the middle score (for an odd number of scores)• the average of the two middle scores (for an even number of scores).

The mode is the score that occurs most often.



The medianThe median is often used as a measure of location when there are extreme values. For example, when referring to houses sold during a month, the median house price (not the mean price) will be used as an indicator of house prices.

1 Collect a page of house prices from a capital city daily newspaper. Why do you think the median is used when referring to house prices in preference to the mean or mode?

2 If one house sold for $5 million, explain how this would affect the mean and median sales values that week.

Just for the record

In this exercise, round your answer to one decimal place where necessary.

1 Find the range of each of these sets of scores:a 6, 1, 8, 4, 4, 5, 8, 10 b 23, 28, 18, 33, 25, 22, 32, 19, 25, 24, 31c 8.3, 10.1, 9.7, 9.6, 8.8, 9.2, 9.8 d 8, −2, 4, 5, −3, −5, 2, 3, 2, 1, 0, 3, −4

2 Find the mean of each of the following sets of scores:a 8, 3, 5, 6, 2, 7, 10, 2 b 55, 68, 42, 57, 67, 83, 46, 79c $1.15, $1.25, $1.10, $1.30, $1.40, $1.15, $1.25d 10.1, 11.5, 9.8, 10.0, 11.2, 10.3, 9.9 e 3, 5, −4, −6, −7, 0, 1, 6, −3, −3, 4

3 Find the median of each of the following sets of scores:a 23, 20, 12, 17, 22, 16, 28, 19, 14 b 5, 3, 8, 7, 1, 12, 15, 5, 7, 7, 6c 2, 1, 9, 8, 4, 5, 5, 6, 9, 12, 8, 3 d −3, 4, −2, −1, 2, 0, 5, 2e 8.8, 8.6, 9.3, 7.2, 7.2, 5.6, 10.1, 9.4, 8.1, 8.9

4 Find the mode of each of these sets of scores:a 15, 3, 17, 28, 15, 8, 23, 12, 24, 18 b 5, 6, 2, 1, 8, 4, 9, 6, 10, 5c 23, 24, 24, 25, 25, 25, 26, 30, 32, 32, 35d 6.2, 5.8, 7.1, 8.4, 7.1, 8.9, 7.6, 7.6, 7.1

5 Find the range, mean, median and mode of each of the following sets of scores:a 4, 6, 2, 8, 5, 8, 6, 1, 9 b 15, 18, 12, 12, 18, 17, 20, 15c 53.5, 50.0, 49.7, 52.0, 54.2, 49.8, 51.6d 15, 8, 12, 11, 16, 14, 7, 18, 9, 12, 14, 17, 15, 14, 14e 84, 68, 70, 54, 46, 60, 67, 75, 82

6 a Find the mean, median and mode of this set of scores:5, 7, 3, 9, 4, 6, 4, 7, 5, 5, 8

b Which measure is the easiest to find? Why?

7 a Find the range, mean and median of this set of scores:20, 24, 23, 26, 25, 27, 18, 32

b Does the data in part a have a mode? Explain your answer.

Exercise 10-04

Worksheet 10-05

Stem-and-leaf plots

Skillsheet 10-01

Statistical measures

SkillBuilder 15-11Mean


Median

SkillBuilder 15-14Mode



8 For each of the following sets of data, find the:i range ii mean iii median iv mode.

a b

9 This stem-and-leaf plot represents the points scored by a basketball team.a Find the range of the scores.b Find the mode.c Find the mean.d i There are 23 scores in the stem-and-leaf plot. The median

will be the twelfth score. Explain why.ii Find the median.

10 This stem-and-leaf plot shows the marks of 27 students in ascience test.a Construct an ordered stem-and-leaf plot.b Find the range of the marks.c Find the mode.d Find the median.e Find the mean.

11 This stem-and-leaf plot represents the runs made by George in 30 innings of cricket.a What was George’s highest score?b Find the mode of George’s scores.c Explain why the median is the average of the fifteenth score

and the sixteenth score.d Find the median of the scores.e Find the mean.

12 The monthly rainfall (in millimetres) for a town is as follows:

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

Rainfall (mm) 550 640 685 420 360 172 0 0 310 473 595 625

48 49 50 51 52 53 54

Stem Leaf

1 3 5

2 0 1 1 3 7

3 2 4 4 6 8 9 9 9

4 5 6 6 8

5 0 1 2 3 5 6

6 2 4 5

Stem Leaf

3 0 7

4 1 1 3 5

5 4 6 6 8 9

6 3 5 8 8 8 9

7 2 4 5 5

8 0 2

Stem Leaf

4 0 1 5

5 3 1 5 6 8

6 7 7 8 6 4 3 5 7

7 9 1 7 4 2

8 8 2 7

9 2 0 4

Stem Leaf

1 1 2 2 7

2 0 1 3 4 4 7

3 5 7 8 8 8 9 9

4 1 1 4 5 6 8

5 0 7 8 8

6 0 1 2



a Find the median of the data.b Find the mode.c Find the mean.d Why is the mode of this data a misleading measure?e Which measure (median, mode or mean) is the most representative of the data? Why?

13 The average points scored by a basketball team for the 30 games played during the season is 92. What is the total number of points scored?

14 A class of 30 recorded the number of brothers and sisters each student in the class had. The results are displayed in the histogram on the right.a What is the range of this data?b What is the mode?c How many brothers and sisters are there

altogether?d Calculate the mean number of brothers and sisters.e Explain how you would calculate the median for

this data. Calculate it.

15 Arrange each of the following sets of data into an ordered stem-and-leaf plot and then find the median.a 35 21 62 42 43 28 47 51 45 36 44 54 60 44 38b 12 26 66 50 45 37 21 23 48 49 34 10 54 57 22

16 a Construct an ordered stem-and-leaf plot for the following set of data.22 5 6 6 10 22 20 13 26 21 1515 18 7 15 26 21 17 18 24 11 16

b Find the median.

1

0

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6

Fre

qu

en

cyNumber of brothers and sisters

Summary statisticsThe members of a basketball team would like to compare the number of points they each have scored (over 10 games) and to compare them with the number of points the team has scored.

The numbers of points each player has scored are as follows:

Amanda 5 8 2 10 6 8 4 4 6 10

Colin 4 8 6 7 7 9 10 6 5 8

Eloise 12 8 8 16 4 5 7 10 11 8

Diana 4 3 5 2 6 8 4 3 3 6

Tran 10 14 7 8 12 16 10 10 8 6

Michelle 12 4 6 10 9 9 5 16 8 12

Shaheid 4 7 6 8 7 5 3 8 9 2

Greg 14 12 8 7 10 9 5 12 15 8

Using technology

Spreadsheet 10-04

Summary statistics



Set up your spreadsheet as follows:

• Enter the function =sum(D8:M8) in cell N8 and copy down to N15.• Enter the function =average(D8:M8) in cell O8 and copy down to O15.• Enter the function =sum(D8:D15) in cell D16 and copy across to N16.• Enter the function =average(D8:D15) in cell D17 and copy across to O17.• Make a copy of your spreadsheet and paste it in your workbook.

1 Which player had the best average?

2 What was the average number of points scored per game by the team?

3 Describe the meaning of the results in cells:a D17 to M17 b N17 c O17

C D E … M N O

6 Game

7 Name 1 2 10 Total Average

8 Amanda 5 8 10 =sum(D8:M8) =average(D8:M8)

9 Colin 4 8 8

10 Eloise 12 8 8

�

15 Greg

16 Total =sum(D8:D15)

17 Average =average(D8:D15)

Working mathematicallyReasoning and reflecting: Ranges, modes, medians and means1 a If the mean of a set of scores is 15, does 15 need to be one of the scores? Give an

example to illustrate your answer.b If the median of a set of scores is 46, does 46 need to be one of the scores? Make up

a set of scores to illustrate your answer.c If the mode of a set of scores is 20, does 20 need to be one of the scores? Provide an

example to illustrate your answer.d Can two sets of scores have the same mean but different ranges? Make up an example

to illustrate your answer.

2 In a diving competition, the judges awarded the following scores:

7.5, 7.0, 8.0, 8.0, 7.5, 7.0, 8.5, 4.5, 8.0

a Find the mean, median and mode of these scores.b Which measure in part a best describes the data? Why?

3 A student scored the following marks in eight maths tests:

67, 70, 73, 65, 28, 75, 68, 65

a Find the mean, median and mode of the data.b Which measure in part a do you think best describes the student’s results? Why?



The mean from a frequency tableFor a large number of scores, we usually make up a frequency distribution table. The table can be extended by adding an extra column so that the mean can be easily found.

Example 5

Find the mean of the following sets of scores:

4 6 3 3 8 6 5 8 5 6 6 8 7 9 4

4 5 5 3 6 8 9 9 5 8 8 7 7 6 6

SolutionThe scores are organised in a frequency table:

Σ is the Greek letter sigma which, in mathematics, means the sum of.We use the symbol x̄ (read as ‘x bar’) to represent the mean.

For a frequency table, the mean isx̄ = (sum of all scores divided by number of scores)

∴ x̄ =

= 6.13

Score, x Frequency, f fx

3 3 9

4 3 12

5 5 25

6 7 42

7 3 21

8 6 48

9 3 27

Σf = 30 Σfx = 184

ΣfxΣf--------

18430---------

.

Extra columnfx = f × x

which gives thesum of the scores.

Σfx is the sum of the fx column (that is, the

sum of all scores).

Σf is the sum of the frequency column (that is, the total

number of scores).

1 Copy and complete the following tables. Calculate each mean.

a Scorex

Frequencyf fx

2 5

3 3

4 7

5 6

6 4

Σf = Σfx =

b x f fx

22 3

23 8

24 7

25 9

26 5

27 4

Σf = Σfx =




Using a calculator to find the meanWhen finding the mean for a small number of scores, it is very easy to just add the scores and divide the sum by the number of scores. However, when working with a large number of scores, it may be easier to use the statistics mode on a calculator.• Switch your calculator to the statistics mode (STAT or SD may be displayed).• Before starting a new statistics calculation, press the appropriate key to clear all data from the

previous calculation. (This could be the SAC, SCL, CA, … function key.)

• To enter data, press the number followed by the key (also called the data or data entry key).

• The function gives the mean of the scores.• Most calculators will have a function that gives the number of scores entered.

It is important to learn which keys to use on your calculator. Ask your teacher (or refer to the calculator manual) if necessary.

2 The following data is the number of times 40 children have gone to the movies in one month:

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

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

Make up a frequency distribution table and then calculate the mean.

3 Find the mean of each of the following sets of data.

4 A TV repair service received the following numbers of calls per day over 31 days:

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

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

Arrange the data in a frequency distribution table and find the mean number of calls per day.

d x f fx

0 2

1 4

2 5

3 8

4 10

5 6

6 5

Σf = Σfx =

c x f fx

7.0 6

7.1 10

7.2 8

7.3 4

7.4 2

Σf = Σfx =

a x 0 1 2 3 4

f 11 17 20 6 1

b x 10 11 12 13 14 15 16

f 4 7 11 12 8 6 2

M+

x



Example 6

Find the mean of these scores: 14, 11, 10, 9, 16, 16, 14, 15.

SolutionPut the calculator in statistics mode and then press the keys shown below to enter the scores.

Enter the scores:

14 11 10 9 16 16

14 15

Find the mean of the data: (answer should be 13.125)

Find the mean of this frequency distribution.

SolutionThe scores (x) and frequencies can be entered into a calculator when it is in STAT or SD mode. Ask your teacher, or refer to the calculator manual if necessary.

The mean of the data is x̄ = 22.825.

M+ M+ M+ M+ M+ M+M+ M+

x

Example 7

x f

21 6

22 8

23 15

24 9

25 2

1 Use your calculator to find the mean of each of the following sets of data (correct to one decimal place where necessary).a 5, 7, 2, 8, 8, 4, 3, 6, 7, 8, 10 b 12.1, 13.4, 13.6, 12.8, 14.8, 15.8, 11.9, 14.0, 13.8c 47, 55, 48, 58, 52, 49, 51, 62, 44 d 64, 71, 48, 83, 60, 55, 70, 77, 85, 35

2 Find the mean of each of the following frequency distributions (correct to one decimal place where necessary).

a Scorex

Frequencyf

32 5

33 8

34 12

35 9

36 7

37 4

b x f

10 12

11 18

12 27

13 37

14 23

15 15

16 9

c x f

6.0 5

6.5 14

7.0 19

7.5 13

8.0 7

d x 55 56 57 58 59 60

f 23 28 35 17 14 8

e x 47 48 49 50 51 52

f 7 18 23 25 20 12


Example 7

Worksheet 10-06

Statistical calculations

Worksheet 10-07

Statistical mode (graphics calculator)



3 The heights of 30 students were recorded to the nearest centimetre. These are the results:

143 156 152 164 170 159 173 179 168 173

156 168 170 172 169 145 155 154 148 178

179 167 168 156 154 158 149 152 163 175

Find the mean height, to the nearest centimetre.

Averages and totalsThe average or mean of a set of scores is given by the formula:

Mean x̄ =

1 Consider these examples.a Find the average of the following scores: 11, 15, 7, 9, 16.

Sum of scores = 11 + 15 + 7 + 9 + 16 = 58Number of scores = 5

Average = = 11.6

b Find three numbers that have an average of 18.Working backwards, number of scores = 3, mean = 18, so we require a sum of scores that, when divided by 3, equals 18.∴ the sum of the scores = 18 × 3 = 54.Choose any three numbers that have a total of 54 to be your answer, for example: 20, 20 and 14.

c In his last four cricket matches, Anand scored 48, 76, 58 and 60 runs. If Anand wants to achieve an average score of 65, how many runs does he need to make in his next match?We want the five scores to have an average of 65.∴ the sum of the scores needs to be 65 × 5 = 325.The sum of the first four scores = 48 + 76 + 58 + 60 = 242.∴ the required fifth score = 325 − 242 = 83.Anand needs to score 83 in his next match to achieve an average of 65.

d Three girls with a mean height of 170 cm are joined by another girl whose height is 165 cm. What is the mean height of the girls now?We can solve this problem, even if we don’t know the individual heights of the original three girls.The original three girls have a mean height of 170 cm.∴ the sum of the three girls’ heights = 170 × 3 = 510 cm.∴ the sum of the four girls’ heights = 510 + 165 = 675 cm.

∴ the mean height of the four girls = = 168.75 cm.

2 Now solve the following problems:a Find the average of the following scores: 34, 20, 16, 25, 30, 31.b Find three numbers that have an average of 13.c Find four numbers that have an average of 22.d Melinda scored 77%, 81% and 65% in her last three maths tests. What mark must she

score in her next maths test to achieve an average score of 80%?

sum of scoresnumber of scores-----------------------------------------

585------

6754

---------

Skillbank 10SkillTest 10-01

Averages and totals



Grouping dataWhen measuring the heights of students, there could be a large number of possible scores. In a class of 30 students you could have 30 different heights, and constructing a frequency table would not provide much useful information. To overcome this problem, the scores can be grouped together into class intervals.

e Three numbers have an average of 50. Two of them are 17 and 45. What is the third number?

f Four boys with an average mass of 72 kg are joined by a fifth boy whose mass is 77 kg. What is their average mass now?

g Eight workers have a mean weekly wage of $627. They are joined by another worker whose wage is $582. What is the mean wage of the workers now?

h In her last four basketball games, Siobhan scored an average of 38 points. If she wants to increase her average to 40 points, how many points must she score in her next game?

i Three friends have an average age of 16. They are joined by another friend who decreases their average age to 15. How old is the fourth friend?

j Five boys have an average height of 175 cm. They want to find another boy whose height will make their average height increase to 180 cm. How tall must this sixth boy be?

Example 8

Here are the marks of 50 students scored for a history test, marked out of 100.

65 56 26 34 67 70 34 59 72 89

37 75 48 64 72 61 53 48 57 68

19 34 56 66 84 90 71 66 37 26

48 29 78 63 59 54 68 74 81 69

45 63 67 74 68 54 44 38 57 60

a Arrange the scores in class intervals (0–9, 10–19, 20–29, etc.) and make up a frequency table with class centres.

b Find the modal class.c Find the median class.d Find an estimate for the mean mark from the frequency distribution table.

Solutiona The range for the test scores is: 90 − 19 = 71. This is so large that the data should be grouped

together into class intervals.We will take the class intervals to be 0–9, 10–19, 20–29, …, 100–109. Since the lowest score is 19, and the highest score is 90, the intervals 0–9 and 100–109 can be omitted.In order to find the mean, we need to find the middle score of each class, which is called the

class centre. For the class 20–29, the class centre is 24.5 . (The class centre is

also used when drawing the histogram.)

20 29+2

------------------ 24.5=( )



Histograms and frequency polygons using grouped dataThe information from a grouped frequency distribution can also be shown graphically in a histogram.

The class centres are used on the horizontal axis when constructing histograms and frequency polygons.

Using the data from the Example 16, we obtain the following histogram and frequency polygon.

The graph shows that nine students scored around 54.5% (actually 50–59) while 14 students scored around 64.5%. How many students scored around 44.5%?

The frequency distribution for the history test mark is:

Grouping the history test marks into class intervals has made them more manageable. It is now easier to see how the test marks are distributed or spread.

b The modal class is the class with the highest frequency. For the history test, the modal class is 60–69.

c In the history test there are 50 scores, so the middle score is the average of the 25th and 26th scores. Since these scores fall in the class 60–69, the median class is 60–69.

d An estimate for the mean is obtained by considering the class centres as the scores. So the mean (x̄ ) is given by:

x̄ =

=

= 57.1Therefore, the mean (average) history mark was 57.1.

Classinterval

Class centrex

TallyFrequency

f fx

10–19 14.5 | 1 14.5

20–29 24.5 ||| 3 73.5

30–39 34.5 |||| | 6 207.0

40–49 44.5 |||| 5 222.5

50–59 54.5 |||| |||| 9 490.5

60–69 64.5 |||| |||| |||| 14 903.0

70–79 74.5 |||| ||| 8 596.0

80–89 84.5 ||| 3 253.5

90–99 94.5 | 1 94.5

Σf = 50 Σfx = 2855

ΣfxΣf--------

285550

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

2

4

6

8

10

12

14

14.524.5 44.5 64.5 84.5

034.5 54.5 74.5 94.5

Histogram and frequency polygon

Test marks (class centres)

Fre

qu

en

cy



1 Find the class centre of each of the following class intervals.a 15–23b 21–30c 45–50

2 a Copy and complete the following frequency distribution table.

b Find the mean of the distribution.c What is the modal class?d What is the median class?

3 Copy and complete the following frequency distribution table and then find the mean.

4 The pollution level of a large Australian city was tested over a period of 30 days. The levels of sulphur dioxide (in parts per million) were as follows:

0.07 0.17 0.18 0.11 0.10 0.09 0.08 0.08 0.04 0.15

0.12 0.06 0.12 0.13 0.18 0.14 0.07 0.16 0.19 0.23

0.22 0.26 0.25 0.19 0.13 0.05 0.12 0.03 0.03 0.11

a Construct a frequency distribution table using these class intervals:0.00–0.04, 0.05–0.09, 0.10–0.14, 0.15–0.19, 0.20–0.24, 0.25–0.29

b Construct a histogram and frequency polygon of this data.c Find the mean.

Classinterval

Class centrex

TallyFrequency

f fx

18–24 21 |||

25–31 |||| |||

32–38 |||| |||| |

39–45 |||| |||| ||||

46–52 |||| ||||

53–59 ||||

60–66 ||

Σf = Σfx =

Classinterval

Class centrex f fx

10–19 4

20–29 10

30–39 18

40–49 25

50–59 14

60–69 8

70–79 1

Σf = Σfx =



Frequencies and histograms

Worksheet 10-08

Analysing data



Cumulative frequency and the medianThe cumulative frequency is a progressive total of the frequency. The cumulative frequency for a particular score is the sum of the frequencies for that score and for all the scores less than it.

The cumulative frequency enables us to answer questions such as: ‘How many students scored less than 7 out of 10’?

The cumulative frequency can also be used to find the median.

The last figure in the cumulative frequency (cf ) column must equal the total of the frequency column.

Using the cumulative frequency column, we can see (among other things) that:• 15 scores are less than or equal to 2• 18 scores are less than 4.

Finding the median

5 The following results are the marks obtained by a class in a maths test.71 65 95 70 52 33 87 72 69 7652 73 62 47 56 63 73 53 41 4847 68 58 87 68 71 76 70 60 67a Organise the data into a grouped frequency table,

using the intervals 30–39, 40–49, …, 90–99.b Draw a frequency histogram of the data.c Construct an ordered stem-and-leaf plot, using

stems of 30, 40, etc.d Which display of this data do you prefer? Give

reasons.

6 This histogram on the right shows the heights of students in a high school.a How many students had their height measured?b What are the class intervals?c Calculate the mean height of the students.

2

4

6

8

10

12

143 150 157 164 171 178 185

Heights of studentsat high school

Heights of students (cm)

Fre

qu

en

cy

Scorex

Frequencyf

Cumulativefrequency (cf )

0 2 2

1 5 7

2 8 15

3 3 18

4 1 19

5 1 20

Σf = 20

Example 9

1 Find the median of the scores displayed in this frequency table.

SolutionThe median is the middle score. For 31 scores, the middle score is the sixteenth score. From the cumulative frequency column, it can be seen that the middle score lies between the 12th and 22nd scores.

x f cf

3 4 4

4 8 12

5 10 22

6 5 27

7 3 30

8 1 31

Σf = 31



The first 22 scores are less than or equal to 5, but the first 12 of these are less than or equal to 4, so the median must be 5.∴ median = 5.


SolutionThere are 40 scores, so the median must be halfway between the 20th and 21st scores.

The first 29 scores are less than or equal to 4, but the first 17 scores are less than or equal to 3, so the 20th and 21st scores must both be 4.

∴ median=

= 4


SolutionSince there are 20 scores, the median is halfway between the 10th and 11th scores.

The first 17 scores are 3 or less and the first 10 scores are 2 or less, so the 10th and 11th scores are 2 and 3 respectively.

∴ median=

= 2.5

4 Find the median class of the scores displayed in this frequency table.

SolutionSince there are 50 scores, the median must be halfway between the 25th and 26th scores. The first 34 scores are in the intervals 30–39 or less, and the first 17 scores are in the intervals 20–29 or less.

∴ the 25th and 26th scores are both in the 30–39 class interval.

∴ the median class is 30–39.

x f cf

1 2 2

2 5 7

3 10 17

4 12 29

5 7 36

6 4 40

Σf = 40

4 4+2

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

x f cf

0 1 1

1 3 4

2 6 10

3 7 17

4 2 19

5 1 20

Σf = 20

2 3+2

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

Classinterval

Class centrex f cf

0–9 4.5 2 2

10–19 14.5 5 7

20–29 24.5 10 17

30–39 34.5 17 34

40–49 44.5 12 46

50–59 54.5 4 50

Σf = 50



1 Copy and complete each of the following frequency distribution tables and then find the median or median class of each distribution.

a x f cf

0 7 7

1 8

2 10

3 5

4 9

5 6

6 5

b x f cf

45 2

46 6

47 7

48 10

49 3

50 3

cClass

intervalClass

centre x f cf

1–5 2

6–10 7

11–15 10

16–20 8

21–25 10

26–30 3

dClass

intervalClass

centre x f cf

0–9 3

10–19 7

20–29 11

30–39 19

40–49 7

50–59 2

e x f cf

12 5

13 16

14 11

15 7

16 4

fClass

interval xClass

centre x f cf

35–39 3

40–44 5

45–49 10

50–54 15

55–59 28

60–64 27

65–69 22

70–74 18

75–79 15

80–84 9




2 The blood alcohol levels of 50 drivers who tested positive when breathalysed are as follows:

0.18 0.01 0.02 0.05 0.06

0.03 0.10 0.07 0.07 0.15

0.06 0.20 0.24 0.08 0.10

0.03 0.24 0.09 0.16 0.18

0.12 0.25 0.03 0.15 0.16

0.12 0.13 0.04 0.06 0.21

0.28 0.25 0.11 0.05 0.17

0.08 0.05 0.10 0.14 0.17

0.23 0.08 0.15 0.06 0.09

0.20 0.22 0.19 0.16 0.04

a Make up a frequency distribution table using these class intervals:0.00–0.04, 0.05–0.09, 0.10–0.14, …, 0.25–0.29.

b What is the modal class?c Use the table to calculate the mean.d Find the median class.

3 The following data shows the speeds in kilometres per hour of semi-trailers travelling towards Liverpool.

127 93 110 104 102 130 101 124 109 125

88 139 105 102 143 90 120 113 91 99

121 110 100 138 150 117 132 98 104 107

85 95 103 115 98 126 133 94 97 101

a Make up a frequency distribution table using these class intervals:81–90, 91–100, …, 141–150

b What is the modal class?c Use the table to calculate an estimate for the mean.d Find the median class.e If the speed limit for semi-trailers is 100 km/h, what percentage of semi-trailers were

speeding?

4 The marks of 50 students in a science test are as follows:

47 64 58 55 40 72 75 81 68 70

39 49 55 56 68 67 72 79 60 63

74 54 50 56 46 48 66 65 76 51

45 63 67 38 46 51 51 85 77 64

52 62 75 68 60 70 50 78 87 71

a Use the class intervals 30–39, 40–49, 50–59, …, 80–89 to make up a frequency distribution table.

b What is the modal class?c Find the median class.d Use the table to calculate an estimate for the mean.e What percentage of students scored a mark above the mean?f How many students scored:

i less than 50? ii 70 marks or more?



Cumulative frequency polygons and the medianA histogram and frequency polygon are drawn using the ‘score’ and ‘frequency’ columns. In the same way, a cumulative frequency histogram and polygon can be drawn using the ‘score’ and ‘cumulative frequency’ columns. The cumulative frequency polygon can then be used to find the median.

Example 10

Draw a cumulative frequency histogram and polygon from the following table, and then find the median.

SolutionA feature of a cumulative frequency histogram is that each successive column is higher. Why?

In drawing the cumulative frequency polygon (which is also called the ogive), the top right-hand corners of the columns are joined instead of the midpoints.

To determine the median using this graph, follow the steps below.

Step 1: Find the 50% (halfway) point on the cumulative frequency axis.

Step 2: Draw a horizontal line from this point to meet the polygon (ogive).

Step 3: From this point, draw a vertical line to meet the horizontal axis.

• If the vertical line falls within a column, then the median is the score for that column.

• If the vertical line falls on the border between two columns, then the median is the average of the two columns.

The median for this data is 23.

Score x

Frequency f

Cumulative frequency cf

21 3 3

22 5 8

23 9 17

24 7 24

25 4 28

26 2 30

2

0

4

6

8

10

12

14

16

18

20

22

24

26

28

30

21 22 23 24 25 26

halfway point

ogive

50%

Cumulative frequencyhistogram and polygon

Score

Cu

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qu

en

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1 Draw a cumulative frequency histogram and polygon for each of the following distribution tables and then find the median (or median class).

2 The following data represents the number of sandwiches sold daily in a school canteen.

25 28 22 35 36 40 18 43 37 25 26 34 36 42 28 48 33

41 32 27 22 35 45 28 41 37 37 27 47 38 23 35 39 24

a Use the class intervals 16–21, 22–27, 28–33, … to make up a frequency distribution table (include the cumulative frequency).

b Draw a cumulative frequency histogram and polygon.c Use the graph from part b to find the median class.d Use the frequency distribution table to calculate the mean.

3 This cumulative frequency histogram represents the data from a grouped frequency distribution.a What are the class intervals?b Draw the ogive to find the median

class.c Which class interval is the modal

class?d What percentage of employee ages

were less than 45?

a x f cf

6 4 4

7 10 14

8 15 29

9 7 36

10 3 39

11 1 40

b x f cf

30 2 2

31 3 5

32 5 10

33 6 16

34 3 19

35 1 20

c Classinterval

Class centre x f cf

5–9 6 6

10–14 10 16

15–19 17 33

20–24 15 48

25–29 9 57

30–34 3 60

0

10

20

30

40

50

28 35 42 49 56 63

The age of employees

Class centres

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4 This table shows the blood-alcohol levels of 50 drivers who tested positive when breathalysed.

a Copy and complete the table.b Draw a cumulative frequency histogram and polygon.c Use the graphic from part b to find the median class.d Use the frequency distribution table to calculate the mean.

Class interval

Class centre x f cf

0.00–0.04 6

0.05–0.09 18

0.10–0.14 11

0.15–0.19 10

0.20–0.24 4

0.25–0.29 1

Working mathematicallyReasoning and reflecting: Which method is better?1 The following scores are marks out of 100 for a test taken by 40 students.

72 62 35 47 28 93 27 48 55 64 84 68 49 52 43 33 37 76 78 55

34 88 76 60 67 56 40 69 73 80 53 29 84 74 75 66 70 82 33 20

a Copy and complete this distribution table, using the data given above.

b What is the modal class? c What is the median class?

2 a Organise the 40 test marks from Question 1 into an ordered stem-and-leaf plot.b What is the mode? c Find the median.d Use the stem-and-leaf plot to find:

i the modal class ii the median class.

3 Compare the two methods (frequency distribution table and stem-and-leaf plot) of displaying data used in Questions 1 and 2.a List the advantages and disadvantages of each method.b Which method is easier to use when finding:

i the mode? ii the median?c Which method is easier to use when finding:

i the modal class? ii the median class?

Classinterval

Class centrex

Frequencyf

Cumulative frequencycf

20–29

30–39

90–99



Measures of spreadThe statistical measures considered so far are the range, mode, median and mean. The mode, median and mean are measures of location because they give an indication of a central value.

The range, however is a measure of spread or dispersion. Measures of spread indicate how much a set of data is spread out or dispersed around a central value, often the mean.

QuartilesWhen a set of data is arranged in ascending order, it can be divided into four equal parts. The values that divide the data into the four equal parts are called the first, second and third quartiles.

The first quartile is often referred to as the lower quartile and is represented by the symbol Q1 or QL. It is the value below which 25% of the scores lie.

The second quartile (Q2), since it is the middle value, is also the median. It is the value that separates the lower 50% of scores from the upper 50% of scores.

The third quartile is often called the upper quartile and is represented by the symbol Q3 or QU. It is the value above which 25% of the scores lie.

Note: The lower quartile is the median of the lower half of the scores.The upper quartile is the median of the upper half of the scores.

d For the set of 40 marks, which is more useful — the mode or the modal class? Why?

4 Which would be easier to use when finding the mean — a frequency distribution table or a stem-and-leaf plot? Give reasons.

Lowest score(or lower extreme)

First quartile(Q1 or QL)

Second quartile(Q2 or median)

Third quartile(Q3 or QU)

Highest score(or upper extreme)

x x x x x x x x x x x x x x x x x x x x

Scores (in order)

Example 11

Find the quartiles for each of the following sets of data.a 65 84 75 82 97 70 68 76 93 48

79 54 80 79 82 96 63 85 72 70b 9 3 8 7 6 8 4 6 2 10 9c 15 18 7 16 23 9 15 20 16 14 13 11 19

Solutiona Arranging the 20 scores in ascending order, we have:

48 54 63 65 68 70 70 72 75 76 79 79 80 82 82 84 85 93 96 97

Q1 (lower quartile) = 69 (halfway between 68 and 70)Q2 = median = 77.5Q3 = (upper quartile) = 83

Q1 =

= 69

68 70+2

------------------ Q2 (median)=

= 77.5

76 79+2

------------------ Q3 =

= 83

82 84+2

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



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

The interquartile range takes into account the middle 50% of scores and ignores very low or very high scores (outliers).

b Arranging the 11 scores in ascending order, we have:2 3 4 6 6 7 8 8 9 9 10

Lower quartile Median Upper quartileQ1 = 4 Q2 = 7 Q3 = 9

c Arranging the 13 scores in ascending order, we have:7 9 11 13 14 15 15 16 16 18 19 20 23

Lower quartile Median Upper quartile

Q1 = Q2 = 15 Q3 =

= 12 = 18.5

11 13+2

------------------ 18 19+2

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

Interquartile range= upper quartile − lower quartile

= Q3 − Q1

lower quartileQ1

medianQ2

upper quartileQ3

50%

interquartile range

25% 25%

Example 12

A pizza restaurant received the following orders for pizzas from its tables.

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

Find the interquartile range.

SolutionFirst arrange the scores in order:

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

Lower quartile Median Upper quartile

Q1 = Q2 = Q3 =

= 2 = 3 = 4.5

Interquartile range = Q3 − Q1

= 4.5 − 2

= 2.5

2 2+2

------------ 3 3+2

------------ 4 5+2

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



Example 13

Find the interquartile range for each of the following sets of data.

a b

Solutiona There are 18 scores so the median is

‘between’ the 9th and 10th scores∴ median, Q2 = 9The lower quartile is the middle (or median) of the lower half of scores.∴ lower quartile, Q1 = 7The upper quartile is the middle of the upper half of scores.∴ upper quartile, Q3 = 9∴ interquartile range= Q3 − Q1

= 9 − 7 = 2b There are 21 scores so the median is the 11th of the ordered scores.

∴ median, Q2 = 34Lower quartile, Q1 = 24

Upper quartile, Q3 =

= 40.5∴ interquartile range= 40.5 − 24

= 16.5

Find the interquartile range for the data presented in the cumulative frequency histogram and polygon on the right.

5 6 7 8 9 10 11 12

Stem Leaf

1 2 7

2 0 3 4 4 5

3 1 2 2 4 6 8 8 9

4 0 1 3 7

5 1 2

5 6 7 8 9 10 11 12

Q1

Q2

Q3

Q1

Q2

Q3

Stem Leaf

1 2 7

2 0 3 4 4 5

3 1 2 2 4 6 8 8 9

4 0 1 3 7

5 1 2

40 41+2

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

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4 5 6 7 8 9 10Score

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Example 14

10_NC_Maths_9_Stages_5.2/5.3 2 Friday, February 6, 2004 2:23 PM


The interquartile range focuses on the middle 50% of scores and is unaffected by extreme scores.

SolutionThe vertical axis is rescaled, showing the 25th, 50th and 75th percentiles (Q1, Q2 and Q3). Lines are drawn across from the 25% and 75% marks to the ogive and then the lower and upper quartiles are located on the horizontal axis.

∴ Q1 = 5 and Q3 = 8

∴ interquartile range= Q3 − Q1

= 8 − 5= 3

Use the interquartile range to determine which set of data (A or B) has the smaller spread.A: 32 15 26 42 37 18 29 20B: 15 24 23 26 28 42 29 24

SolutionArranging the scores in ascending order, we have:A:

Q1 = 19 (halfway between 18 and 20), Q3 = 34.5∴ interquartile range= 34.5 − 19

= 15.5

B:

Q1 = 23.5, Q3 = 28.5∴ interquartile range= 28.5 − 23.5

= 5Since B’s interquartile range is much smaller than that of A, B is less spread out than A.

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100%

50%

75%

25%

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ogive

Me

dia

n

Q1

Q3

Example 15

Q1 median Q3

15 18 20 26 29 32 37 42

Q1 median Q3

15 23 24 24 26 28 29 42

1 Find the quartiles for each of the following sets of data.a 2 5 8 3 7 8 7 5b 24 20 18 26 25 28 26 30 23




c 15 16 12 18 18 16 15 14 11 17 20 12 15d 29 35 42 38 21 64 49 58 30 40e 25 28 27 36 40 18 43 25 26 27 45 41 37 22 23 39 24 48

2 For each of the following sets of data, find:i the median, Q2 ii the lower quartile, Q1

iii the upper quartile, Q3 iv the interquartile range.a 4 5 5 7 10 12 14 15 15b 20 23 23 24 26 28 30 31 31 31 33c 56 41 50 42 42 50 55 60 45 35 26 42 52 50 58 41

3 Find the interquartile range of each of the following sets of data.a b

c d

e f

4 Find the interquartile range for the data represented in each of the cumulative frequency histograms below.a b

10 11 12 13 14 15 16 17 6 7 8 9 10 11

Stem Leaf

3 2 7

4 0 3 3 5

5 2 4 5 6 7 8 8

6 3 4 7

7 2

Stem Leaf

1 3 5 8 9

2 0 1 3 3 4 5 6

3 5 8 9 9

4 1 3

5 4

Stem Leaf

10 3 5 5 6 6

11 0 1 2

12 3 4 6 7 8

13 4 7

14 1

48 49 50 51 52 53

2

0

4

6

8

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12

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16

18

20

42 43 44 45 46 47Score

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1 2 3 4 5 6 7Score

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Example 12

Example 14

Example 13

Worksheet 10-09

Statistical match-up



c

5 In testing the life of torch batteries, the Durable Battery Co. loaded 50 torches with new batteries and left them on until they failed. The times (in hours) to failure were as follows:16.7 15.5 14.6 19.2 18.3 17.6 15.4 20.7 8.2 8.610.8 12.5 11.0 9.8 17.1 20.2 18.5 12.2 15.5 19.810.7 4.8 16.8 17.0 13.6 9.2 19.8 14.2 10.6 18.121.0 14.8 18.4 18.0 23.5 15.4 18.4 13.7 9.8 8.616.7 15.4 19.9 21.0 22.3 8.4 11.8 13.5 13.6 14.2a Using class intervals of 0–5, 5–10, 10–15, 15–20 and 20–25, construct a grouped

frequency distribution table (include a cumulative frequency column).b Draw a cumulative frequency histogram and polygon.c Use the graph in part b to find the median and interquartile range.

6 Two groups of students have their pulse rates taken. The results are as follows: A: 82 81 72 96 62 66 91 69 77 67 80 75 72 79 77 58B: 82 61 51 79 73 64 93 81 74 75 64 59 86 90 81 72a Draw a back-to-back, ordered stem-and-leaf plot to display the data.b Find the median pulse rate of each group.c Find the pulse rate range of each group.d Find the interquartile range of each group.e Which group has less spread in their pulse rates? Justify your answer.

7 Two netball teams were compared to determine which one was more consistent in goal scoring. The results are as follows:Team 1: 47 55 32 64 22 57 73 35 46 51Team 2: 23 48 55 46 24 29 42 35 19 47a Find the range of each team’s score.b Is there a significant difference between the ranges?c Find the interquartile range of each team.d Which team is more consistent? Justify your answer.

0

50

40

30

20

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60

70

15 16 17 18 19 20 21Score

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Example 15



Summary statistics on a graphics calculatorUsing a TI-82 or TI-83 graphics calculator

All the summary statistics we have considered can be found using a graphics calculator. An ordinary calculator will usually be able to calculate the mean, but not the median or the quartiles. The steps on a TI-82 or TI-83 are shown below. Use with this data:

6 4 5 3 2 6 5 6 4 3 2 4 2 4

Step 1: Enter the data:

1 : EDIT 6 4 and so on for the rest of the data.You should see the screen shown to the right.

Step 2: Find 1-Var Statistics:

1: 1-Var Stats You should see the screen shown to the right.

Use the down arrow to scroll down to the screen shown to the right.

Step 3: The summary statistics are:Mean = 4.0Minimum = 2Maximum = 6Lower quartile (Q1) = 3.0Median (Q2) = 5Upper quartile (Q3) = 5.0

Step 4: Clear memory:

4 : ClrAllLists


Step 1: Enter the data:

6 4 5 and so on for the rest of the data.You should see the screen shown to the right.

STAT ENTER ENTER

STAT ENTER

+ ENTER

STAT EXE EXE EXE

Using technology



Box-and-whisker plotsThe lower extreme (lowest score), lower quartile, median, upper quartile and upper extreme (highest) score together make a five-point summary. These points can be shown on a box-and-whisker plot (sometimes called a boxplot).

Step 2: Find 1-Variable statistics:

(CALC) (1 Var)You should see the screen shown to the right.

Step 3: Use the down arrow to scroll down to the screen shown to the right.

F2 F1

lower extremeor lowest score

upper extremeor highest score

whisker

lowerquartile median

interquartile range

upperquartile

whisker

Example 16

Represent the following data on a box-and-whisker plot.

6 7 8 5 10 13 7 3 12

SolutionFirst arrange the scores in order.

Lower extreme = 3 Upper extreme = 13Median = 7 Lower quartile, Q1 = 5.5 Upper quartile, Q3 = 11

Q1 = 5.5 median Q3 = 11

3 5 6 7 7 8 10 12 13

3 4 5 6 7 8 9 10 11 12 13 14

Score

lower extreme

upper extreme

medianQ1 Q3



Example 17

The boxplot below represents the number of houses sold per week by a real estate agency over a period of 16 weeks.

a What was the range in the number of houses sold?b What was the interquartile range?c What was the median number of houses sold?d What percentage of weeks had sales of more than 14 houses?e How many weeks had sales of more than 11 houses?

Solutiona Range= upper extreme − lower extreme

= 23 − 9= 14 houses

b Interquartile range= Q3 − Q1

= 18.5 − 11= 7.5 houses

c Median number sold = 14 housesd 14 houses is the median, so 50% of weeks had sales of more than 14 houses.e 11 is the lower quartile, so 75% × 16 = 12 weeks had sales of more than 11 houses.

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Number of houses

1 The number of hours per day that Sandra worked for three weeks before Christmas are as follows:5 5 4 8 8 10 6 3 4 6 108 2 2 6 7 7 10 9 6 8a Find the median. b What is the lower quartile?c What is the upper quartile? d Calculate the interquartile range.e Draw a box-and-whisker plot for the data.

2 Draw box-and-whisker plots for the following sets of data.a 9 6 3 9 8 6 10 8 9 2 8 9b 9 5 13 12 13 11 14 12 10 11 14 9 8 15c 40 52 47 46 53 46 45 44 49 48 48 49 54 48 47 42 45

3 The box-and-whisker plot below summarises the number of hours worked in one week by employees.

20 21 22 23 24 25 26 27 28 29 30 31 32

Hours worked


Example 17

Spreadsheet 10-05

Box-and-whisker plots


Box and whisker plots



a What is the median number of hours worked?b What is the lower quartile?c What is the upper quartile?d Find the interquartile range.e What percentage of employees worked between 26 to 30 hours?

4 The following data represents average monthly temperatures (in °C).Alice Springs: 28 27 25 20 16 12 12 14 18 23 25 28Hobart: 16 16 15 12 11 8 8 9 11 12 14 15a What is the median temperature of:

i Alice Springs?ii Hobart?

b Find the interquartile ranges of both cities.c Draw box-and-whisker plots to compare the temperatures of the two cities.

5 These box-and-whisker plots show the numbers of points scored by two players during the season.

a Which player has the highest point score for a single game?b What is the range of the points scored by:

i player X?ii player Y?

c By just looking at the range, which player would seem to be more consistent? Justify your answer.

d Find the median scores of both players.e Find the interquartile ranges for both players.f Which player is more consistent?g In what percentage of games did player X score 9 or 10 points?h In what percentage of games did player Y score more than 12 points?

6 Find the five-point summary and draw a boxplot for each of the following sets of data.

a b

4 5 6 7 8 9 10 11 12 13 14 14 16

Points scored

Player X

Player Y

Stem Leaf

1 2 4 5

2 0 1 6 6 8

3 2 3 3 7 8 9 9

4 0 1 3 8

5 2 7

1 2 3 4 5 6 7



Drawing boxplots on a graphics calculatorUsing a TI-82 or TI-83 graphics calculator

Follow the steps below to draw a boxplot for this set of scores:8 7 9 5 6 8 5 7 9 4 7 8 8 8

Step 1: Open and enter the data in column L1.

Step 2: Turn PLOT on for Plot 1.

Step 3: For Plot 1, select the boxplot icon, L1 as the Xlist, and 1 as the frequency.

Step 4: Select , 9: ZoomStat, then .

Step 5: Select to move around each boxplot to compare the key measures.


Step 1: Open and enter the scores.

Step 2: Press (GRPH) (SET) and use the down arrow to make the following settings.

GRAPH TYPE BOX ( )

X LIST LIST 1 ( )

FREQUENCY 1( )

Step 3: Press to return to the screen of lists and select (GPH1) to graph the boxplot.

Draw a boxplot for the data in the following table.

Score 3 4 5 6 7 8

Frequency 5 8 17 10 6 3

STAT

STAT

ZOOM GRAPH

TRACE

STAT

F1 F6

F6 F2

F1

F1

EXE F1

Using technology



1 The following table gives the tide movement for Sydney over a 48-hour period.

a With time on the horizontal axis, plot the information given in the table. Join the points of your line graph with a smooth curve.

b Why should the curve joining the points be smooth?

2 Types of random samplingThere are other ways of selecting random samples so that cost is reduced and reliability is not sacrificed. Two of these include:• Stratified sampling

The population is divided into groups and a number is then selected from each group in proportion to the size of that group. For example, if a sample of 20 is to be taken from a population of 80 boys and 120 girls, then 8 boys and 12 girls should be selected.

• Systematic samplingThe first item of a sample is selected at random and subsequent items are selected at regular intervals, for example every hundredth item. This is often used when a product, such as torch batteries, is tested for quality.

a The table shows the number of students in each year level at Southwest High School.

i What is the school population?ii What percentage of the school population is

formed by each year level?iii 100 students are to be surveyed about the

number of hours they spend watching TV on Monday afternoon. How many students from each year should be selected? Why?

iv What type of sample is this?

b What type of random sampling — stratified (STR) or systematic (SYS) — would you use to obtain information about each of the following?

i Whether the student council is doing a good job.ii Determining whether Australia needs a new flag by surveying people from each state.iii Checking repair kits for 10-speed bicycles.iv What new books the school library should buy.v Checking the quality of furniture made by a manufacturer.vi The quality of a pair of running shoes.vii Recreational venues in a local suburb.viii The quality of calculators being produced by Comcalc Pty Ltd.

3 The mean of the scores 63, 56, 77, 45 and x is 62. Find the value of x.

4 Nicole’s average mark for five maths tests was 64. She did another test and, as a result, her average mark for the six tests increased to 70. What mark did Nicole obtain in the sixth test?

TimeMid-night

3:00am

6:00am

9:00am

Noon3:00pm

6:00pm

9:00pm

Mid-night

3:00am

6:00am

9:00am

Noon3:00pm

6:00pm

9:00pm

Mid-night

Tide (m)

1.25 0.75 0.45 1 1.4 1 0.6 0.8 1.25 0.9 0.5 0.8 1.3 1.25 0.6 0.7 1

YearNumber of students

7 180

8 190

9 150

10 200

11 130

12 90

Power plus



5 The heights (in centimetres) of 30 students are as follows:164.3 166.5 172.3 169.3 164.2 170.8 171.8 179.0 175.6 163.8169.8 179.2 177.4 171.3 163.6 173.5 181.3 163.5 182.4 174.6174.2 160.1 163.4 160.9 172.4 171.3 179.8 164.7 168.1 163.2a Round the heights to the nearest centimetre, and use intervals of 5 cm to arrange the data in

an ordered stem-and-leaf plot.b Find the median.

6 Seeds were sown in 40 large boxes. The number of seedlings that sprouted per box are as follows:15 23 18 7 33 28 10 19 14 25 34 30 9 5 12 17 23 20 31 618 17 28 24 25 27 17 15 9 3 16 14 11 25 22 24 30 8 18 22a Organise the data into ordered stem-and-leaf plots using intervals of:

i 10 ii 5b Find the median of this data.c Find the modal class of both stem-and-leaf plots.d Which of the stem-and-leaf plots gives a better indication of how the seedlings have

grown? Explain your answer.

7 An outlier is a value or score that is separated from the rest of the scores. We may decide to ignore an outlier when drawing conclusions from a survey.Using the upper and lower quartiles as well as the interquartile range, we can calculate whether a score is an outlier or not.If a score is:• bigger than the upper quartile + 1.5 × (the interquartile range)• smaller than the lower quartile − 1.5 × (the interquartile range)then that score is an outlier.a Determine whether the following sets of scores have outliers.

i 11 8 1 7 12 6 8 7 10 12ii 12 12 14 20 6 10 11 15 17 18 22 8 16

b The temperatures (in °C) for 30 consecutive days are as follows:22 23 26 25 27 28 30 31 33 33 26 28 25 24 2527 28 26 30 32 33 36 24 25 30 17 18 25 26 30Determine which of the temperatures are outliers.

Language of mathsbias boxplot box-and-whisker plot

categorical census class centre class interval

cluster continuous cumulative data

discrete dot plot frequency grouped data

histogram interquartile range mean measure of location

measure of spread median mode ogive

outlier polygon population quantitative

quartile random range sample

statistic stem-and-leaf plot survey

1 The word range can be used in many non-statistical ways. Use it in three sentences that show its different meanings.

Worksheet 10-10

Statistics crossword



Topic overview• Rate yourself on the work in this chapter by copying and completing the following scales.

a Able to design and trial a questionnaire

b Able to obtain a random sample.

c Able to display data using a variety of methods.

d Able to analyse data using the mode, mean, median, range and interquartile range.

e Able to select an appropriate method for organising data.

• Copy and complete this overview, making any changes that you find necessary. Remember to use different coloured pens or pencils and pictures and key words.

2 A newspaper reported that ‘A boy almost drowned in a local lake. The average depth of the lake is just 30 cm.’ Explain why we need to be careful when using the average or mean.

3 Explain the difference between a sample and a census.

4 What is an outlier? Given an example of one.

5 What is the meaning of random selection? Describe a way in which a sample could be randomly selected.

0

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Low

Low

High

High

High

High

High

0 7 8 4 2 2 0 1 21 2 3 4 4 3 1 1 3 0 3 52 0 2 5 7 8 43 1 4 4 7 9 5

Stem Leaf A B

Displaying/organising dataCollecting data

Analysing data

Exploring data

• sample population• questionnaires• random••

• range• mode, modal class••••

•••



1 State whether the following data is categorical, discrete or continuous.a The temperature at Thredbo. b The number of times you go to the movies.c Shirt size.d The make of cars passing through an intersection.

2 The following data gives the number of children per house in a street.2 0 1 3 2 2 0 4 3 2 5 3 3 4 30 1 0 3 3 5 2 3 4 2 3 0 3 2 1a Construct a frequency distribution table of this data.b Draw a frequency histogram and polygon.c What is the mode of this data? d Calculate the mean.e Draw a cumulative frequency histogram and polygon, and use them to find the median.

3 The following scores are marks (out of 10) for a spelling test.6 3 7 8 5 7 6 5 7 6 10 9 5 0 3 4 6 6 5 4

a Display the data as a dot plot.b Are there any outliers? Give reasons for your answer.c Are the scores clustered about any mark? Give reasons for your answer.

4 The following scores are marks in a test out of 100 obtained by a group of 10 students.38 67 76 84 54 47 68 77 89 60a Find the range of these marks. b What is the mean?c What is the median?

5 Calculate the mean of the following sets of data (correct to one decimal place where necessary).

6 Use your calculator to find the mean of each of the following sets of data (correct to one decimal place where necessary).

Chapter 10 Review

Ex 10-01

Ex 10-02

Ex 10-03

Ex 10-04

Ex 10-05

a Score Frequency

8 15

9 24

10 37

11 20

12 9

b x 47 48 49 50 51 52 53

f 6 13 15 28 19 14 8

c x 17 18 19 20 21 22

f 4 9 14 15 12 7

Ex 10-06

a Score Frequency

32 7

33 15

34 18

35 27

36 13

37 6

b x 15 16 17 18 19 20

f 23 47 85 76 53 30

Topic testChapter 10



7 Golfers in a major tournament received the following scores.71 72 70 73 67 66 78 65 79 78 8074 75 75 69 68 66 68 67 70 70 7164 69 65 66 76 69 70 73 77 76 81a Organise the data in a grouped frequency distribution table, using the intervals 60–64,

65–69, …, 80–84.b Use the grouped frequency distribution table to find an estimate for the mean of the scores.c What is the modal class?d Draw a grouped frequency histogram and polygon.e Find the median class.

8 Find the median, or median class, of each of the following distributions.

9 The heights (in centimetres) of 50 students are given below.139 163 142 155 173 138 174 168 155 147147 181 177 164 168 176 184 180 171 163163 147 158 150 146 159 170 163 166 154168 152 158 164 175 163 177 170 150 156183 174 163 157 159 162 172 167 178 161a Use the class intervals 131–140, 141–150, 151–160, … to make up a frequency

distribution table. (Include the cumulative frequency.)b Draw a cumulative frequency histogram and polygon.c Use the graph from part b to find the median class.d What percentage of students were:

i shorter than 160 cm? ii taller than 170 cm?

10 The following scores are marks in a test out of 100 obtained by a group of 10 students.38 67 76 84 54 47 68 77 89 60Find:a the median b the lower quartile, Q1

c the upper quartile, Q3 d the interquartile range.

11 The box-and-whisker plot below summarises the weights of students in a Year 8 class.

a What is the median?b What is the interquartile range?

c What percentage of students weighed 36–40 kg?d What percentage of students weighed less than 46 kg?

Ex 10-07

Ex 10-08

a x f cf

3 5

4 12

5 18

6 6

7 3

8 1

b Class interval f cf

0–9 3

10–19 8

20–29 15

30–39 27

40–49 14

50–59 9

Ex 10-09

Ex 10-10

Ex 10-11

32 34 36 38 40 42 44 46 48 50 52 54 56

Weight (kg)


chapter 10

Documents

set of data

categorical data

ungrouped data

data statistics

sets of data

numerical data

stopping distance

grouped data nd