12-1 Asking statistical questions
KEY IDEAS
Statistics is the study of the collection, organisation,
analysis, interpretation, and presentation of data. Data
is information and is often numeric. Statisticians are
data detectives. They carry out statistical investigations
using the statistical enquiry cycle.
Conclusion
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Problem
The five steps of the PPDAC statistical investigation
cycle and what they involve:
• P is for defining the Problem by asking a question.
• P is for Plan. This involves deciding what data is
needed, how to collect it and who to collect it from.
Plan
Analysis
• D is for Data. This is information which needs to be
collected and recorded.
Data
• A is for Analysing the data.
• C is for the Conclusion to answer the question.
The population is the group that is the focus of the statistical investigation.
A sample is a subset or smaller part of the population. It is important that the sample has similar
characteristics to the whole population.
A census involves collecting data from everyone in the population.
Investigative questions are divided into different types: summary, comparison, relationship, change
over time questions.
• A summary question focuses on a single variable; e.g. I wonder: What is the height of the tallest
student in the class?
• A comparison question compares a single variable between two categories; e.g. I wonder: Are
the boys taller than the girls in this class?
• A relationship question asks about a link or relationship between two variables; e.g. I wonder:
What is the relationship between heights and ages for students in the class?
• A change over time question asks about changes or trends in one variable over time;
e.g. I wonder: How has the amount of homework done by students in the class changed
during the year? I wonder: What are the trends in the amount of homework done by students in
the class?
Investigative questions should clearly identify the following three aspect:
• the variable to investigate; e.g. heights, reaction time
• the population; e.g. year 9 students in this class, all students in our school
• the whole group not just an individual; e.g. what are typical heights NOT what is Max’s height.
1
Statisticians have asked a range of questions that involve investigating the following datasets.
For each dataset decide whether the data is categorical or numerical.
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a
Favourite flavour of icecream
b
Height
c
Number of people whose favourite flavour of icecream is chocolate
d
Distance between home and school
e
Month of birth
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Chapter 12
Gender
g
Number of Toyotas belonging to students
h
Hours of TV watched
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f
Statistics
2
i
Favourite TV programme
j
Rating for canteen food
k
Year level (9, 10, 11, 12 or 13)
On the first day of term 2, students from Ms Tepuni’s class collected information from a sample of students from
all year levels at their school. They collected information about the following.
·
Age in years
·
Number of hours of TV watched last night
·
Gender – male/female
·
Number of times they had breakfast last week
·
Place of birth
·
Number of times they were on time to school last
week
·
Number of siblings
·
Like school: Yes/No
The students wrote five questions to investigate the information. Ms Tepuni said only one of the questions
identifies all three aspects needed in an investigative question. Below are the questions they wrote:
QUESTION 1: Where were students born?
QUESTION 2: Is there a link between having breakfast and the gender for the year 9 students in the sample?
QUESTION 3: How many hours of TV were watched altogether in the week?
QUESTION 4: How many students were on time to school?
QUESTION 5: How many students have siblings?
a
What are the three aspects an investigative question needs to identify clearly?
b
Which question identifies all three of these?
c
For each question decide if it is a summary question, a comparison question or a relationship question.
QUESTION 1
QUESTION 2
QUESTION 3
QUESTION 4
QUESTION 5
d
For the information collected by Ms Tepuni’s students, write another:
i
summary question
ii
comparison question
iii relationship question.
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Consider these investigative questions. For each question:
i
state the population or if not clear, define a suitable population
ii
decide if a sample or a census would be best and give a reason for your decision.
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3
iii state what sort of question it is.
a
Is there a relationship between the hours of homework and achievement for students in our class?
i
b
ii
iii
ii
iii
ii
iii
How many hours do students in our school spend playing sport in a week?
i
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iii
How many New Zealand families own their own home?
i
g
ii
Are year 12 students taller than year 11 students?
i
f
iii
What is the maximum number of texts sent in a week by a year 9 student at our school?
i
e
ii
Do our students usually bring lunch to school?
i
d
iii
How does the amount a family spends on food each week compare between New Zealand and Australia?
i
c
ii
ii
iii
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Chapter 12
Statistics
12-2 Data, data everywhere
KEY IDEAS
Data is another word for information. What data we need depends on the variable in our
investigative question. There are two types of data:
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• Categorical data: has values which are organised into separate groups; e.g. eye colour and gender. Also called qualitative data.
• Numerical data: has values that result from counting or measuring; e.g. height and armspan.
Also called quantitative data.
If data are already available, we need to search for reliable sources such as Statistics New Zealand,
local councils, newspapers and some internet sites.
If data are not already available, we need to collect and record using tallies, frequency tables and
databases. Data are collected by:
• taking measurements — this involves the use of instruments such as tape measures, scales,
speedometers, light sensors and voltmeters.
• carrying out observations — this involves watching, measuring, counting and recording data.
• using a questionnaire to ask questions — this can involve a written or online questionnaire or an
interview (either face to face or telephone).
Questionnaires must be designed so that:
• questions are clear and unambiguous
• questions allow for all possible answers
• data collected make sense
• the questionnaire is as short as possible
• only information that is needed is asked
• questions that some people may not want to answer are avoided
• answers are easy to analyse.
A statistical survey involves giving a questionnaire to a sample of people taken from a population.
1
For each of the following sets of data decide whether:
a
data is numerical (N) or categorical (C)
b
data should be collected by observation (O), by measurement (M) or by using a questionnaire (Q).
i
Favourite icecream flavours a
ii
Weights of school bags carried by year 9 students compared to year 11 students
a
b
b
iii Hand spans of year 9 students a
b
iv The number of cars who drive through a red traffic light
a
v
b
The number of students who eat lunch at school
a
b
vi The number of teachers who usually drive to school
a
b
vii Number of texts sent this week compared to the same week last year
a
b
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2
Natalila wrote the following survey to find out how students in her class travelled to school each day and how
long it took them.
Survey question 1: What is your name?
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Survey question 2: What is your age?
Survey question 3: How long does it take you to travel to school?
Survey question 4: How do you travel to school?
3
a
Which survey questions can be deleted?
b
What is wrong with survey questions 3 and 4?
c
Rewrite survey questions 3 and 4 to make it easier to analyse the data and answer the question.
Robert wants to find out from year 13 students in his class who has part-time work and how much they are
paid per hour.
Write at least two questions that Robert could ask.
4
A mobile phone company wants to find out about the monthly mobile phone use of high school students, to
help them design phone plans for teenagers. Design a questionnaire to find out:
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·
how many phones each student owns
·
how many text messages they send in a week
·
how many calls they make in a week
·
what added extras they like their phone to have.
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Chapter 12
Statistics
12-3 Reading graphs
KEY IDEAS
Statistical graphs (also called charts or plots) are used to display data.
To read a graph you describe what information the graph shows and what story it tells.
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When describing the information given on a graph, you need to consider:
• what the information is about by looking at its heading and keys
• what the axes and the scale tell you
• any trends or information shown by the graph.
Graphs can be misleading — you have to look carefully at their scales to interpret the information
correctly.
The graph shows the speeds of cars travelling along a main road in km/h.
a
How many cars travelled at 59 km/h?
b
State the total number of cars.
c
What speed would be the: i
Frequency
(number of cars)
1
ii
mode?
median?
iii mean?
Speeds of cars
10
8
6
4
2
0
55 56 57 58 59 60 61 62 63 64 65
Speed (km/h)
2
The following pie graph is from the New Zealand Statistics website.
Divorces by duration of marriage1
30 years and over
7.9%
Under 5 years
13.7%
25–29 years
8.9%
20–24 years
10.7%
5–9 years
25.1%
15–19 years
14.4%
10–14 years
19.3%
1 in complete years
a
Write a sentence explaining what the graph is showing.
b
Does the graph tell us how many divorces there were in 1999? Explain your answer.
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What length of marriage had the highest percentage of divorces?
d
A newspaper article stated ‘If you are married longer than 15 years you have a higher chance of getting a
divorce.’ Do you agree or disagree? Explain your answer.
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c
This bar graph shows the number of domestic and international passengers travelling between 1986 and 1995.
Domestic and international passengers
1400
1200
1000
800
600
400
200
0
6
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
Domestic
International
19
8
Number of passengers
3
Years
a
In what year was the total number of passengers greatest?
b
In what year was the total number of domestic passengers greatest?
c
In what year was the total number of international passengers greatest?
d
Estimate the number of domestic passengers travelling in:
i
e
ii 1994
1991
Estimate the number of international passengers travelling in:
ii 1994
1987
Number of students
i
9
8
7
6
5
4
3
2
1
0
Hours spent on sporting activities
1
2
3
4
5
6
7
8
9 10 11
Hours
4
The above graph shows the results from a survey of time spent on sporting activities during the Labour
weekend by some year 9 students.
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a
How many students answered the survey?
b
How many students spent less than one hour on sporting activities?
c
How many students spent more than 6 hours on sporting activities?
d
Can you tell from this graph who spent exactly 8 hours on sporting activities? Explain your answer.
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Chapter 12
Statistics
12-4 Graphing categorical data
KEY IDEAS
Categorical data can be displayed in many different types of graphs.
Pictograms, bar graphs, and dot plots all show the frequency of items in a group. The frequency of a
value is the number of times that value occurs.
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Pie charts show the proportions (or fractions) of the total in each category but they don’t help to see
the numbers of items involved.
1
The table shows the free-to-air TV channel watched most often by 140 people during a particular week.
a
What TV channel was the most popular?
b
What TV channel was the least popular?
c
Complete the table and calculate the angles in order to draw a pie chart.
TV channel
No. of students
Angle for pie chart
1
10
10
× 360° =
140
2
50
3
35
4
30
5
15
d
Draw the pie chart.
e
Draw a bar chart to represent the data.
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2
The following data shows the number of absences each student from a year 9 class had from Maths in term 1.
1, 3, 1, 2, 2, 3, 0, 2, 2, 2, 3, 7, 2, 1, 2, 3, 2, 0, 3, 1, 3, 2, 4, 6, 4, 13
Draw a dot plot to show the number of absences.
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a
b
3
Write at least one sentence to describe what the dot plot shows about the data. What do you notice?
Ms Steven’s maths class interviewed all the year 9 students at their school. They recorded the information they
collected in the frequency table below.
Question: How do you usually get to school?
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Travel to school
Frequency
Bus
56
Cycle
37
Car
24
Walk
18
Other
5
Total
140
a
Draw a suitable graph to represent the information.
b
Draw a suitable graph different from the type of graph you used in a to represent the information.
c
Write 1–2 sentences to describe the story told by the graph.
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Chapter 12
Statistics
12-5 Calculating statistics
KEY IDEAS
The word statistics is singular, as in ‘Statistics is a science and an art.’
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This is not the same meaning as the word statistic, referring to a quantity (such as mean or median)
calculated from a set of data, whose plural is statistics (e.g. ‘this statistic seems wrong’ or ‘these
statistics are misleading’).
We calculate statistics in order to answer statistical questions about a data set.
Different statistics tell us different things about the data set.
Statistics calculated from a set of data include the following:
• Mean is the average of the values. It is calculated by adding all the values and dividing this total
by the number of values. It involves every piece of the data and so can be affected by extreme
values.
• Mode is the value that occurs most often.
• Median is the middle number. If there are an odd number of data values the median is the middle
number. If there are an even number of data values the median is halfway between the two
middle numbers. The data needs to be in order for the median to be calculated. The median is not
affected by extreme values.
• Range is the difference between the highest value and the lowest value. It gives an indication on
how spread out the data is.
• Extreme values are data that stand out because they are much higher or lower than the rest of
the data.
Frequency tables list the data in order and show the number of times each value occurs. We can use
these to calculate most statistics.
1
The number of times that Jason visited the gym each month for the year is shown below.
12, 14, 11, 12, 15, 11, 12, 12, 10, 11, 10, 12
a
b
2
Calculate the mean, median, mode and range for the number of times Jason went to the gym.
mean
median
mode
range
For how many months did Jason visit the gym fewer than 11 times?
The heights (in cm) of players in a team of footballers are:
168, 175, 174, 168, 162, 168, 170, 180, 175, 168, 182, 174
3
a
What is the modal height?
b
What is the range?
c
What is the mean height of the footballers?
d
How tall must a new player be to increase the mean by 1 cm?
The maths quiz results for 10 students are listed below.
85, 75, 100, 65, 55, 90, 85, 78, 99, 100
a
Place the data in order from smallest to largest.
b
Find the mean test result.
c
Find the median test result.
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4
A businessman has two video shops. The sales in each shop during a one-hour period over 15 days were
monitored and recorded as follows.
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Shop A: 8, 27, 28, 29, 10, 22, 23, 26, 14, 9, 14, 24, 15, 12, 11
Shop B: 12, 17, 27, 13, 12, 14, 16, 4, 6, 16, 7, 7, 8, 9, 6
5
a
Calculate appropriate statistics to allow you to compare the sales for Shop A and Shop B.
b
Write at least two sentences comparing the sales from Shop A and Shop B.
What number would you add to the following numbers to get a mean of 6?
4, 6, 8, 10:
6
7
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a
Write five numbers with a range of 10.
b
Write three numbers that have a mean of 20 and a range of 20.
c
Write four numbers with a median of 25 and a range of 30.
Give an example of a set of data for which:
a
the median is two more than the mean
b
the median is $4000 more than the mean.
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Chapter 12
Statistics
12-6 Data distribution
KEY IDEAS
To understand a numerical data set we need to know about its distribution.
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• A data set is divided into quarters by the quartiles. The median is another name for the
middle quartile.
• The lower quartile (LQ) is the middle of the bottom half of the data.
• The upper quartile (UQ) is the middle of the top half of the data.
• The middle 50% of the data (often called the middle 50) lies between the upper and lower
quartiles.
Bar graphs, dot plots and hat plots all show the distribution of a data set.
• Bar charts and dot plots both show the frequency of each item but dot plots also show the exact
scores for each data value.
• A hat plot has a brim and a crown. The brim is a line from the smallest to the largest data values.
The crown extends between the upper and lower quartiles.
Lower quartile
Upper quartile
↑Minimum
Maximum↓
Brim
includes 25% of values
Crown
Brim
includes 50% of values includes 25% of values
Use I notice statements to describe distribution of data. Describe the shape of a data distribution by
commenting on:
• aymmetry – if the hat is in the middle of the brim, the distribution is symmetrical
• the overall spread – where the brim begins and ends
• the middle 50% – this is where the crown of the hat begins and ends
• any clusters – where data points are grouped or bunched
• any unusual features – any data points that stand out as extreme in any way.
1
For the following data sets find:
i
the median
ii
the lower quartile
iii the upper quartile.
Hint: remember the data sets need to be in order first.
a
4, 5, 6, 8, 9, 11, 13, 16, 16, 18, 20, 21, 25, 30, 31, 33, 36, 37, 40, 41
i
b
ii
iii
13, 18, 6, 20, 25, 11, 9, 18, 3, 30, 16, 9, 8, 23, 26, 17
i
d
iii
8, 11, 20, 10, 2, 17, 15, 5, 16, 15, 25, 6
i
c
ii
ii
iii
160, 154, 154, 150, 156, 154, 156, 154, 142
i
ii
iii
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These dot plots are of heights (measured to the nearest cm) for two different groups. For each group, find:
i
the median
ii
lower quartile
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iii upper quartile.
a
b
15
14
12
10
8
6
4
2
10
5
0
0
151 152 153 154 155 156
Height (cm)
3
151 152 153 154 155 156
Height (cm)
i
i
ii
ii
iii
iii
For each of the data sets in question 2:
i
draw the hat plot
ii
describe the data distribution in terms of symmetry, spread, any clustering and any extreme values.
a
b
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Chapter 12
4
Statistics
The armspans of a sample of year 9 students are shown on the dot plot and the hat plot below.
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PA T
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4
2
0
135
140
145
150
155
160
165
170
175
180
185
190
Armspan (cm)
a
How many students are in the sample?
b
What is the range of the data?
c
How many values are in the middle 50?
d
What two values does the middle 50 lie between?
e
Are there any unusual data points? Explain.
f
Comment on the symmetry of the distribution.
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12-7 Being a data detective
KEY IDEAS
Scientists, policy-makers, and the public need to be able to interpret complex information and
recognise both the benefits and pitfalls of statistical analysis.
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The five steps of the PPDAC cycle are useful for any statistics investigation.
Problem: Write an investigative question.
Plan: State what you are planning to do — what data will be used and what you will do with the data.
Data: Inspect the data. Is there ‘dirty data’?
Analysis: Display the data with appropriate graphs and calculate statistics. Describe the data
distribution: I notice … shape, middle 50, spread, unusual features.
Conclusion: Answer your question and reflect on the investigation.
‘Dirty’ data arises from human errors. Errors can occur with observations or measurements; e.g
data recorded incorrectly by putting the decimal point in the wrong place. Errors also occur with
questionnaires, such as people skipping questions, making mistakes or giving silly answers.
Dirty data needs to be cleaned/fixed, but how this is done depends on the data and the context.
Options include removing errors or correcting them.
1
Information was collected from a group of 200 year 9 students. The data are displayed in the dot plots on the
next page. Some statistics have also been calculated.
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a
Examine the data sets. For some of the data sets, both comparative and summary questions could be
asked. Which data sets are these?
b
For each data set, carry out a statistical investigation. In order to do this you need to:
·
Problem: Write an investigative question (either summary or comparison).
·
Plan: State what you are planning to do − what you will do with the data?
·
Data: Inspect the data. Is there ‘dirty data’?
·
Analysis: Display the data with appropriate graphs and calculate statistics.
·
Describe the data distribution: I notice … shape, middle 50, spread, unusual features.
·
Conclusion: Answer your question and reflect on the investigation.
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Chapter 12
Number of months 200 year 9 students have owned their current mobile phone.
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i
Statistics
0
10
20
30
40
50
60
Mobile phone (months)
Mobile phone (months)
Mean
9.95
Minimum
1
LQ
4
Median
6
UQ
13
Maximum
52
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Right foot length for boys and girls
boy
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ii
20
15
20
25
30
35
40
25
30
35
40
girl
gender
15
Right foot (cm)
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Right foot (cm) boys
Right foot (cm) girls
Mean
25.6
23.4
Count
92
95
Minimum
19
15
LQ
24
23
Median
25
23
UQ
27
24
Maximum
36
28
Mathematics & Statistics for the New Zealand Curriculum Workbook: Year 9
Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400
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Chapter 12
Statistics
U
N
SA C
O
M R
PL R
E EC
PA T
E
G D
ES
iii Neck circumference for 200 year 9 students
24
26
28
30
32
34
36
38
40
42
44
Neck (cm)
Neck (cm)
Mean
32.2
Minimum
25
LQ
30
Median
32
UQ
34
Maximum
43
Chapter 12
Statistics
139
Uncorrected third sample pages • Cambridge University Press © Fagan, Goodey & Lawrence 2014 • ISBN 978-1-107-65357-3 • Ph 03 8671 1400
9781107653573txt_03pp.indd 139
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