Developing statistical
thinking capabilities for
postgraduates across
disciplines
Helen MacGillivray
Vice-president, International Statistical Institute
Immediate past-president, IASE (International Association for Statistics Education)
Senior Learning and Teaching Fellow
Past president & honorary life member, Statistical Society of Australia Inc
Outline of presentation
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Symposia in Statistical Thinking for Postgraduates across
disciplines
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Why, when, how
What
 From participants
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Statistical thinking and data investigations
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Let’s plan a data investigation!
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From statistical consultants and statistical educators
How research questions become statistical investigations
Key statistical thinking and hands-on planning
Key points from symposia on statistical analyses
What about secondary data?
Results from pre-symposia questionnaire
Implications for ug
How can universities develop/support statistics for pg’s
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Models and ideas: feedback
QUT Maths Access Centre established 2004
Aim: to develop and provide university-wide learning
support for students in any area for which numeracy,
mathematical and/or statistical confidence are needed.
The programmes gradually developed to include:
 drop-in facilities (3 campuses), with resources & schedule of duty tutors.
 student-driven support sessions for a range of courses or topics.
 provision & development of general and course-specific resources and
questionnaires for student self-diagnostic support.
 areas supported include engineering, chemistry, nursing, education, human
movement, psychology, maths, stats
 an apprenticeship-model tutor development program with mentoring and
training for talented undergraduates
 collaboration with academic & learning support programs.
 website and Facebook.
 research: data collected and analysed
 links with UK; ALTC leadership project 2006-2007
In 2013 became part of STIMulate – Science, IT, Maths & Stats
Like other maths and stats learning support centres,
QUTMAC oriented to helping at undergraduate
level, esp. first years & anyone with lack of
confidence in background or foundation skills
But
In agreeing to providing some funds (very modest),
DVC recommended “doing something” for
postgraduates
!!
In 2005, Manager of Research Students Centre
approached me asking if I could help in statistics
for commencing postgraduates.
Me, very warily:
* pg’s tend to say they want individual help, &
* people often think pg’s need workshops in specific
advanced methods,
but, before any of that, my experience is that
* they need to understand role of statistics in research, &
* they need sufficient understanding of core statistical data
analysis methods to have some confidence for their
research & possibly discipline-specific or more
advanced methods
Manager: exactly!
(me, thinks: wonderful to have someone who understands!)
So me & the manager plotted & planned a trial to see if
“Statistical Thinking for Postgraduates Across Disciplines”
would fly.
Trial & evaluation, 2005
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Offered trial initial session, described by
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Session Objectives: Learning about
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Your statistical thinking
Planning a statistics-friendly data investigation.
 Statistical questions and research questions.
 Choosing, using and interpreting statistical graphs and procedures
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The session will refer to real data investigations in contexts that do
not depend on discipline-specific knowledge.
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A pre-session “diagnostic” questionnaire sent to registrants to
help them (& me!)
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included stats literacy questions from PhD work by Therese Wilson
(now Maths and Stats Coordinator STIMulate)
 main emphasis on questions on core stats methods oriented to
misconceptions & even mistakes often seen in pg/research in other
disciplines
 see later for details & some interesting results
Trial & evaluation, 2005
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74 registered, from all faculties, 51 attended
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Included staff, particularly from Health
Focus on
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Planning an investigation
Importance of identifying variables & their types, and
subjects (“design your spreadsheet for your raw
data”)
Turning research questions into statistical questions
General principles of estimation & error of estimation,
prediction, modelling, fitting, testing, checking
assumptions
Choice of graphs
What is needed to choose analysis method(s)
Trial & evaluation, 2005
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Evaluation included asking what they’d like for
next session & in future
38 evaluations (across faculties) with lots of
comments
66% agreed or strongly agreed it addressed
their information needs
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Surprisingly large given it was just an introduction
85-90% agreed or strongly agreed that

Objectives were clear
 Content was relevant
 Well structured & presented
 They learned something useful to their pg research
Trial & evaluation, 2005
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Comments generally indicated that they
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wanted more
ranged from having very little background (“What’s a ttest?” “What’s ANOVA?”) to having one ug stats course
wanted basics: “chisq, ANOVA, regression, confounders,”
“what to use when & where” “when to use different graphs”
General comments such as:
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“a pleasure to attend, very informative & delivered in a very
friendly, comfortable, clear manner”
“basic info made easy to understand – should teach health
stats”
Conclusions: yes, worth doing, and yes, they did want to understand
basics from a research point of view. And they wanted it delivered in clear,
simple & friendly manner
Follow-up session: trial & evaluation, 2005
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Session Objectives
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Focus on using key techniques of chisquare, ANOVA, and
multiple regression in detecting what is happening in a dataset,
including understanding of statistical hypothesis testing and its
value for research, and awareness of possible problems and
how to use diagnostics to check for problems.
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Techniques demonstrated and discussed through real datasets
with interesting statistical features that are explored within
context, emphasizing the importance of understanding the data
context.
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Follow-up to the first one, focussing on topics most commonly
requested by the participants.
26 attended: almost unanimous approval - “very useful” “wonderful” “very
good pace”; strong support for planned series of 4 symposia
Symposia in Statistical Thinking for Postgraduates
2006 
Series of 4 symposia, 2.5 hours each (incl coffee break!) given
twice a year.

Timetable, registrations & evaluations done by Research
Students Centre
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40-50 registrations each time; approx 65%-70% attendance

Types of comments & approval ratings generally consistent
over the years (approx 90% approval)
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Some staff in first couple of years; now mostly pg students in
their first semester/year, masters as well as PhD. Across
faculties
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Pre-symposia questionnaire retained with slight adaptations
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Associated SPSS workshops 2007-
Symposia in Statistical Thinking for Postgraduates
Aim: assist pg students with statistical understanding essential to
planning, carrying out and analysing data investigations using
core statistical tools.
 For those with some statistical background
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For those with little statistical background
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help to see what can be achieved with core statistical tools.
For others
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help consolidate and improve confidence in using core statistical tools in
real & sometimes challenging applications, & provide indicators of further
statistical tools you may wish to learn.
help identify and fill in gaps in your statistical knowledge and
understanding.
Series modularised - optional questionnaire organised under
the headings of the four symposia.
Real datasets in contexts that do not require discipline-specific
knowledge, but sufficiently complex to allow demonstration of
choice, use, interpretation and synthesis of statistical tools and
thinking.
Computer output is used throughout the symposia but the
symposia do not include teaching the use of a particular
statistical package (See SPSS workshops).
Symposia in Statistical Thinking for Postgraduates
Titles & key points
1. Planning investigations, handling & exploring data
 Research question(s)
planning
pilot
 Identify subjects, variables, types
planning
2. Categorical data
 Estimating proportions; error of estimation; general concepts
of confidence intervals
 Statistical hypothesis testing. Turning research questions into
statistical questions
 1 & 2 categorical variables & chisq test
 More than two?
3. Continuous response, categorical predictors
 Confidence interval for mean vs. prediction/tolerance interval
for individual values
 ANOVA, experimental design, interactions, residuals,
assumptions, diagnostics, transformations
 More?
Symposia in Statistical Thinking for Postgraduates
Titles & key points
4. Continuous response, quantitative predictors.
Regression (multiple)
 Models, interpreting output, residual plots & diagnostics
 Changing model
 Dependence & interaction in regression
 R-sq & cautions
 Categorical predictors?
 General linear model
 More?
Participants ask questions throughout. Referring to their own
context within symposia facilitates statistical understanding for
all.
Any extra discussions/points depend on group & questions.
Developments & SPSS w’shops
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Questions and opportunities to find out areas of participants
help to orient the important ad hoc examples
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Aspects known to be needed from start require even more
emphasis and attention - but benefits are great
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Variables & their types, subjects
Scientific hypotheses vs. statistical hypotheses
Scientific experiments vs. statistical investigations
Essential roles of graphs in exploration & diagnostics
Dangers of one-explanatory-variable-at-a-time
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Worst problem from standard/conventional introductory courses
Ban ‘two-sample t’ in research!
Beware of correlations!
Coping with discipline requirements/habits without compromising
statistics
Three SPSS hands-on workshops linked with & immediately
following symposia topics 2, 3 & 4
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Limited to 30 registrants
< 20 participants
Some participant comments
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Fantastic! I arrived being “scared of numbers” and left the
session feeling empowered.
It may sound crazy but I really “enjoyed” this learning
experience. Thanks.
…you motivated me to want to understand
I’m starting to understand.
Really oriented to our needs
I did not understand it (regression) but I realize now what I need
to know.
Really good to refresh knowledge of statistics & learn how to
apply to our research.
Found ANOVA interesting.
Clear, hit points.
I really enjoyed it. It is helping me to understand stats well.
I become clearer about the purpose of using statistics for my
research.
Some participant comments
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…..a joy to be there
….real-life examples of the data and the interpretation of the
results was simply great
It was fantastic. ………. explained this heavy subject with
appropriate examples. Now I am confident to use the
techniques I had learnt in analyzing my data
It is becoming more and more interesting. I would definitely use
the knowledge in the evaluation phase of my experiments.
Excellent resources. Very useful for research students.
I found that it was a very good and succinct summary of the
approach needed for different data sets and in different types
of analysis…. …. being a science student ……. I am very happy
I attended all sessions as I found them to be a very helpful
revision of where and how different tests should be used and
has given me enormous confidence in reading published work
and also considering stats while designing my experiments.
Only a few negative comments: Pre-2007, wanted SPSS workshops
Others from extremes: “too fast” “more time to digest” “more depth please”
From consultant statisticians

Barnett
(1976) describes statisticians as “translators and
communicators”
 (1986) “we see, tied up together, the role of the
statistician as consultant, consultancy as the stimulus
for research in statistics, and consultancy as the basis
for teaching statistics”.
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From (extensive) literature on statistical consulting,
comes emphasis on ‘greater statistics’ and the whole data
investigation process
From consultant statisticians
Cameron (2009) considers
 desirable key components of university-based training
of a statistician to prepare for a research consulting
career of “entwined collaboration” and “serial
collaboration”
 consults what many “wide and experienced”
statisticians have written (Box, 1976, Chambers, 1993)
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identifies
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formulating a problem so that it can be tackled statistically
preparing data (including planning, collecting, organising and
validating)
analysing data
presenting information from data
researching the interplay of observation, experiment and
theory.
From consultant statisticians
Kenett & Thyregod (2005)
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describe the 5 steps in statistical consulting
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problem elicitation
data collection and/or aggregation
data analysis using statistical methods
formulation of findings & consequences
presentation of findings and conclusions/recommendations.
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“important to take part in collection of data, or at least have
the opportunity to watch data being collected or generated.”
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“Our long-term objective is to encourage academic courses
to cover the full 1–5 cycle.”
From statistical educators
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Statistical education advances advocating such
approaches for all disciplines started across the world
over the past two decades
Both statistical consultants and statistical educators
also champion
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Garfield (1995) comments that
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authentic experiential learning as vital
constructivist approaches.
students will understand material only after they have
constructed their own meaning for what they are learning.
I add to that:
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Students understand the whole data investigation process
only when they have lived it. This applies to all disciplines.
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Students understand probabilistic thinking and its modeling
only when they construct meaning that links with everyday
contexts and with data, and builds on previous experiences.
Data investigations: school & tertiary
RSS Centre for
Statistical Education
problem-solving cycle
(from 1970’s)
Plan
Collect
Process
Discuss
Authentic learning of data investigations: tertiary
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My courses,1994-2011: semester-long free-choice full
data investigation embedded in introductory statistics
courses in all engineering, science and mainstream
mathematics and statistics programs
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Student-centred “inquiry” (Anderson and Sungur, 2010)
“Inquiry oriented learning” IOL in Science (Kirkup et al, 2010)
Authentic student-centred discovery of K&T’s Steps 1-5
Research questions
statistical questions
Early software use helps students discover power of statistics
in many-variabled & messy real world situations
Thousands of projects, datasets and examples
“Excellent foundation for statistical research” (PhD student)
embody everyday experiences, constructivist principles and
“emotionally & cognitively supportive” problem-solving
environment Gal et al (1997)
Extract from Symposium 1
Planning and design of data investigations involve
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identifying the topics of interest
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identifying the variables to be
observed
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identifying the subjects of the study
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planning the practicalities
Planning and design of data investigations
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The planning of collection of data can
include
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design of experiments or studies in a lab situation
design of field experiments or studies – outside a lab
situation
design of an observational study
design of surveys
In all quantitative problem-solving,
identification of variables is important
To plan your own data investigation or
understand others, and to choose and use
statistical techniques, identification of
variables is essential
Just a few examples of practical challenges
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What do we want to find out about?
What can we find out about?
What can we observe?
What can we measure?
Can we measure what we want. Is there
anything we should observe/record ….. just
in case?
Should we do a pilot study or preliminary
experiment?
Beware the traps of the rigid version of the
scientific method!
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Might start with a single research question, but must
turn into investigation
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Many/most discoveries are made by observation
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of something unexpected
of something during investigating something else
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and/or thinking outside the original box
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Don’t be a slave to identifying & investigating a single
idea/thought/hypothesis – investigate objectively
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Research hypothesis vs statistical hypothesis
Example of research hypothesis “There are little green men out there!”
 Statistical hypothesis “There are no little green men out there”
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Qualitative & quantitative - Not versus
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Not alternatives
Form the continuum of investigation
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Consider the continuum of interviews, questionnaires,
surveys
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In qualitative, often searching for what to measure &
how to measure … how to describe, how to classify,
what is different and what is similar
In quantitative, must know and understand measures
and variables, and how procedures depend on this
knowledge
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Some general points
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Don’t pre-categorise variables such as time, age,
monetary amounts, etc, without good reason and
thought
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Identifying variables ALWAYS helps in planning
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identifying subjects of the study
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identifies what the rows of your spreadsheet will correspond
to
can also help in deciding what you are interested in
Report how, when, where and possible problems of
the data collection
Keep raw data.
Importance of randomness
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In an experiment
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In a survey
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Design, controls and random allocation within design
Randomness essential to investigate causation
Question/instruction design AND random selection
(possibly within sampling frame for more sophisticated
designs) AND obtaining responses
Randomness essential to infer to population
In an observational study

Of what can we consider our data to be randomly
representative with respect to questions of interest?
Random representativeness of our data
with respect to questions of interest
Data entry - much more to it than one may think
– and value in the thinking about it
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Each variable has a column
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Each “observational/experimental”
unit/item/subject has a row.
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Identifying “experimental unit” = each row in
spreadsheet corresponds to …...?
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Enter in order of observation - in case you
want to check for any trends or changes in
conditions.
Some examples of data collected by tertiary
students
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Egg crush test (A. Warr and S.Yu, 2001)

The students designed an experiment to test the
strength of egg shells of various types and brands.
Research hypothesis: brown eggs are stronger than
white
Their home-made apparatus held the egg steady on its
side under a container that they gradually filled with
water until the shell first cracked. Some eggs could not
be used. They obtained 202 observations
Variables:
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Response variable = crush load (=weight of water when egg
cracked
Other variables: colour, brand, housing, egg length, egg width
Egg crush apparatus
Egg crush procedure
Egg crush after the experiment
Some examples of data collected by tertiary
students
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Bike track (Ford, Hungerford, Johnson and Swain)
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Students wanted to investigate use of a bike track.
Motivation: speeds of cyclists vs pedestrian usage
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They carried out their study on 1 day from 8am-6pm.
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They recorded time of day, speed, type of travel, direction, and
gender.
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Time of day was recorded in 15min intervals due to “collecting
pressure”. They used one collection form for each 15mins. After
exploring the data, they further grouped the times into am peak, am
off-peak, lunch, pm off-peak and pm peak.
Types of data - types of variables
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Considering types of data is also
considering types of variables
In all quantitative problem-solving
identification of variables is important.
In statistics, identification of the TYPES
of variables helps in choosing appropriate
methods of exploring and analysing data.
Variables are of two broad types: discrete
and continuous.
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Discrete variables take individual or separated
values.
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Discrete variables are further divided into
categorical and count.
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Categorical data fall into categories which can be named or
coded. Codes have no meaning in themselves - just labels.
Count data record a number of items, events, people etc.
Some categorical variables have a natural ordering of their
categories. The order of the codes matter but not their
actual values e.g. 1=strongly disagree, 2=disagree, ….
5=strongly agree. Called ordinal variables.
Continuous variables
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Usually have units
Examples include time, height, weight, distance, age,
amounts of money ...
Money? Consider exchange rates, hourly wage
‘Height is 1.5m’ = height is between 1.45m and
1.50m. ‘Height is 1.52m’ = height is between 1.515m
and 1.525m.
Continuous variables take values in intervals –
typically continuous data have values given “to the
nearest….”
Variables can be
Discrete
Categorical
Nominal
Continuous
Count
Ordinal
Other words that are sometimes used for data are
“qualitative” and “quantitative”:
categorical are qualitative data
continuous and count are quantitative data.
Graphs and tables help us organise,
explore, highlight features and present data
Identifying types of variables helps us in
choosing types of plots and graphs
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Categorical: tables, barcharts, (piecharts)
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Piecharts can do only single categorical variables
Continuous: dotplots, histograms, stem-andleaf plots, boxplots, scatterplots, time series
Count:
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often very small (tables, barcharts) or
large (dotplots, histograms, stem-and-leaf, boxplots)
Beware: incorrect 3D graphs; barcharts that
start above 0; scatterplots that start at 0
Let’s design a data investigation!
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What’s of interest
Identify variables
Identify subjects of study (observations are
per what?)
Practicalities – how, when, where, who.
Have we forgotten anything?
Anything else we should record?
What will our spreadsheet look like?
What do we need to check in our pilot?
Random representativeness of our data?
Let’s design a data investigation
Groups choose a topic & what to investigate (at least 4
variables), identify variables and subjects, outline data
collection, spreadsheet, conduct preliminary pilot, report.
1. Stretchiness of snakes (confectionery
snakes!)
OR
2. “When people clasp their hands, they tend to
have left thumb on top”
OR
3. How often do people blink?
“Various studies have shown the average person
blinks at a rate of approximately 20 to 25 times per
minute.”
From symposium 2: Categorical data
Proportions are everywhere!
Estimating proportions: conditional proportions
One categorical variable – testing a set of %’s
Testing statistical hypotheses – understanding the
principles and p-values; interpreting p-values
Two categorical variables – testing independence
Confidence intervals for proportions
Confidence intervals as estimates of parameters
More than two categorical variables?
Estimated proportions are everywhere!
“Rich keenest on hand-out for insulation”
In a story by Eloise Gibson in the New Zealand Herald on
15th September, 2009,
(http://www.nzherald.co.nz/nz/news/article.cfm?c_id=1&o
bjectid=10597307 ) reported the results of an online
survey conducted by the Business Council for
Sustainable Development of 1578 homeowners on floor
and ceiling insulation subsidies under which home
owners must pay up to $3500 themselves.
The newspaper report includes the information that
“Homeowners making more than $200,000 a year made
up only 1 per cent of respondents, but 45 per cent of
them planned to get help with home insulation costs, as
did 35 per cent of those making $70,000 to $100,000 and
29 per cent of those on $100,000 and $150,000.”
One categorical variable - testing a set of %’s
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A City Council brochure states that on a typical summer
weekend, users of a pedestrian bridge fall into the following age
groups.
Age group
0-10
11-20
21-30
31-40
41-50
51-60
61-70
70+
Percentage of users
5
20
40
15
8
5
5
2
One typical summer weekend, 100 people were observed
crossing the bridge. Which of the data sets given below would
cause you to question the information in the brochure?
Age group
% of users
Age group
% of users
Age group
% of users
0-10
5
0-10
10
0-10
7
11-20
13
11-20
20
11-20
19
21-30
32
21-30
45
21-30
43
31-40
14
31-40
10
31-40
13
41-50
13
41-50
5
41-50
10
51-60
12
51-60
5
51-60
3
61-70
11
61-70
3
61-70
4
71+
0
71+
2
71+
1
Two categorical variables
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A study considered the aspect that cocaine addicts need
the drug to feel pleasure. Perhaps giving them a medicine
that fights depression will help them stay off cocaine. A
three-year study compared an antidepressant called
desipramine with lithium (a standard treatment for cocaine
addiction) and a placebo. The subjects were 95 chronic
users of cocaine who wanted to break their drug habit. Each
treatment had subjects randomly assigned to it. The results
are below. Is there evidence of any treatment effect?
Relapse
No relapse
Desipramine
13
18
Lithium
22
10
Placebo
24
8
Two categorical variables
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Lift or stairs?

In 2007, a group of engineering students decided to observe the
busway stop at the Brisbane Cultural Centre to investigate the use
of stairs or lift to provide information on design issues for public
transport stops. They recorded data on users in three time periods
– morning peak hour, afternoon off-peak, and evening peak hour.
For each person using the bus station in these times, they
recorded a number of variables, including gender, direction of
travel (up or down), and whether stairs or lift was used.

The table shows the observed frequencies for choices of lift or
stairs when going up at the bus station in the three different times
of day. Do the data provide evidence that the choice for those
going up depends on the time of day?
Choice
Time
Lift
Stairs
Total
Morning_peak
222
210
432
Off-peak
232
130
362
Evening_peak
300
213
513
Total
754
553
1307
Testing hypotheses: research vs null

No matter what the research hypothesis, the
statistical null hypothesis is that ‘nothing’s going
on’, ‘there’s no effect’.

The ‘null’ reduces the unknowns or complexity

The research hypothesis is often the alternative
hypothesis.

So we are trying to see if there is sufficient
evidence to say that

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
There is something going on
We need to go to a more complicated model
There is evidence of the research hypothesis
Testing statistical hypotheses: p-value

Need a test statistic to summarise data or the
‘difference’ between data and model
We find the chance of obtaining our data (strictly
speaking the value of the test statistic) IF there’s
‘nothing going on’ – that is, assuming the null.

This probability is called the p-value of the test
for our data.

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A small p-value says that we had little chance of
getting our data if the assumption is true

So a small p-value says our data do not tend to
support the assumption.
Testing statistical hypotheses: p-value

A not-small p-value says there was a reasonable
chance of getting our data if null true, so insufficient
evidence against null, for alternative.

Sliding scale of strength of evidence
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p-value < 0.01 (or 1% if you prefer): very strong evidence
against the null hypothesis
p-value between approx 0.01 and 0.03: strong or reasonably
strong evidence against the null
p-value between approx 0.04 and 0.06: evidence (or
reasonable evidence) against null
p-value between approx 0.06 and 0.1: slight or some evidence
against null
p-value > 0.1: not sufficient evidence against the null
Just think where you’d be prepared to stand up and say
there’s evidence against the null
“Rich keenest on hand-out for insulation”
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The only proportion for which we can obtain a
confidence interval in this example, is the
proportion of homeowners making more than
$200,000 a year who plan to get help with home
insulation. For these homeowners, the data we
have is n = 15.78 (1% of 1578) which we’ll take as
16, and X = 7 (45% of 1% of 1578). Based on
these data, a 95% confidence interval is:
(19.4%, 68%)
From symposium 3: Analysis of Variance:
continuous response, categorical predictors
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Confidence interval for mean and
tolerance/prediction interval for individual values
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Continuous response and 1 categorical predictor:
1-way ANOVA (2 sample t), multiple comparisons
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Continuous response and categorical predictors:
ANOVA, interaction, residual plots, assumptions,
diagnostics, modelling, interpreting. R-sq.
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Experimental design
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Where to go for more
The experiments/investigations
Car prices
Car prices were collected from
www.carpoint.com.au on one day for one
model from one manufacturer. Amongst other
variables, the year of manufacture (2001-2005)
and whether the advertiser was a dealer or
private, were noted. Also state of sale and
colour of car.
Three possible predictors.
The experiments/investigations
Soluble aspirin
To investigate the time to dissolve different types of
soluble aspirin tablets, an experiment was conducted
with 5 brands, using two water temperatures
(room/fridge), two pH’s of water (neutral/acidic), and
two water types (normal/mineral). Aspirin was
classified as dissolved once it had broken up and
dissociated from the surface of the water.
Four experimental factors.
Research questions: are brands different? does water
type, temp or pH matter?
Egg crush
Three categorical and two continuous predictors
Research hypothesis: brown are stronger than white
The experiments/investigations
Burn Rate of Candles
Party candles are long and thin and available in four colours. The
interest was in whether time to burn is affected by colour and
angle. To hold the candles, two rows of holes were drilled in a
block of wood. The first row was vertical to the surface of the
block, whereas the second row was drilled at a 60 angle.
Four packets were tested. There are 24 candles in a packet - 6 each of
pink, yellow, white and blue. A stopwatch was started as the first
candle was lit, and then they were lit at intervals of 10 seconds.
They were then put out with a wet sponge after they had each
burned for 5 minutes.
The same person lit all of the candles with the same technique and
same lighter (gas). The experiment was done all on one day, to
avoid any climatic effects.
The length and weight before burning were measured, and the length
after burning. That is, the rate of burning was measured by amount
of burning that happened in 5 mins.
Length before & after, weight before, colour, packet, angle
Research question: do colours burn differently to each other?
Key points in analysing
1. Never comment before checking residuals!
2. Interaction
Example: Car Prices
Interaction between seller and year looks at whether the
difference in average prices between dealers and
private sellers changes over the years.
Must have replication to be able to include interaction.
Choose a context to explain interaction.
3. Multiple comparisons are used to look at which means
are different e.g. which years are different in car prices.
Note: Big data can have a lot of variables or a lot of
observations or both.
Key points: 1 predictor at a time mistakes
Dissolving aspirin
1 predictor at a time:
only some evidence that temp matters (strange?)
no evidence that pH matters (what?)
strong evidence that water type matters
strong evidence that brands are different
When all predictors used together & interaction included:
strong evidence that each factor matters, averaged over
the others;
evidence that different brands react differently to pH
and to type of water, but not to temperature
Can you see a possible problem in this experiment?
Key points: 1 predictor at a time mistakes
Egg crush experiment
1 predictor at a time:
strong evidence that colour matters
strong evidence that housing matters
strong evidence that brands are different
When all predictors used together & interaction included:
strong evidence that brand and housing matter, but no
evidence that colour matters, averaged over others.
And no evidence that difference between caged and
free range depends on brand.
Can you think why? How could you investigate?
Key points:
Understanding confidence intervals for means
What is the most common mistake in interpreting
confidence intervals for means?
Example: the pH’s of 20 rainfalls in an area gave:
* a 95% confidence interval for mean pH of (5, 5.6);
* a 95% prediction interval for 90% of the rainfalls of
(3.82, 6.78)
Confounding and experimental design
In the candle burning experiment, the students used 3
packets of candles for the vertical ones and 1 packet
for the ones at 60 degrees.
What is the problem?
From symposium 4:
continuous response, quantitative predictors;
then general linear model
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Fitting model, interpreting output, residual plots
and diagnostics
Effects of a variable after allowing for/after
fitting/in presence of others
Changing model: allowing for curvature
Dependence and interaction in multiple regression
“Unusual” observations
R-square and caution with correlation
Can we include categorical predictors? Indicators
Transforming data
General linear models
Examples
Nutritional content of breakfast cereals
There are many different breakfast cereals with
different purposes. Measures that are commonly
provided per 100 gms serve include energy (in
KJ, standing for KiloJoules), protein (in gms),
carbohydrate (in gms), dietary fibre (in gms),
amount of iron (in mgs). How are they related?
And how closely are they related?
Examples
What affects textbook prices?
Data were collected on all textbooks from a university
bookshop. Course notes were not included nor were
general reference books. Where multiple editions of a
book were available, only the most recent was included.
Each book was classified according to whether the cover
was hard or soft, if it came with a CD and whether it had
colour or not. Each book was weighed, its thickness
measured, and its price and year of publication noted.
The main response variable in this dataset is price. There
are three categorical variables, each with two categories
(cover type, CD, and colour), two other continuous
variables (weight, thickness) and the variable year of
publication which could be used as a quantitative or a
categorical variable.
Examples: general linear model; “Analysis of
covariance”
Egg crush
Burn time of candles
Secondary data
Example: Plasma Study investigating the determinants of plasma
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concentrations. 315 observations on the following 14 variables
Age (years)
Sex
Smoking status. Never, Former or Current Smoker
Quetelet (weight/(height2))
Vitamin Use. Yes, fairly often, Yes, not often or No
Calories consumed per day
Grams of fat consumed per day
Grams of fibre consumed per day
Number of alcoholic drinks consumed per week
Cholesterol consumed (mg/day)
Dietary beta-carotene consumed (mcg/day)
Dietary retinol consumed (mcg/day)
Plasma beta-carotene (ng/mL)
Plasma Retinol (ng/mL)
Secondary data
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Example: Very low birth weight infants Study on infants with birth
weights less than 1600 grams collected at Duke University Medical
Center. Collected from 1981-1987, including 671 instances of very low
birth weights. Information collected on 27 variables
Variables:
Date of birth (admission) plus a fraction of the year
Date of death or discharge plus a fraction of the year
Hospital stay (days)
lowest pH in first 4 days of life
Platelet count
Race
Birth weight (grams)
Gestational age (weeks)
born at Duke or transported
If multiple gestation observed
Duration of labour (hours)
If mother treated with MgSO4
Secondary data
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Example: Very low birth weight infants (cont)
Variables:
If mother treated with beta-methasone
If mother treated with beta-adrenergic drug
Delivery abdominal or vaginal
Appearance, Pulse, Grimace, Activity, Respiration (APGAR) at one minute
If assisted ventilation used
If pneumothorax occurred
If patent ductus arteriosus detected
If on supplied oxygen at 30 days
Periventricular hemorrhage, absent, possible or definite
Intraventricular hemorrhage, absent, possible or definite
intraparenchymal echodensity, absent, possible or definite
Year of birth plus a fraction of the year, eg. 84:591 is August 3, 1984
Sex
If patient alive or dead
Some pre-symposia questionnaire results
(DK = don’t know or no response)
 9 stat lit questions:
scores from 2 to 9, median 6
 2D vs 3D graphs:
34% correct, 20% DK
 Identifying continuous variables:
9% correct, 11% DK
 Identifying categorical variables:
20% correct, 9% DK
 Identifying subjects:
23% correct, 11% DK
 Interpreting boxplots (out of 7):
scores from 0 to 7, median 5
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Incorrect coding of N/A:
60% correct, 14% DK
Estimating conditional probabilities (2 questions):
80% correct
71% correct, 6% DK
Some pre-symposia questionnaire results
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Situtation for (chisq) test of independence
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Identify which test needed:
State null:
Interpret p-value (in general terms):
23% correct, 43% DK
20% correct, 31% DK
63% correct, 31% DK
Situtation for testing mean:
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Identify null:
Interpret p-value (in general terms):
26% correct, 66% DK
57% correct, 34% DK
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Identify problem with multiple t-tests:
6% correct, 51% DK
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Compare multiple t-tests with multiple predictor
ANOVA:
3% correct, 54% DK
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Problem of doing 1-way ANOVA on carefully
designed randomised block experiment:
6% correct, 71% DK
Some pre-symposia questionnaire results
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Identify mistake in interpreting confidence interval
for mean as prediction interval for individual values
(quality control context):
6% correct, 66% DK
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Identify problem in judging regression model just on
p-value & R-sq:
0% correct, 69% DK
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Identify problem in using categorical predictor
(>2 categories) in regression:
23% correct, 66% DK
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Identify problem in using ordinal variable as
response in regression:
17% correct, 71% DK
Some implications for ug curricula
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Service courses in statistics should go to many
variables ASAP & avoid focus on 2-sample & simple
linear regression
Experiential learning
Whole data investigation process
Investigations/projects should avoid design-forprocedure
Focus on essential concepts of statistical inference
& data modelling
Contexts must not dominate statistical learning
Contexts must be familiar or readily accessible to
students
Must use technology as used in practice of statistics
Real data and no toy datasets
Staff research interests must be controlled
Beware teacher-centred or too-complex case studies
How can universities develop/support
statistics for pg’s?
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Courses for credit plus rationed consultations?
Statistical consultants with faculty ‘buy-in’?
Online plus rationed consultation?
Consider a tree analogy for combining development
of graduate capabilities and support for pg work
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Curriculum of symposia described here is common ‘trunk’
of tree
‘Branches’ are special topics
‘Leaves’ are (rationed) individual consultations with student
and supervisor
Thank you and here’s to statistics!