PY1PR1 lecture 1: Describing data
Dr David Field
General Information
• The Research Methods course consists of statistics
lectures, workshop exercises, and laboratory practicals
• Bring calculators to workshops, not mobile phone!
• You should have two handouts for this lecture containing
• Handout 1
– The schedule for Autumn term Psychological Research Methods
PY1PR1
– Details of Assessment for this module
• Handout 2
– Lecture handout – “Describing data”
General Information
• PowerPoint presentations for this lecture series will be
available to download from my web page
– http://www.personal.rdg.ac.uk/~sxs02dtf/home.html
• and also on BlackBoard
• There is additional information in the “notes” sections at
the bottom of the slides that you won’t see projected on the
screen today
• So, no need to write everything down
• Today’s slides contains some questions that you should try
to answer at home using the course textbook
– Discovering Statistics using SPSS. 3rd Edition. Andy Field
• The questions are repeated at the end of your printed
handout
Using the course textbook
• The material in today’s lecture is covered in Andy Field
chapters 1 and 2
• My slides and handouts will indicate which specific
sections of the textbook you need to read
– e.g. calculating the mean is covered in section 1.7.2
• But reading whole chapters is a good idea
• A guide to the meanings of symbols and Greek letters
is given on page XXXI, just before before Chapter 1
• Occasionally I will point out to you an issue where my
teaching will diverge from the book
– There is subjectivity in statistics
– For purposes of this course, my procedure should be followed
• If you have studied ‘A’ Level Psychology then you will
be familiar with some of today’s topics
– But you might find that things are covered in more depth here
• If you have not studied ‘A’ level, you might be
wondering “Why on earth am I taking a course in
statistics?”
Help with statistics
• University of Reading Maths Support Centre
• Located on the first floor of the Main Library
• Specialist statistics tutor available every
Wednesday afternoon in term time from 2.00pm4.00pm
• Alternatively, in a form with your question on the
website and get a reply by email
– http://www.reading.ac.uk/mathssupport/
What is data?
• Data is made up of variables
• A variable is something that can take different values
between individuals or in the same individual at
different time points
– Gender can take the value “male” or
“female”
– Age can take a minimum numeric value of
zero, and a maximum numeric value of
many years
– Time to react to your name being called out
is an example of a variable that would vary
if you measured it in the same individual at
several time points
• It is usual in Psychology to measure the value
of a variable in many separate individuals
What does statistics do to data?
• Describe – today’s topic
– Different types of variables
• categorical, ordinal, continuous (interval and ratio)
– If you have measured the same variable in many
individuals you need a way of summarising the data
– What’s the “average” value?
– How much variation is there in the data?
• Compare – ask if one group differs from another
on the value of a variable
• Relate – ask how one variable changes as a
function of another one
Variables are classified according to their
level of measurement
• Country of birth
– Example values are France, UK, Germany
– this is an unordered category because France is not
more or less than the UK
– We may assign numbers to category values for
convenience (e.g. 1 = UK, 2 = France), but you cannot
meaningfully add or subtract the numbers
– This severely restricts the type of statistics we can use
with categorical variables
Variables are classified according to their
level of measurement
• Finishing position in a running race
– this is an ordinal variable because 1st is better (more)
than 2nd
– but you can’t finish 1.5th (no decimals)
Variables are classified according to their
level of measurement
• Finishing position in a running race
– this is an ordinal variable because 1st is better (more)
than 2nd
– but you can’t finish 1.5th (no decimals)
– it is not meaningful to say that 3rd is twice as good as
6th because gaps between positions are not equal
– therefore, you can’t add, subtract, multiply, or divide
the values of ordinal variables and statistics should be
calculated based on ranks
Variables are classified according to their
level of measurement
• Annual salary
– this is a continuous variable because the gap
between £20,000 and £21,000 is the same as that
between £40,000 and £41,000
– it makes sense to add and subtract, and decimal
places make sense too
– Annual salary has a true zero that refers to the
absence of the quantity under consideration (money)
• Ratio level measurement
– Zero does not mean absence for all continuous
variables (e.g. zero celsius is not the absence of
temperature)
• Interval level measurement
Working with variables
• The following examples are based on an
imaginary set of data
• The following variables have been measured in
a sample of 30 people
– Country of birth
– Intelligence Quotient (IQ)
– Extroversion
Measures of central tendency
• If we have values on a variable for a sample of 30 people
(or 300 people) one thing we might need to do is
summarise the values in a shorter form
• The aim is to find a single number that characterises the
typical value of the variable in the sample
• The options we will consider are the
– Mode
– Mean
– Median
• Which one you use depends in part on the level of
measurement of the variable
Measures of central tendency
• The mode can be used with all data types, and is the only
measure applicable to unordered categories
• The mode is the most frequently occurring score, and may
be illustrated with a pie chart
• In the example data set the variable “birthCountry”
contains 15 instances of “France”, 13 instances of “UK”,
and 2 instances of “Germany
France
UK
Germany
Questions to answer at home
• What is the modal birth country for a sample
containing 20 UK, 23 French, 50 Indian, and 50
Chinese?
– What word describes this sample?
Central tendency for ordinal, interval and
ratio level variables
• Before calculating a measure of central
tendency you should first visually inspect the
variable using a frequency histogram
• Histograms are most informative for large
sample sizes of several hundred cases or
more
– but they are still an essential step for small
samples
• The first step in producing a histogram is to
sort the cases in the variable from lowest to
highest
• The second step is to count the frequency of
occurrence of each value
The 30 IQ values from earlier
• 109
77
79
109
• Sorted:
90
101
97
97
134
103
101
103
115
124
105
118
114
90
117
68
• 68 72
97
104
117
77
97
105
118
79
100
105
124
82
101
109
134
90 90 96 97
101 101 103 103
109 109 114 115
140
100 82 140
72 104 109
101 96 105
97
The IQ score 101
occurs 3 times in
the sample
Histogram x axis intervals or “bin sizes”
• In the previous example the interval was equal to
one unit on the IQ scale
• Typically, the interval will be wider than a single
unit of the scale
• Be aware of the interval, because a bad interval
choice can make a histogram misleading
– often every score contained in a variable is slightly
different, so a histogram with very small bin sizes will
just look flat
With the same data,
the interval is now
5 IQ points
Note that the y axis
maximum has now
changed
With the same data,
the interval is now
50 IQ points
Note that the y axis
maximum has now
increased
dramatically
The mean (commonly “average”)
• To calculate the mean you sum all the scores
(e.g. IQ’s 109 + 90 + 134 + 115 + 114 +….)
• Then you divide by the number of scores you
added together (30, in the example data set)
• This gives an indication of the typical score
The mean IQ in this
sample is 101.9
The median
• The median is the score that lies in the middle of the
sample, which therefore has an equal number of scores
higher and lower than it
• To calculate the median you first sort the scores, as for
making a histogram
3
13
10.5
7
6
8
8
12
4
3
4
6
7
8
8
10.5
12
13
1
2
3
4
5
6
7
8
9
The median
• Then assign ranking positions in the list and locate the
score corresponding to the middle rank
• At home, find out how the is procedure modified when the
number of scores in the variable is even?
3
13
10.5
7
6
8
8
12
4
3
4
6
7
8
8
10.5
12
13
1
2
3
4
5
6
7
8
9
The mean IQ in this
sample is 101.9
The median IQ is
102
The mean Extroversion
score in this sample is
36.17
The median is 33
When to choose the median
• Firstly, if the histogram is not symmetrical about its
peak (most frequently occurring value) then the
median and mean will differ, and you can make the
case that the middle ranking score (median) is a
more appropriate description of central tendency
• Secondly, if the histogram reveals a few outlying
values that seem to be quite different from the rest
of the sample, then these outlying values will have
a large and disproportionate influence on the
mean, but not on the median
• Always calculate both and compare them
These outliers will
“drag” the mean
away from the
median
Measures of dispersion
• Imagine we contact the example sample and
use a questionnaire to assess their attitude to
the European Union
• The questionnaire produces scores ranging
from 5 (very negative) to 50 (very positive).
• We can compare French and British attitudes to
the European Union
• There are only 2 Germans in the sample, and
intuitively this is too few to assess German
attitudes to the European Union
The first 10 cases from the 30 in the example. Note
missing data for Germany
Mean 22.20
Median 23
Mean 22.54
Median 23
The range
• The simplest measure of dispersion is obtained
by subtracting the minimum score from the
maximum score
– French sub-sample attitudeEurope has a range of 22
– UK sub-sample attitudeEurope has a range of 31
• Reporting the mean and the range is adequate
as a way of comparing UK and French attitudes
to Europe in this sample
• But the range fails to capture dispersion properly
in some cases, which is why the standard
deviation is normally preferred
– At home, find out what the weaknesses of the range as
a measure of dispersion are
The standard deviation
• This is a measure of how much all the scores in a data set
vary around the mean in the same units as the mean itself
(e.g. years, grams)
– A big SD implies very spread out data
– If the SD is small the data is clustered close to the mean
• Understanding what the standard deviation means, and
how to calculate it, is very important
• It will be mentioned frequently in the next two lectures
The standard deviation
• For each score in the sample, subtract the mean of
the sample to produce “deviation scores”
scores
1
4
5
6
9
11
deviations
-5
-2
-1
0
3
5
• 1 – 6 = -5, 4 – 6 = -2…………………….11 – 5 = 5
• Intuitively, the mean of the deviation scores will be
a measure of the amount of variation in the sample
But the mean deviation is always zero because the
positives deviations exactly cancel the negative ones
The standard deviation
• The negative signs are removed by squaring the deviation
scores
• 22 = 4, -22 = 4, 32 = 9, -32 = 9, -42 = 16 etc
• An important statistic called the variance is obtained by
assessing the central tendency in the squared deviation
scores
• Sum the squared deviations
– The squaring process increases the relative contribution of scores
that are far from the mean to the variance, compared to those
scores that are close to the mean
• To calculate the variance you divide the sum of squared
deviations by the number of original scores minus 1
The standard deviation
scores
1
4
5
6
9
11
deviations
-5
-2
-1
0
3
5
squared
deviations
25
4
1
0
9
25
• The sum of the squared deviations is 64
• The mean deviation (variance) is therefore
– 64 /(6 – 1) = 12.8
• If the units of the scores is Kg, what is the units of
the variance?
The standard deviation
scores
1
4
5
6
9
11
deviations
-5
-2
-1
0
3
5
squared
deviations
25
4
1
0
9
25
• The sum of the squared deviations is 64
• The mean deviation (variance) is therefore
– 64 /(6 – 1) = 12.8
• If the units of the scores is Kg, what is the units of
the variance?
The standard deviation
• To convert the variance back into units we can
understand intuitively we take the square root of
the variance and call it the standard deviation
– In the worked example the square root of 12.8 is 3.58
• The standard deviation (SD) is in the same units
as the sample mean, so, for example, you can
write that the mean weight of adult domestic cats
in the sample is 5.0 Kg (SD 1.0 Kg)
• If the population of cat weights is normally
distributed then 68% of cats will weigh 5.0 Kg +/one SD from the mean
– 68% of cats weigh between 4Kg and 6Kg
Mean 22.20
SD 6.5
Mean 22.54
SD 8.7
List of questions to answer at home
• What is the modal birth country for a sample
containing 20 UK, 23 French, 50 Indian, and 50
Chinese?
– What word describes this sample?
• How the is procedure for calculating the median
modified when the number of scores in the variable is
even compared to when there are an odd number of
scores?
• The range fails to capture dispersion properly in some
cases, which is why the standard deviation is normally
preferred
– Find out what the weaknesses of the range as a measure of
dispersion are
• Below is a list of statistical terms that you should know the
meaning of in order to be sure you have understood the
material from today’s lecture. Note that the technical
meaning of terms in statistics is not always the same as
the everyday meaning of the words. You can use this list to
help you with your exam revision.
• Variable
• Level of measurement
– Categorical
– Ordinal
– Continuous
• Interval
• Ratio
• Measures of central tendency
– Mode
– Mean
– Median
• Frequency histogram
– Bin sizes
• Measures of dispersion
– Range
– Variance
– Standard deviation
Variance (s2) formula
The square (2) of the average difference between each individual score and
the mean for that sample
Each score
in sample
Mean of
sample
Formula:
s
2
(X  X )


The sum of..
2
N 1
Number of
scores in
sample
minus 1
Standard deviation formula
Formula:
s
(X  X )
2
N 1
Step 1. Calculate the variance
Step 2. Take the square root of the variance