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## Introduction to Data Analysis.

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### Introduction to Data Analysis. Levels of measurement and Descriptive statistics – PowerPoint PPT presentation

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Title: Introduction to Data Analysis.

1
Introduction to Data Analysis.
• Levels of measurement and
• Descriptive statistics

2
• Introduction to the use of quantitative data in
social science.
• The tools we need in order to use numerical data
(i.e. anything we can count) to better understand
the world.
• Very basic introduction, students intending to
write theses using primarily quantitative data
should also attend the intermediate/ advanced
lectures.

3
Why am I here?
• Using quantitative data as an integral part of
• Using quantitative data as supplementary
evidence.
• Making better use of qualitative data.
• Other peoples research.
• Understanding work in your area.
• Criticising work in your area.
• Its compulsory.

4
What is Statistics?
• Methods for
• Designing and carrying out research studies
• Describing collected data
• represented by data

5
Some key terms (1)
• Populationthe total set of individual objects of
persons of interest in a study
• Samplea subset of the population that is
actually observed

6
Key Terms (2)
• Descriptive Stats consist of methods of graphical
and numerical techniques for summarizing the
information in a collection of data
• Inferential stats consist of procedures for
making generalizations about characteristics of a
population, based on info from a sample.

7
Key terms (3)
• Parameters are the characteristics of the
population about which we make inferences using
sample data
• Statistics are the corresponding characteristics
of the sample data, upon which we base our

8
Some key terms (1)
• Populationthe total set of individual objects of
persons of interest in a study
• Samplea subset of the population that is
actually observed

9
Variables and their measurement
• Variable measurement of a characteristic of a
subject (something or someone) that varies across
subjects in a population of subjects.
• Different levels of measurement, which means that
we have to examine different types of data in
different ways.

10
Some key terms (1)
• Populationthe total set of individual objects of
persons of interest in a study
• Samplea subset of the population that is
actually observed

11
Nominal level measures (1)
• Just represent a category.
• e.g. Male Female.
• e.g. Single Married Divorced.
• Since there is no ordering, these are nominal
measures.
• Often called qualitative, since two values differ
in quality not quantity.

12
Nominal level measures (2)
• Can quantify these data by tabulating them.
• Normally represent nominal data in a simple table
with percentages.
• Take the marital status of all of my 25 friends
(i.e. the population we are looking at is all
Ryans friends).

Marital status Number
Single 18 72
Married 6 24
Divorced 1 4
Total 25 100
13
Ordinal level measures
• Categories again, but these categories are
ordered.
• e.g. Many polling/survey questions.
• It was right for Britain to send troops to Iraq
• Strongly agree
• Agree
• Disagree
• Strongly disagree.
• The distance between each category is unknown.
• Strong agreers are more hawkish than agreers,
but we have no idea how much more hawkish they
are.
• We can say on observation is greater in rank than
another.
• Can be ranking in class (for example) or from
naturally ordered categories
• Called quantitative because different values
represent different magnitudes.

14
Interval level measures
• Numbers represent a quantitative variable.
• e.g. Income, number of pupils per teacher, age,
etc.
• There is a specific distance between each level.
• We can not only say that my sister is younger
than I am, but that she is 2 years younger.
• Age is a continuous variable, one can also
subdivide the measure (784 days, 3 hours and 2
minutes younger).
• It is also true that my parents have only 2
children.
• Number of children is a discrete variable, you
cannot sub-divide children, you have 1, or 2, or
3. You cant have 2 ½ children.

15
Descriptive statistics
• Most statistics that we will cover today apply to
variables that are interval level measures.
• Descriptive statistics are just that. They
describe a large amount of data in a summary
form.
• Why bother? Because were often interested in
what a typical person (or country or school or
parliament etc.) looks like.

16
Measuring the central tendency
• What we want to do is reduce a lot of interval
level measurements to a few numbers.
• The salaries of all of my best friends (the
population is Ryans best friends).
• What is the typical annual salary of a best
friend of mine.

Name Salary
Ellen 75,000
Jenny 13,000
Justin 31,000
Andrew 26,000
Mungo 15,000
17
The mean
• The most usual way of measuring the central
tendency is to use the mean (or average).
• This is simply the sum of the measurements
divided by the number of observations.
• For our salaried people
• Mean 32,000

18
A (very) little bit of math
• To introduce some terms which will be useful
later, the mean is calculated as follows. Suppose
we have n observations, with each value denoted
by X1, X2 and so on until Xn. Then the mean is
described as follows

Or, to put it another way
19
The means properties
• Shift of origin of measurement.
• If everyone earns 2000 more, then the new mean
salary is just the old mean salary (32,000) PLUS
2000.
• Change of scale.
• If we calculate salary in dollars (say 1 2),
then the new mean salary is simply twice the old
mean salary.
• Sum of two variables.
• Imagine that income salary savings interest.
• Mean income mean salary mean savings
interest.

20
The median
• Another common way deriving one number to
describe many is to use the median.
• Imagine we ranked all observations, the median is
simply the observation in the middle (½ of
observations above and ½ below).
• In ascending order the salaries are
• 13,000 15,000 26,000 31,000 75,000.
• Median 26,000.
• Median ½(26,00031,000) 28,500.

21
The medians properties (1)
• Shift of origin of measurement. YES
• Change of scale. YES
• Sum of two variables. NO
• The lack of this property is somewhat important
(which will become apparent in the following
weeks), and is related to one of the reasons why
we generally use the mean in most statistical
analysis.
• Nonetheless, the median does have some advantages
over the mean in describing some types of data.

22
The medians properties (2)
• For our salary example, the mean of my best
friends salaries gives a substantially higher
value than the median (6000 more).
• This is due to the distribution of the
observations. For the mean and median to be the
same the distribution of observations needs to be
symmetrical.
• Imagine we now look at all my friends and
acquaintances (the population of 25 people as
before), and plot the frequency of each salary
for all 25.

23
Frequency graph of salaries
Median 26,000
Mean 34,000
24
Positions of the median and mean
• For distributions with a long tail to the right,
the mean will take a higher value than the
median.
• This is generally true across the world for
income distributions, and is captured by Pens
parade of dwarfs and a few giants.
• If such a parade were organised today, then the
person of mean height (and income) would be
taller (and richer) than 65 of the population
and so would pass by after 40 minutes had
elapsed.
• Mean income is 24,000, median income is
16,000.
• For data with outliers the median can give a
better idea of what the typical observation is
like.

25
Ordinal level data
• The median can be used for ordinal level data.
their position on the Iraq war 2 strongly agreed
with sending British troops, one agreed, one
disagreed and one strongly disagreed.
• We can rank these answers and then find the
median.
• Strongly agree strongly agree agree disagree
strongly disagree.
• Thus the median answer is agree.

26
Nominal level data
• In general, we cant use the median or mean for
nominal data.
• Normally use the mode. This is the most commonly
occurring value.
• e.g if 53 people here are politics students, 40
sociology students, and 46 are other subjects,
then the modal value is politics.
• There is one special case in which we can use the
mean for nominal data however

27
Nominal binary data
• binary data is an exception as we can use the
mean. Binary data (e.g. Yes/No, Male/Female) can
be coded as 0 or 1.
• A variable measuring sex, men are coded 1 and
women coded 0.
• The mean score for those 0s and 1s is the
proportion of men. There were 2 women and 3 men
amongst my best friends.
• The median does NOT make sense for binary data.
It just tells us what the majority of the
population is.

28
Measures of dispersion
• The mean (or median) tells us something about the
centre of the distribution, but what about its
dispersion?
• The means/medians of the below distributions of
childrens scores on a maths test in three
different classes are all the same (48
observations, mean of 7, median of 7), but each
tells a quite different story.

29
The range
• The range is simply a measure of the distance
between the largest and smallest observations.
• The range for our salary example is therefore
• 75,000 13,000 62,000.
• Clearly this is not ideal as it relies on only
two observations.
• Say we have 1000 poker players. 999 win nothing,
and 1 wins 1million. The range indicates lots of
variation, when most people are actually
identical.

30
The variance and standard deviation
• A better way of assessing how much values of a
variable vary around the mean is to use the
standard deviation or variance.
• Basic idea is to measure how different individual
values are from the mean value.
• Some of these deviations from the mean will be
positive and some negative, so we square each
deviation.

31
The variance
• Take my 5 best friends. The mean salary was
32,000.
• If we added up all the differences then we would
get zero, so we need to square the differences
(i.e. multiply them by themselves).

Andrew
Ellen (75,000)
Justin
Jenny
Mungo (15,000)
15,000 - 32,000 -17,000
Difference 75,000 - 32,000 43,000
Mean32,000
32
Calculating variance
• Salary example, with 5 obs, and mean of 32,000.

Salary (000s) Deviation from mean Squared deviation
75 75 - 32 43 43 43 1849
13 13 32 -19 -20 -20 361
31 31 32 -1 -1 -1 1
26 26 32 -6 -6 -6 36
15 15 32 -17 -17 -17 289
33
Calculating standard deviation
• The standard deviation is the most common way to
measure deviation from the mean and is simply the
square root of the variance.
• We normally call the variance s2 and the standard
deviation s. Thus for our example, s2 507.2,
and s 22.5.

we usually use n-1 in the denominator
34
Examples of standard deviation
s 1.02 Tight distribution (All children perform
similarly)
s 1.67 Clustered distribution (Most children
perform to a similar level, with some variation)
s 4.01 Dispersed distribution (One group of
geniuses, one group of idiots)
35
But what does it mean?
• Our salary example had a standard deviation of
22.5, but for the distributions above the s
varied between 1 and 4, what does this tell us?
• Best way to think of it is as a kind of rough
average distance of an observation to the mean.
• Thus the standard deviation depends on the units
we are measuring in.

36
Standard deviation summary
• Broadly speaking, high levels of s indicate
greater variation, and the value of s gives a
broad idea of a typical distance from the mean.
• The concept of standard deviation is an important
one, and next week Ill talk more about
particular types of distributions and their
properties.

37
How to (not) lie with statistics
• Even simple descriptive statistics can be misused
• Particularly the case for simple graphs.
• Most examples I will use here are from Edward
Tufte The Visual Display of Quantitative
Information (1983, and later reprints).
• See any copy of any of the Sunday papers for
similar glaring errors however.

38
Too little information
• Presenting too little summary information.
• Example courtesy of Tukey (1979) in JASA.
• Take Washoe County in Nevada, USA. There is a
mean population density of 13 ½ people per square
mile.
• The mean is not informative without information
on the distribution however, for in fact 80 of
the inhabitants live in two cities.
• The cities have population densities of 5000 per
square mile.
• The rest of the county has a population density
of 2 ½ people per square mile.

39
Base years (1)
• Picking your base year (Tufte 1983).

40
Base years (2)
41
Measures over time (1)
42
Measures over time (2)
43
The lie factor
• Are doctors really becoming smaller?

44
Small differences
• Just because somethings top or bottom of a list,
doesnt imply anything.
• The difference between top and bottom might be
very small.
• Close to home, look at the Norrington table for
this. The difference between the middle 10
colleges is essentially zero, but its the
• Ranking of countries by something like literacy
rates is often similarly futile. There has to be
one at the top with 99.9 but all Western
countries will have 99 rates

45
(very) Small samples
• 9 out of 10 cats prefer Whiskers
• We may think that the evidence for this is strong
if thousands of cats had their opinion solicited,
but maybe weak if only 10 cats were questioned
out of the population of millions.
• Knowing when a small sample is too small is one
of the topics we will cover over the next two
weeks and is a critical part of understanding
commonly used statistics.

46
How to talk back to a statistic
• Who says so?
• We all want to prove our own theories correct
• How does he know?
• Is the data reputable?
• Whats missing?
• e.g. means are no use without standard
deviations.
• Does it make sense?
• Social science is the science of the bloody
obvious most of the time. Dont let numbers
confuse or fool you if it sounds wrong, it
probably is.