Quantitative Data Analysis:
Statistics
Sherlock Holmes
"... while man is an insoluble puzzle,
in the aggregate he becomes a
mathematical certainty. You can,
for example, never foretell what
any one man will do, but you can
say with precision what an
average number will be up to.
Individuals vary, but percentages
remain constant. So says the
statistician"
Overview
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General Statistics
The Normal Distribution
Z-Tests
Confidence Intervals
T-Tests
General Statistics
~ THE GOLDEN RULE ~
Statistics NEVER replace
the judgment of the expert.
Approach to Statistical
Research
1.
2.
3.
4.
5.
6.
Formulate a Hypothesis
State predictions of the hypothesis
Perform experiments or observations
Interpret experiments or observations
Evaluate results with respect to hypothesis
Refine hypothesis and start again
(Basically the same as all other research)
Hypothesis Testing
H0 : Null Hypothesis, status quo
HA : Alternative Hypothesis, research question
So, either :
"The data does not support H0"
or
"We fail to reject H0"
Types of Data

Continuous
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Discrete

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# of days worked this week, # leaves on a tree
Ordinal

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height, age, time
{Good, O.K., Bad}
Nominal

{Yes/No}, {Teacher/Chemist/Haberdasher}
Picturing The Data
Pie Charts

Nominal/Ordinal

Only suitable for
data that adds up
to 1

Hard to compare
values in the chart
Bar Charts

Nominal/Ordinal

Easier to compare
values than pie
chart

Suitable for a wider
range of data
Dot Plots

Nominal/Ordinal

Represents all the
data

Difficult to read
Box Plots

Nominal/Ordinal

1IQR, 3IQR

Outliers
Scatter Plots

Excellent for
examining
association
between two
variables
Histograms
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Continuous Data
Divide Data into
ranges
Time-Series Plots
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Time related Data
e.g. Stock Prices
Question 1
In a telephone survey of 68
households, when asked
do they have pets, the
following were the
responses :
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

16 : No Pets
28 : Dogs
32 : Cats
Draw the appropriate graphic to
illustrate the results !!
Question 1 - Solution
Total number surveyed = 68
Number with no pets = 16
=>Total with pets = (68 - 16) = 52
But total 28 dogs + 32 cats = 60
=> So some people have both cats and dogs
Question 1 - Solution
How many? It must be (60 - 52) = 8 people
 No pets = 16
 Dogs =
20
 Cats =
24
 Both =
8
------------------------Total = 68
Question 1 - Solution
Graphic: Pie Chart or Bar Chart
The Literary Digest Poll

1936 US Presidential Election

Alf Landon (R) vs. Franklin D. Roosevelt (D)
The Literary Digest Poll

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Literary Digest had been
conducting successful
presidential election polls
since 1916
They had correctly predicted
the outcomes of the 1916,
1920, 1924, 1928, and 1932
elections by conducting polls.
These polls were a lucrative
venture for the magazine:
readers liked them;
newspapers played them up;
and each “ballot” included a
subscription blank.
The Literary Digest Poll

They sent out 10 million ballots to two
groups of people:


prospective subscribers, “who were chiefly
upper- and middle-income people”
a list designed to "correct for bias" from the
first list, consisting of names selected from
telephone books and motor vehicle registries
The Literary Digest Poll


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Response rate: approximately 25%, or
2,376,523 responses
Result: Landon in a landslide (predicted
57% of the vote, Roosevelt predicted 40%)
Election result: Roosevelt received
approximately 60% of the vote
The Literary Digest Poll
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POSSIBLE CAUSES OF ERROR
Selection Bias: By taking names and addresses from
telephone directories, survey systematically excluded
poor voters.
 Republicans were markedly overrepresented
 in 1936, Democrats did not have as many
phones, not as likely to drive cars, and did not read
the Literary Digest
 “Sampling Frame” is the actual population of
individuals from which a sample is drawn: Selection
bias results when sampling frame is not
representative of the population of interest
The Literary Digest Poll
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
POSSIBLE CAUSES OF ERROR
Non-response Bias: Because only 20% of 10
million people returned surveys, nonrespondents may have different preferences from
respondents
 Indeed, respondents favored Landon
 Greater response rates reduce the odds of
biased samples
Terminology
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Population: is a set of entities concerning which
statistical inferences are to be drawn.
Sample: a number of independent observations
from the same probability distribution
Parameter: the distribution of a random variable as
belonging to a family of probability distributions,
distinguished from each other by the values of a
finite number of parameters
Bias: a factor that causes a statistical sample of a
population to have some examples of the
population less represented than others.
Outliers (and their treatment)

An "outlier" is an observation that does not
fit the pattern in the rest of the data

Check the data
Check with the measurer
If reason to believe it is NOT real, change it if
possible, otherwise leave it out (but note).
If reason to believe it is real, leave it out and note.
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The Mean
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The Mean (Arithmetic)
The mean is defined as the sum of all
the elements, divided by the number of
elements.
The statistical mean of a set of
observations is the average of the
measurements in a set of data
The Variance
But there can be a lot of variance in
individual elements,
e.g. teacher salaries
Average = €22,000
Lowest = € 12,000
Difference = 12,000 - 22,000 = -10,000
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The Variance

Sum of (Sample - Average) = 0, thus we
need to define variance.

The variance of a set of data is a
cumulative measure of the squares of the
difference of all the data values from the
mean divided by sample size minus one.
Standard Deviation

The standard deviation of a set of data is
the positive square root of the variance.
-1
-1
Question 2

Find the mean and variance of the
following sample values :
36, 41, 43, 44, 46
Question 2

Mean: (36 + 41 + 43 + 44 + 46)/5 = 42
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Variance
Difference
Square
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36 – 42 = -6
36
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41 – 42 = -1
1
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43 – 42 = 1
1
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44 – 42 = 2
4
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46 – 42 = 4
16
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---------------------------------------
58
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58 / (5 -1) = 58 / 4 = 14.5
The Normal Distribution
Density Curves: Properties
The Normal Distribution
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The graph has a single peak at the
center, this peak occurs at the mean
The graph is symmetrical about the
mean
The graph never touches the
horizontal axis
The area under the graph is equal to 1
Characterization
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A normal distribution
is bell-shaped and
symmetric.
The distribution is
determined by the
mean mu, m, and the
standard deviation
sigma, s.
The mean mu
controls the center
and sigma controls
the spread.
The Normal Distribution

If a variable is normally distributed,
then:



within one standard deviation of the mean there
will be approximately 68% of the data
within two standard deviations of the mean there
will be approximately 95% of the data
within three standard deviations of the mean
there will be approximately 99.7% of the data
The Normal Distribution
Why?

One reason the normal distribution is
important is that many psychological and
organsational variables are distributed
approximately normally. Measures of
reading ability, introversion, job satisfaction,
and memory are among the many
psychological variables approximately
normally distributed. Although the
distributions are only approximately normal,
they are usually quite close.
Why?

A second reason the normal distribution is
so important is that it is easy for
mathematical statisticians to work with. This
means that many kinds of statistical tests
can be derived for normal distributions.
Almost all statistical tests discussed in this
text assume normal distributions.
Fortunately, these tests work very well even
if the distribution is only approximately
normally distributed. Some tests work well
even with very wide deviations from
normality.
One Tail / Two Tail

Imagine we undertook an experiment
where we measured staff productivity
before and after we introduced a
computer system to help record
solutions to common issues of work


Average productivity before = 6.4
Average productivity after = 9.2
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
0
Before = 6.4
After = 9.2
10
One Tail / Two Tail
σ
0
Before = 6.4
σ
σ
After = 9.2
10
One Tail / Two Tail
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One-Tailed
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H0 : m1 >= m2
HA : m1 < m2
Two-Tailed
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H0 : m1 = m2
HA : m1 <>m2
STANDARD NORMAL
DISTRIBUTION
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Normal Distribution is defined as
N(mean, (Std dev)^2)
Standard Normal Distribution is defined as
N(0, (1)^2)
STANDARD NORMAL
DISTRIBUTION
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Using the following formula :
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will convert a normal table into a standard
normal table.
Exercise

If the average IQ in a given population
is 100, and the standard deviation is
15, what percentage of the population
has an IQ of 145 or higher ?
Answer

P(X >= 145)
P(Z >= ((145 - 100)/15))
P(Z >= 3)
From tables: 99.87% are less than 3
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=> 0.13% of population
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Trends in Statistical Tests used
in Research Papers
Historically
Results in:
Accept/Reject
Currently
Results in:
p-Value
Results in:
Approx. Mean
Confidence Intervals

A confidence interval is used to express the
uncertainty in a quantity being estimated.
There is uncertainty because inferences are
based on a random sample of finite size
from a population or process of interest. To
judge the statistical procedure we can ask
what would happen if we were to repeat the
same study, over and over, getting different
data (and thus different confidence intervals)
each time.
Confidence Intervals

If we know the true population mean
and sample n individuals, we know
that if the data is normally distributed,
Average mean of these n samples has
a 95% chance of falling into the
interval
Confidence Intervals

where the standard error for a 95% CI
may be calculated as follows;
Example 1
Example 1

Does FF-PD-G have more of the
popular vote than FG-L ?

In a random sample of 721 respondents :
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382 FF-PD-G
339 FG-L
Can we conclude that FF-PD-G has
more than 50% of the popular vote ?
Example 1 - Solution

Sample proportion = p = 382/721 = 0.53
Sample size = n = 721
Standard Error = (SqRt((p(1-p)/n))) = 0.02

95% Confidence Interval
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0.53 +/- 1.96 (0.02)
0.53 +/- 0.04
[0.49, 0.57]
Thus, we cannot conclude that FF-PD-G had more of
the popular vote, since this interval spans 50%. So, we
say: "the data are consistent with the hypothesis
that there is no difference"
Example 2
Example 2

Did Obama have more of the popular
vote than McCain ?

In a random sample of 1000 respondents
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532 Obama
468 McCain
Can we conclude that Obama had more
than 50% of the popular vote ?
Example 2 – 95% CI

Sample proportion = p = 532/1000 = 0.532
Sample size = n = 1000
Standard Error = (SqRt((p(1-p)/n))) = 0.016

95% Confidence Interval
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0.532 +/- 1.96 (0.016)
0.532 +/- 0.03136
[0.5006, 0.56336]
Thus, we can conclude that Obama had more of the
popular vote, since this interval does not span 50%.
So, we say : "the data are consistent with the
hypothesis that there is a difference in a 95% CI"
Example 2 – 99% CI

Sample proportion = p = 532/1000 = 0.532
Sample size = n = 1000
Standard Error = (SqRt((p(1-p)/n))) = 0.016

99% Confidence Interval
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0.532 +/- 2.58 (0.016)
0.532 +/- 0.041
[0.491, 0.573]
Thus, we cannot conclude that Obama had more of
the popular vote, since this interval does span 50%.
So, we say : "the data are consistent with the
hypothesis that there is no difference in a 99% CI"
Example 2 – 99.99% CI

Sample proportion = p = 532/1000 = 0.532
Sample size = n = 1000
Standard Error = (SqRt((p(1-p)/n))) = 0.016

99.99% Confidence Interval
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
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0.532 +/- 3.87 (0.016)
0.532 +/- 0.06
[0.472, 0.592]
Thus, we cannot conclude that Obama had more of the
popular vote, since this interval does span 50%. So, we
say : "the data are consistent with the hypothesis
that there is no difference in a 99.99% CI"
T-Tests
One Tail / Two Tail
T-test
Z-test
T-Tests
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powerful parametric test for calculating
the significance of a small sample
mean
necessary for small samples because
their distributions are not normal
one first has to calculate the "degrees
of freedom"
T-Tests


The t-test is often called the Student's t-test.
It was created by a chief brewer named
William S. Gossett who worked for the
Guinness Brewery. He discovered this
statistic as part of his work in the brewery to
compare the different brewing processes for
changing raw materials into beer.
Guinness did not allow its employees to
publish results but the management decided
to allow Gossett to publish it under a
pseudonym - Student. Hence we have the
Student's t-test.
T-Test
~ THE GOLDEN RULE ~
Use the t-Test when your
sample size is less than 30
T-Tests
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If the underlying population is normal
If the underlying population is not skewed
and reasonable to normal
(n < 15)
If the underlying population is skewed and
there are no major outliers
(n > 15)
If the underlying population is skewed and
some outliers
(n > 24)
T-Tests

Form of Confidence Interval with tValue
Mean +/- tValue * SE
-------------as before
as before
Two Sample T-Test:
Unpaired Sample

Consider a questionnaire on computer use
to final year undergraduates in year 2007
and the same questionnaire give to
undergraduates in 2008. As there is no
direct one-to-one correspondence between
individual students (in fact, there may be
different number of students in different
classes), you have to sum up all the
responses of a given year, obtain an
average from that, down the same for the
following year, and compare averages.
Two Sample T-Test:
Paired Sample

If you are doing a questionnaire that is
testing the BEFORE/AFTER effect of
parameter on the same population,
then we can individually calculate
differences between each sample and
then average the differences. The
paired test is a much strong (more
powerful) statistical test.
Choosing the right test
Choosing a statistical test
http://www.graphpad.com/www/Book/Choose.htm
Choosing a statistical test
http://www.graphpad.com/www/Book/Choose.htm