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Statistical Methods
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Variability and Averages
The Normal Distribution
Comparing Population Variances
Experimental Error & Treatment Effects
Evaluating the Null Hypothesis
Assumptions Underlying Analysis of Variance:
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Variability and Averages
Frequency
• Graph 1: Bipolar disorder
• Different variability
• Same averages
Patients
Controls
Depressed
C82MST Statistical Methods 2 - Lecture 2
Patients
Controls
Frequency
• Graph 2: Blood sugar
levels
• Same variability
• Different averages
Manic
Low
High
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The Normal Distribution
• The normal distribution is used in statistical analysis in
order to make standardized comparisons across different
populations (treatments).
• The kinds of parametric statistical techniques we use
assume that a population is normally distributed.
• This allows us to compare directly between two populations
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The Normal Distribution
• The Normal Distribution is a mathematical function that
defines the distribution of scores in population with respect
to two population parameters.
• The first parameter is the Greek letter (m, mu). This
represents the population mean.
• The second parameter is the Greek letter (s, sigma) that
represents the population standard deviation.
• Different normal distributions are generated whenever
the population mean or the population standard
deviation are different
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The Normal Distribution
• Normal distributions with different population variances and
the same population mean
s2 = 1
s2 = 2
f(x)
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s2 = 3
s2 = 4
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The Normal Distribution
• Normal distributions with different population means and
the same population variance
m= 1
m= 2
f(x)
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The Normal Distribution
• Normal distributions with different population variances and
different population means
s 2 = 1 m= 1
s 2 = 3 m= 3
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Normal Distribution
• Most samples of data are normally distributed (but not all)
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Comparing Populations in terms of Shared Variances
• When the null hypothesis (Ho) is approximately true we
have the following:
• There is almost a complete overlap between the two
distributions of scores
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Comparing Populations in terms of Shared Variances
• When the alternative hypothesis (H1) is true we have the
following:
• There is very little overlap between the two distributions
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Shared Variance and the Null Hypothesis
• The crux of the problem of rejecting the null hypothesis is
the fact that we can always attribute some portion of the
difference we observe among treatment parameters to
chance factors
• These chance factors are known as experimental error
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Experimental Error
• All uncontrolled sources of variability in an experiment are
considered potential contributors to experimental error.
• There are two basic kinds of experimental error:
• individual differences error
• measurement error.
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Estimates of Experimental Error
• In a real experiment both sources of experimental error will
influence and contribute to the scores of each subject.
• The variability of subjects treated alike, i.e. within the
same treatment condition or level, provides a measure
of the experimental error.
• At the same time the variability of subjects within each of
the other treatment levels also offers estimates of
experimental error
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Estimate of Treatment Effects
• The means of the different groups in the experiment should
reflect the differences in the population means, if there are
any.
• The treatments are viewed as a systematic source of
variability in contrast to the unsystematic source of
variability the experimental error.
• This systematic source of variability is known as the
treatment effect.
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An Example
• Two lecturers teach the same course.
• Ho: lecturer does not influence exam score.
• Experimental design
• 10 students: 5 assigned to each lecturer.
• IV: Lecturer (A1, A2)
• DV: Exam score
• Results:
• A1: 16, 18, 10,12,19
• A1: Mean=15
• A2: 4, 6, 8, 10, 2
• A2: Mean=6
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Partitioning the Deviations
AS 25
A2
T
Between Subjects
deviation
Within Subjects
deviation
0
2
4
6
8
AS 25 - A 2
AS 25 -
10
A2 - T
T
Total
Deviation
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Partitioning the Deviations
• Each of the deviations from the grand mean have specific
names
• AS25  T is called the total deviation.
• A2  T is called the between groups deviation.
• AS25  A is called the within subjects deviation.
• Dividing the deviation from the grand mean is known as
partitioning
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Evaluating the Null Hypothesis
• The between groups deviation
A2  T
• represents the effects of both error and the treatment
• The within subjects deviation
AS 25  A
• represents the effect of error alone
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Evaluating the Null Hypothesis
• If we consider the ratio of the between groups variability
and the within groups variability
Differences among treatment means
Difference among subjects treated alike
• Then we have
Experimental Error + Treatment Effects
Experimental Error
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Evaluating the Null Hypothesis
• If the null hypothesis is true then the treatment effect is
equal to zero:
Experimental Error + 0
=1
Experimental Error
• If the null hypothesis is false then the treatment effect is

greater than zero:
Experimental Error + Treatment Effect
1
Experimental Error

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Evaluating the Null Hypothesis
• The ratio
Experimental Error + Treatment Effect
1
Experimental Error
• is compared to the F-distribution

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ANOVA
• Analysis of variance uses the ratio of two sources of
variability to test the null hypothesis
• Between group variability estimates both experimental
error and treatment effects
• Within subjects variability estimates experimental error
• The assumptions that underly this technique directly follow
on from the F-ratio.
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Assumptions Underlying Analysis of Variance:
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The measure taken is on an interval or ratio scale.
The populations are normally distributed
The variances of the compared populations are the same.
The estimates of the population variance are independent
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