Download Days Three and Four: Introduction to Statistical Analysis

Day Four: Introduction to Statistical
Analysis
What are Statistics?
Procedures for describing, analyzing, and
interpreting quantitative data
 Your choice of statistical technique should be
guided by your research design and the type
of data you have collected

Reminders:
A variable is simply a phenomenon that is
subject to variation or change
 An independent variable is a presumed
effect or predictor variable
 A dependent variable is an outcome
variable
 Data is simply a collection of
measurements or observations in a study

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Descriptive Statistics
Displaying Data
 Frequency
distributions
 Characteristics of distributions:
◦ Normal distributions
◦ Skewness
◦ Kurtosis
 Used
to describe or summarize sets of
data to make them more interpretable
or understandable
◦ Measures of central tendency
 mean, median, mode
◦ Measures of variability
 range, variance, standard deviation
Central Tendency
 What
is the typical salary in SG?
 Mean - the arithmetic average
 Median - the middle score after
ranking in order
 Mode - the most commonly
occurring score
Measures of Variability
 Range
- the difference between the
highest and lowest score in a set of data
 Variance - the average squared distance
between every score and the mean
 Standard deviation - the average distance
between every score and the mean
Indices of Relative Standing
 Rank
scores
 Percentiles
 Standard Scores
 Variations on Standard Scores
Inferential Statistics
 How
likely is it that the effects I have
seen in my study are true in the full
population from which my sample is
drawn?
 Using probability to make decisions
about persuasiveness of study results
Statistics and Probability
 Probability
simply represents a judgment
about likelihood of outcomes, i.e., how likely
is it that I could obtain a result like this
purely by chance?
 Statistical inferences
◦ significant – very unlikely the effect would occur by
chance, e.g. less than 5%
◦ not significant - results are likely to have occurred by
chance
• inferential statistics...
Tools that permit you to generalize to a
population based upon sample
information
More specifically indicates how likely you
are to have obtained your results by
chance
• The problem of sampling error
Differences in samples due to random
fluctuations within the population
• By simulating the drawing of random
samples of size N from a population
with a specific mean and variance, we
can explore:
– (a) how much error we can expect on average in
drawing a sample of that size and
– (b) how much variation there will be on average in
the errors observed
Sampling distribution: the distribution of a sample
statistic (e.g., a mean) when sampled under known
sampling conditions from a known population.
Standard error is simply the standard deviation of the
sample means (SEx) - tells the researcher by how
much the researcher would expect the sample
means to differ if the researcher used other samples
from the same population
a smaller standard error indicates less sampling error

• a mathematical formula can be used to
estimate the standard error...
SD
SEx = √ N - 1
.
Sample size affects the size of the
standard error of the mean
The size of the population standard
deviation also affects the standard
error of the mean
Statistical Hypotheses: The null
hypothesis (H0)...

the difference between two sample means is
due to random, chance, sampling error
i.e., there is no true difference or
relationship between parameters in the
populations

The alternative hypothesis is aligned with the
research hypothesis (H1), which is generally
that:
One mean will be higher than another
There will be a relationship between
variables
Etc.

In rejecting the null hypothesis, you conclude
that there was an effect or relationship
i.e., that the effect/s observed are due
to something other than random,
chance, sampling error

There are four possibilities:
1. The null hypothesis is true and the
researcher concludes that it is true
A = B…a correct decision
2. The null hypothesis is false and the
researcher concludes that it is false
A ≠ B…a correct decision
3. The null hypothesis is true but the
researcher concludes that it is false
A = B…an incorrect decision
4. The null hypothesis is false but the
researcher concludes that it is true
A ≠ B…an incorrect decision

In quantitative research, a test of significance is
used to determine whether to reject or fail to
reject the null hypothesis
This involves pre-selecting a level of
probability, “α” (e.g., α = .05) that you
will accept in determining whether to
reject or fail to reject the null hypothesis
Steps in using inferential statistics
1. select the test of significance
2. determine whether significance test will
be two-tailed or one tailed
3. select α (alpha), the probability level
4. compute the test of significance
Significance tests
 statistical
procedures that enable the
researcher to determine if there was a
real difference between the sample
means
different tests of significance should be used
depending on the scale of measurement
represented by the data, number of groups
being compared or associated
Parametric vs. Non-parametric
Tests
• parametric test
Assumes that the variable measured is
normally distributed in the population
The data must represent an interval or
ratio scale of measurement
The selection of participants is
independent
The variances of the population
comparison groups are equal
A “more powerful” test in that it is more
likely to reject a null hypothesis that is
false, that is, the researcher is less likely
to commit a Type II error, and depends
on the four factors discussed yesterday
• A nonparametric test:
makes no assumption about the
distribution of the variable in the
population, that is, the shape of the
distribution
These are used when the data represent a
nominal or ordinal scale, when a
parametric assumption has been greatly
violated, or when the nature of the
distribution is not known
A “less powerful” test in that it is less
likely to reject a null hypothesis at a
given level of significance

Common tests of significance are:
Correlation
t-test
ANOVA
Chi Square
Correlation
 What
is the relationship between
two variables of interest, e.g., high
school performance and university
performance?
Correlation Coefficient

Measures whether two variables change in
a related way
Can be positive (max +1.00)
 Can be negative (min -1.00)
 Can be zero, indicating that the variables are not
related (0.0)

T-tests
• t-tests:
Are used to determine whether the
difference between two sample
means is likely to reflect a real
population difference
Compare the actual mean difference
observed to the difference expected
by chance
Forms a ratio in which the numerator is
the difference between the sample
means and the denominator is the
chance difference that would be
expected if the null hypothesis were
true
After the numerator is divided by the
denominator, the resulting t value is
compared to a theoretical sampling
distribution (of t), against the probability
level and the degrees of freedom
…if the t value is equal to or greater than
a certain critical level, then the null
hypothesis is rejected because the
difference is greater than would be
expected due to chance
two types of t-test:
the t-test for independent samples
(randomly formed)
the t-test for related samples (e.g.,
performance on a pre-/post- test,
different treatments)
Analysis of Variance
This is used to determine whether two
or more means are significantly
different at a selected probability level
This avoids the need to compute multiple ttests to compare groups – important for
maintaining overall alpha level
The basic approach of ANOVA is that
total variation, or variance, can be
divided or “partitioned” into two
sources
(a) treatment variance (“between
groups,” variance caused by the
treatment groups) and
(b) error variance (“within groups”
variance)
The F ratio represents the ratio of
treatment variance as the numerator
(between group variance) and error
variance as the denominator (within
group variance)
A key assumption is that randomly
formed groups of participants are
chosen and are essentially the same
at the beginning of a study on a
measure of the dependent variable
If the treatment variance is sufficiently
larger than the error variance, a
significant F ratio results, that is, the
null hypothesis is rejected and it is
concluded that the treatment had a
significant effect on the dependent
variable
If the treatment variance is not
sufficiently larger than the error
variance, an non-significant F ratio
results, that is, the null hypothesis is
not rejected and it is concluded that
the treatment had no significant effect
on the dependent variable
When the F ratio is significant and more
than two means are involved,
researchers can use multiple
comparison procedures (e.g., Scheffé
test, Tukey’s HSD test, Duncan’s
multiple range test) to look at
differences between specific pairs of
means

Assumptions for analysis of variance
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Groups randomly and independently sampled
Homogeneity of variance
Normality
Set of groups fixed (rather than random)

Two-factor analysis of variance
Basis of two-factor classifications
 Main effects and interactions
 Assumptions and conditions

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Random and independent sampling
Normality
Homogeneity of variance
Fixed levels
Equal number of cases in cells
Cell mean comparisons
Chi-Squares
• Chi Square (2)
This is a nonparametric test of
significance appropriate for nominal or
ordinal data that can be converted to
frequencies
Compares proportions actually observed
(O) to the proportions expected (E) to see
if they are significantly different