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Understanding Statistics
Note: Bring exam review
questions next week. Please do
not provide answers.
Descriptive vs. Inferential
• Descriptive statistics
– Summarize/organize a group of numbers from
a research study
• Inferential statistics
– Draw conclusions/make inferences that go
beyond the numbers from a research study
– Determine if a causal relationship exists
between the IV and DV
Descriptive statistics
• A set of tools to help us exam data
– Descriptive statistics help us see patterns.
• 49, 10, 8, 26, 16, 18, 47, 41, 45, 36, 12, 42, 46, 6,
4, 23, 2, 43, 35, 32
– Can you see a pattern in the above data?
• Imagine if the data set was larger.
– 100 cases
– 1000 cases
• What could we do?
What are Inferential Statistics?
• Refer to certain procedures that allow researchers to
make inferences about a population based on data
obtained from a sample.
• Obtaining a random sample is desirable since it ensures
that this sample is representative of a larger population.
• The better a sample represents a population, the more
researchers will be able to make inferences.
• Making inferences about populations is what Inferential
Statistics are all about.
Statistics vs. Parameters
• A parameter is a characteristic of a population.
– It is a numerical or graphic way to summarize data
obtained from the population
• A statistic is a characteristic of a sample.
– It is a numerical or graphic way to summarize data
obtained from a sample
Sampling Error
• It is reasonable to assume that each sample will
give you a fairly accurate picture of its population.
• However, samples are not likely to be identical to
their parent populations.
• This difference between a sample and its population
is known as Sampling Error.
• Furthermore, no two samples will be identical in all
their characteristics.
Hypothesis Testing
• Hypothesis testing is a way of determining the probability
that an obtained sample statistic will occur, given a
hypothetical population parameter.
• The Research Hypothesis specifies the predicted
outcome of a study.
• The Null Hypothesis typically specifies that there is no
relationship in the population.
Practical vs. Statistical Significance
• The terms “significance level” or “level of
significance” refers to the probability of a sample
statistic occurring as a result of sampling error.
• Significance levels most commonly used in
educational research are the .05 and .01 levels.
• Statistical significance and practical significance are
not necessarily the same since a result of statistical
significance does not mean that it is practically
significant in an educational sense.
Correlational Research
The Nature of Correlational
Research
• Correlational Research is also known as
Associational Research.
• Relationships among two or more
variables are studied without any attempt
to influence them.
• Investigates the possibility of relationships
between two variables.
• There is no manipulation of variables in
Correlational Research.
Purpose of Correlational Research
• Correlational studies are carried out to explain
important human behavior or to predict likely outcomes
(identify relationships among variables).
• If a relationship of sufficient magnitude exists between
two variables, it becomes possible to predict a score
on either variable if a score on the other variable is
known (Prediction Studies).
• The variable that is used to make the prediction is
called the predictor variable (independent).
Purpose of Correlational Research
(cont.)
• The variable about which the prediction is made is
called the criterion variable (dependent).
• Both scatterplots and regression lines are used in
correlational studies to predict a score on a criterion
variable
• A predicted score is never exact. Through a prediction
equation, researchers use a predicted score and an
index of prediction error (standard error of estimate) to
conclude if the score is likely to be incorrect.
Correlation Coefficients
• Pearson product-moment correlation
– The relationship between two variables of
degree.
• Positive: As one variable increases (or decreases)
so does the other.
• Negative: As one variable increases the other
decreases.
– Magnitude or strength of relationship
• -1.00 to +1.00
– Correlation does not equate to causation
Positive Correlation
Negative Correlation
No Correlation
Prediction Using a Scatterplot
More Complex Correlational
Techniques
• Multiple Regression
• Technique that enables
researchers to determine
a correlation between a
criterion variable and the
best combination of two or
more predictor variables
• Discriminant Function
Analysis
• Rather than using multiple
regression, this technique
is used when the criterion
value is categorical
• Factor Analysis
• Allows the researcher to
determine whether many
variables can be
described by a few factors
• Path Analysis
• Used to test the likelihood
of a causal connection
among three or more
variables
• Structural Modeling
• Sophisticated method for
exploring and possibly
confirming causation
among several variables
Path Analysis Diagram
What Do Correlational Coefficients
Tell Us?
• The meaning of a given correlation coefficient depends
on how it is applied.
• Correlation coefficients below .35 show only a slight
relationship between variables.
• Correlations between .40 and .60 may have theoretical
and/or practical value depending on the context.
• Only when a correlation of .65 or higher is obtained, can
one reasonably assume an accurate prediction.
• Correlations over .85 indicate a very strong relationship
between the variables correlated.
Magnitude of effect
• Coefficient of
determination
– Also known as
• Shared variance
• The proportion of
variance accounted for
• Percentage of variance
accounted for
• Coefficient of
nondetermination
– Proportion of variance
not accounted for
r
2
1 r
2
Threats to Internal Validity
in Correlational Research
• Subject
characteristics
• Mortality
• Instrument decay
• Testing
• History
• Data collector
characteristics
• Data collector bias
Causal-Comparative
Research
Similarities and Differences Between
Causal-Comparative and
Correlational Research
• Similarities
– Associative research
– Attempt to explain
phenomena of interest
– Seek to identify variables
that are worthy of later
exploration through
experimental research
– Neither permits the
manipulation of variables
– Attempt to explore
causation
• Differences
– Causal studies compare
two or more groups of
subjects
– Causal studies involve at
least one categorical
variable
– Causal studies often
compare averages or use
crossbreak tables instead
of scatterplots and
correlations coefficients
The Basic Causal-Comparative Designs
Independent
variable
Dependent
variable
I
C
(Group possesses
characteristic)
O
(Measurement)
II
–C
(Group does
not possess
characteristic)
O
(Measurement)
I
C1
(Group possesses
characteristic 1)
O
(Measurement)
II
C2
(Group possesses
characteristic 2)
O
(Measurement)
Group
(a)
(b)
Examples of the Basic CausalComparative Design
Threats to Internal Validity in
Causal-Comparative Research
• Subject Characteristics
• The possibility exists that the groups are not equivalent on
one or more important variables
• One way to control for an extraneous variable is to match
subjects from the comparison groups on that variable
• Creating or finding homogeneous subgroups would be
another way to control for an extraneous variable
• The third way to control for an extraneous variable is to use
the technique of statistical matching
Other Threats
•
•
•
•
Loss of subjects
Instrumentation
History
Maturation
• Data collector bias
• Regression
Evaluating Threats to Internal Validity in
Causal-Comparative Studies
• Involves three sets of steps as shown below:
– Step 1: What specific factors are known to affect the variable
on which groups are being compared or may be logically be
expected to affect this variable?
– Step 2: What is the likelihood of the comparison groups
differing on each of these factors?
– Step 3: Evaluate the threats on the basis of how likely they
are to have an effect and plan to control for them.