Understanding Statistics

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Understanding Statistics

• Note Bring exam review questions next week.
  Please do not provide answers.

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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

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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?

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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.

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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

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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.

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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.

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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.

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Correlational Research
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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.

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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).

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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.

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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

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Positive Correlation
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Negative Correlation
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No Correlation
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Prediction Using a Scatterplot
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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

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Path Analysis Diagram
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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.

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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

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Threats to Internal Validityin Correlational
Research

• Subject characteristics
• Mortality
• Instrument decay

• Testing
• History
• Data collector characteristics
• Data collector bias

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Causal-Comparative Research
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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

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The Basic Causal-Comparative Designs
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Examples of the Basic Causal-Comparative Design
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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

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Other Threats

• Loss of subjects
• Instrumentation
• History
• Maturation

• Data collector bias
• Regression

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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.