Determining and Interpreting
Associations between Variables
Cross-Tabs
Chi-Square
Correlation
Types of Relationships between
Variables
Non-monotonic - there is no direction to the
relationship.
Monotonic - there is a general direction of the
relationship.
Increasing - +ve relation.
Decreasing - -ve relation.
Linear - y=a+bx
Curvilinear - exponential, log, s shape, etc.
Characteristics of Relationships
Presence - is it significance.
Direction - monotonic or non-monotonic; -ve or +ve
sign.
Strength of association
Cross-Tabulations
Cross-tabulations (cross-tabs) consist of
rows and columns defined by the
categories classifying each variable.
 Frequencies
Table
 Raw Percentages Table
 Column Percentages Table
 Row Percentages Table
Cross-Tabs and Chi-Square
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Cross-Tabs and Chi-square (x2) are used to assess
whether or not a relationship exists between two
nominally scaled variables.
Null hypotheses is that the two variables under
investigation are NOT associated.
Chi-square statistics is calculated based on the
difference between observed and expected
frequencies
Chi-Square
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Chi-square distribution is
determined by its degrees of
freedom
Degrees of freedom = (r-1)(c-1)
r
= number of rows
 c = number of columns
Chi-sq = S (observed-expected)Sq/Expected
Correlation Coefficient
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An index number constrained to fall between -1.0 and
1.0 that communicates both the strength and direction
of association between two variables.
The greater the absolute size of the correlation
coefficient, the greater the covariation between the
two variables or the stronger is their association.
Is the correlation
statistically significant?
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If the correlation is not statistically
significant, then it has very little meaning.
Null hypotheses states that the
population correlation is zero; therefore,
the null hypotheses needs to be rejected.
A correlation indicates the strength of
association between variables by its
size. The sign indicates the direction of
the association.
Correlation Coefficient Ranges
Coefficient Range
(absolute value)
.81-1.00
Strength of
Association
Strong
.61-.80
Moderate
.41-.60
Weak
.21-.40
Very weak
.00-.19
None
Pearson Product
Moment Correlation
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The Pearson Product Moment Correlation (
r) measure the degree of linear association
between two variables.
Not only indicates degree of association but
direction as well.
Measures the “tightness” of measured
points on a straight line (linear)
Interval scaling required
Pearson Product
Moment Correlation

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Only considers the two variables - all other
factors are considered to not have any
relationship on the two variables
Does not demonstrate cause and effect
Only expresses linear relationships (no
curvilinear patterns)
Spearman Rank Order
Correlation
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Indicates strength and direction of
a relationship between two rank
(ordinal) variables.
Other forms of correlation that are
specifically designed for ordinal
and nominal data.