Chapter 6: Relationships Between
Two Variables: Cross-Tabulation
• Independent and Dependent Variables
• Constructing a Bivariate Table
• Computing Percentages in a Bivariate Table
• Dealing with Ambiguous Relationships Between
Variables
• Reading the Research Literature
• Properties of a Bivariate Relationship
• Elaboration
• Statistics in Practice
Chapter 6 – 1
Introduction
• Bivariate Analysis: A statistical method
designed to detect and describe the
relationship between two variables.
• Cross-Tabulation: A technique for
analyzing the relationship between two
variables that have been organized in a
table.
Chapter 6 – 2
Understanding Independent and
Dependent Variables
• Example: If we hypothesize that English
proficiency varies by whether person is
native born or foreign born, what is the
independent variable, and what is the
dependent variable?
• Independent:
• Dependent:
nativity
English proficiency
Chapter 6 – 3
Constructing a Bivariate Table
• Bivariate table: A table that displays the
distribution of one variable across the categories
of another variable.
• Column variable: A variable whose categories are
the columns of a bivariate table.
• Row variable: A variable whose categories are the
rows of a bivariate table.
• Cell: The intersection of a row and a column in a
bivariate table.
• Marginals: The row and column totals in a
bivariate table.
Chapter 6 – 4
Chapter 6 – 5
Chapter 6 – 6
Chapter 6 – 7
Percentages Can Be Computed
in Different Ways:
1. Column Percentages: column
totals as base
2. Row Percentages: row totals as
base
Chapter 6 – 8
Absolute Frequencies
Support for Abortion by Job Security
Abortion
Yes
No
Column Total
Job Find Easy Job Find Not Easy
24
25
20
26
44
51
Row Total
49
46
95
Chapter 6 – 9
Column Percentages
Support for Abortion by Job Security
Abortion
Yes
No
Column Total
Job Find Easy Job Find Not Easy
55%
49%
45%
51%
100%
100%
(44)
(51)
Row Total
52%
48%
100%
(95)
Chapter 6 – 10
Row Percentages
Support for Abortion by Job Security
Abortion
Yes
No
Column Total
Job Find Easy Job Find Not Easy
49%
51%
43%
57%
46%
54%
Row Total
100% (49)
100% (46)
100%
(95)
Chapter 6 – 11
Properties of a Bivariate
Relationship
1. Does there appear to be a relationship?
2. How strong is it?
3. What is the direction of the relationship?
Chapter 6 – 12
Existence of a Relationship
IV: Number of Traumas
DV: Support for Abortion
If the number of traumas were unrelated to
attitudes toward abortion among women,
then we would expect to find equal
percentages of women who are pro-choice
(or anti-choice), regardless of the number of
traumas experienced.
Chapter 6 – 13
Existence of the Relationship
Chapter 6 – 14
Determining the Strength of
the Relationship
• A quick method is to examine the
percentage difference across the different
categories of the independent variable.
• The larger the percentage difference across
the categories, the stronger the association.
• We rarely see a situation with either a 0
percent or a 100 percent difference.
Chapter 6 – 15
Direction of the Relationship
• Positive relationship: A bivariate relationship
between two variables measured at the ordinal
level or higher in which the variables vary in the
same direction.
• Negative relationship: A bivariate relationship
between two variables measured at the ordinal
level or higher in which the variables vary in
opposite directions.
Chapter 6 – 16
A Positive Relationship
Chapter 6 – 17
A Negative Relationship
Chapter 6 – 18
Elaboration
• Elaboration is a process designed to further
explore a bivariate relationship; it involves
the introduction of control variables.
• A control variable is an additional variable
considered in a bivariate relationship. The
variable is controlled for when we take into
account its effect on the variables in the
bivariate relationship.
Chapter 6 – 19
Three Goals of Elaboration
1. Elaboration allows us to test for nonspuriousness.
2. Elaboration clarifies the causal sequence
of bivariate relationships by introducing
variables hypothesized to intervene
between the IV and DV.
3. Elaboration specifies the different
conditions under which the original
bivariate relationship might hold.
Chapter 6 – 20
Testing for Nonspuriousness
• Direct causal relationship: a bivariate
relationship that cannot be accounted for by other
theoretically relevant variables.
• Spurious relationship: a relationship in which
both the IV and DV are influenced by a causally
prior control variable and there is no causal link
between them. The relationship between the IV
and DV is said to be “explained away” by the
control variable.
Chapter 6 – 21
The Bivariate Relationship
Between Number of Firefighters
and Property Damage
Number of Firefighters 
(IV)
Property Damage
(DV)
Chapter 6 – 22
Chapter 6 – 23
Process of Elaboration
• Partial tables: bivariate tables that display
the relationship between the IV and DV
while controlling for a third variable.
• Partial relationship: the relationship
between the IV and DV shown in a partial
table.
Chapter 6 – 24
The Process of Elaboration
1. Divide the observations into subgroups on the
basis of the control variable. We have as many
subgroups as there are categories in the control
variable.
2. Re-examine the relationship between the original
two variables separately for the control variable
subgroups.
3. Compare the partial relationships with the original
bivariate relationship for the total group.
Chapter 6 – 25
Chapter 6 – 26
Chapter 6 – 27
Intervening Relationship
• Intervening variable: a control variable that
follows an independent variable but
precedes the dependent variable in a causal
sequence.
• Intervening relationship: a relationship in
which the control variable intervenes
between the independent and dependent
variables.
Chapter 6 – 28
Intervening Relationship:
Example
Religion  Preferred Family Size  Support for Abortion
(IV)
(Intervening Control Variable)
(DV)
Chapter 6 – 29
Conditional Relationships
• Conditional relationship: a relationship in
which the control variable’s effect on the
dependent variable is conditional on its
interaction with the independent variable.
The relationship between the independent
and dependent variables will change
according to the different conditions of the
control variable.
Chapter 6 – 30
Conditional Relationships
• Another way to describe a conditional
relationship is to say that there is a
statistical interaction between the control
variable and the independent variable.
Chapter 6 – 31
Conditional Relationships
Chapter 6 – 32
Conditional Relationships
Chapter 6 – 33