Basic Practice of
Statistics
7th Edition
Lecture PowerPoint Slides
In Chapter 6, We Cover …
 Marginal distributions
 Conditional distributions
 Simpson’s paradox
Categorical Variables
 Review: Categorical variables place individuals into
one of several groups or categories.
 The values of a categorical variable are labels for the
different categories.
 The distribution of a categorical variable lists the
count or percent of individuals who fall into each
category.
 When a dataset involves two categorical variables,
we begin by examining the counts or percents in
various categories for one of the variables.
Two-way table – Describes two categorical variables,
organizing counts according to a row variable and a
column variable.
Two-Way Table
Job outside
home
Stay
home
No
preference
Total
Women, no college
81
104
10
195
Women, with college
173
115
15
303
Men, no college
92
32
2
126
Men, with college
299
81
8
388
Total
1012
 What are the variables described by this
two-way table?
 How many young adults were surveyed?
Marginal Distribution
 The marginal distribution of one of the categorical
variables in a two-way table of counts is the
distribution of values of that variable among all
individuals described by the table.
 Note: Percents are often more informative than
counts, especially when comparing groups of
different sizes.
To examine a marginal distribution:
1. Use the data in the table to calculate the marginal
distribution (in percents) of the row or column totals.
2. Make a graph to display the marginal distribution.
Marginal Distribution
Job
outside
home
Stay
home
No
preference
Total
Examine the
marginal
distribution
of gender/
education.
Women, no college
81
104
10
195
Women, with college
173
115
15
303
Men, no college
92
32
2
126
Men, with college
299
8
81 Survey Participants
388
by Gender/Education
1012
Response
Percent
Women, no college
195/1012= 19.3%
Women, with
college
303/1012 =
29.9%
Men, no college
126/1012 =
12.5%
Men, with college
388/1012 =
38.3%
Percent
Total
45
40
35
30
25
20
15
10
5
0
Women no
college
Women
Men no
college
college
Survey Response
Men college
Conditional Distribution
 Marginal distributions tell us nothing about the relationship
between two variables.
 A conditional distribution of a variable describes the values of
that variable among individuals who have a specific value of
another variable.
To examine or compare conditional distributions:
 Select the row(s) or column(s) of interest.
 Use the data in the table to calculate the conditional distribution
(in percents) of the row(s) or column(s).
 Make a graph to display the conditional distribution.
 Use a side-by-side bar graph or segmented bar graph to
compare distributions.
Conditional Distribution
Job
outside
home
Stay
home
No
preference
Total
Women, no college
81
104
10
195
Women, with college
173
115
15
303
Men, no college
92
32
2
126
Men, with college
299
81
8
388
Total
1012
Men no
college
Job outside
home
81/195 = 41.5%
92/126 =
73.0%
Stay home
104/195 =
53.3%
32/126 =
25.4%
10/195 = 5.1%
2/126 = 1.6%
No
preference
preference,
men
versus
JobJob
preference,
men
versus
women,
no no
college
Job
preference,
women
nowomen,
college
college
100%
80%
80
60
70
50
60 60%
50
40
40
30
30
20 40%
20
10
10
0
0 20%Job outside home
No preference
Percent
Percent
Percent
Women no
college
Response
Calculate the
conditional
distribution of
job preference
for women and
men with no
college.
Job outside home
Stay home
Men
Stay home
Stay home
0%
Men
Opinion
Opinion
Opinion
Women
Job outside home
Women
No preference
No preference
Simpson’s Paradox
 When studying the relationship between two variables, there
may exist a lurking variable that creates a reversal in the
direction of the relationship when the lurking variable is
ignored, as opposed to the direction of the relationship when
the lurking variable is considered.
 The lurking variable creates subgroups, and failure to take
these subgroups into consideration can lead to misleading
conclusions regarding the association between the two
variables.
An association or comparison that holds for all of several groups
can reverse direction when the data are combined to form a
single group. This reversal is called Simpson’s paradox.
Simpson’s Paradox
 Consider the survival rates for the following groups of
victims who were taken to the hospital, either by
helicopter, or by road:
Counts
Victim
died
Victim
Survived
Total
Helicopter
64
136
200
Road
Percents
Died
Survived
Helicopter
32%
68%
Road
24%
76%
260
840
1100
 A higher percentage of those transported by helicopter
died. Does this mean that this (more costly) mode of
transportation isn’t helping?
Simpson’s Paradox
Consider the survival rates when broken down by type of
accident.
Serious accidents
Counts
Helicopter
Road
Died
48
60
Survived
52
40
Total
100
100
Percents
Died
Survived
Helicopter
48%
52%
Road
60%
40%
Less-serious accidents
Counts
Helicopter
Road
Percents
Died
Survived
Died
Survived
Total
16
84
100
200
800
1000
Helicopter
16%
84%
Road
20%
80%
Simpson’s Paradox
 Lurking variable: Accidents were of two sorts—serious
(200) and less serious (1100).
 Helicopter evacuations had a higher survival rate within both
types of accidents than did road evacuations.
 This is not evidence of the inefficacy of helicopter
evacuation!
 This is an example of Simpson’s paradox.
 When the lurking variable (type of accident: serious or less
serious) is ignored, the data seem to suggest road
evacuations are safer than helicopter.
 However, when the type of accident is considered, the
association is reversed and suggests helicopter evacuations
are, in fact, safer.