1. CHAPTER 6: Two-Way Tables*
Basic Practice of Statistics - 3rd Edition
CHAPTER 6: Two-Way Tables*
*This material is important in statistics, but it is needed later in this book only for Chapter 24. You
may omit it if you do not plan to read Chapter 24 or delay reading it until you reach Chapter 24.
Basic Practice of Statistics
7th Edition
Lecture PowerPoint Slides
Chapter 5

2. In Chapter 6, We Cover … Marginal distributions
Conditional distributions
Simpson’s paradox

3. 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.

4. Two-Way Table What are the variables described by this 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 8
388
1012
What are the variables described by this two-way table?
How many young adults were surveyed?

5. 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:
Use the data in the table to calculate the marginal distribution (in percents) of the row or column totals.
Make a graph to display the marginal distribution.

6. Marginal 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 8
388
1012
Examine the marginal distribution of gender/ education.
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%

7. 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.

8. 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 8
388
1012
Calculate the conditional distribution of job preference for women and men with no college.
Response
Women no college
Job outside home
81/195 = 41.5%
Stay home
104/195 = 53.3%
No preference
10/195 = 5.1%
Men no college
92/126 = 73.0%
32/126 = 25.4%
2/126 = 1.6%

9. 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.

10. 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:
A higher percentage of those transported by helicopter died. Does this mean that this (more costly) mode of transportation isn’t helping?
Counts
Helicopter
Road
Victim died 64
260
Victim Survived
136
840
Total
200
1100
Percents
Died
Survived
Helicopter
32%
68%
Road
24%
76%

11. 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
Percents
Died
Survived
Helicopter
48%
52%
Road
60%
40%
Less-serious accidents
Percents
Died
Survived
Helicopter
16%
84%
Road
20%
80%
Counts
Helicopter
Road
Died 16
200
Survived 84
800
Total
100
1000

12. 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.