Displaying Distributions –
Qualitative Variables –
Part 2
Lecture 16
Sec. 4.3.3
Wed, Feb 11, 2004
Studies with Two
Qualitative Variables



Typically, the purpose of studying two
variables is to see whether there is a
relationship between them.
Also, when working with qualitative
data, percentages are the numerical
measure of choice.
The next-most-common measure is
frequency (or count).
Relationships between
Two Qualitative Variables

Frequency table – A table where
The rows represent values of one
variable,
 The columns represent values of the
other variable,
 And the cells show the frequency of the
row-column combinations of values.


A frequency table is also called a
contingency table.
Example



Let the row variable be the student’s
year in college.
Let the column variable be whether
the student is from Virginia or is from
out of state.
This will be a 4 x 2 frequency table.
Example
(frequency)
Virginia
Out of State
Freshman
0
0
Sophomore
0
0
Junior
0
0
Senior
0
0
Frequency Tables



If there is a relationship between the
variables, then perhaps it will be
apparent from the table.
Perhaps not.
Do we see any relationship between
year in college and state of residence?
Example
(percentage)
Virginia
Out of State
Freshman
0
0
Sophomore
0
0
Junior
0
0
Senior
0
0
Example

See example on page 199.
Nutritional Status
Poor Adeq. Exc.
Below
Academic
Average
Performance
Above
70
95
35
130
450
30
90
30
70
Example



Is there any apparent relationship
between academic performance and
nutritional status?
It is hard to say (in my opinion).
A possible relationship is that students
with better nutrition perform better
academically.
Excel Bar Graph
Academic Performance vs. Nutritional Status
500
Frequency
400
Below average
300
Average
200
Above average
100
0
Poor
Adequate
Nutritional Status
Excellent
Excel Bar Graph
Academic Performance vs. Nutritional Status
500
Frequency
400
Poor
300
Adequate
200
Excellent
100
0
Below average
Average
Above average
Academic Performance
The Marginal Distribution



Each variable has a marginal
distribution.
To find the marginal distribution of a
variable, find the total frequency of
the cells for each value of that
variable.
Then express each total frequency as
a percentage of the grand total for all
cells.
The Marginal Distribution
of Nutritional Status
Below
Academic
Average
Performance
Above
Total
Nutritional Status
Poor Adeq. Exc.
70
95
35
130
450
30
90
30
70
290
575 135
Example


The grand total of frequencies is
1000.
The marginal distribution for
nutritional status is
Poor
Adequate
Excellent
29%
57.5%
13.5%
The Marginal Distribution
of Academic Performance
Below
Academic
Average
Performance
Above
Nutritional Status
Total
Poor Adeq. Exc.
70
95
35 200
130
450
30 610
90
30
70 190
Example

The marginal distribution for
academic performance is
Below
Average
Above
???
???
???
The Marginal Distribution

The marginal distribution shows us the
distribution of one variable
independently of the other variable.
Conditional Distributions

In the example,
What percentage of all students are
below average academically and have
poor nutrition?
 What percentage of students who are
below average academically have poor
nutrition?
 What percentage of students who have
poor nutrition are below average
academically?

Conditional Distributions

The answers are
70/1000 = 7%
 70/200 = 35%
 70/290 = 24%

Conditional Distributions

To get the conditional distribution of
academic performance given
nutritional status,

For each category of nutritional status
(i.e., for each column), divide the various
frequencies in that category by the total
for that category.
Conditional Distributions

The conditional distribution of academic
performance given nutritional status is
Nutritional Status
Poor
Academic
Performance
Okay
Good
Below
24%
17%
26%
Average
45%
78%
22%
Above
31%
5%
52%
Conditional Distributions

The conditional distribution of nutritional
status given academic performance is
Nutritional Status
Academic
Performance
Poor
Okay
Good
Below
???
???
???
Average
???
???
???
Above
???
???
???
Let's Do It!
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
Let's do it! 4.8, p. 203 – Beer Tastes.
Let’s do it! 4.9, p. 205 – About Your
Class.

Use the data concerning year in college
vs. whether in or out of state.
Assignment


Page 206: Exercises 12 – 17.
Page 249: Exercises 62 – 66.