Stat 301 – Lecture 33
Categorical Data

A random sample of students
taking introductory statistics at
a large public university are
asked two questions.
1
Categorical Variables

Response: With whom is it
easiest to make friends?
Opposite gender
 Same gender
 No difference

2
Categorical Variables

Explanatory: What is your
gender?
Female
 Male

3
Stat 301 – Lecture 33
Contingency table
With whom is it easiest to make friends?
Opposite Same
No
Gender Gender Difference
Total
Female
58
16
63
137
Male
15
13
40
68
Total
73
29
103
205
4
Bar Graph
With whom is it easiest to make friends?
Distributions
Answer
Frequencies
50.2
35.6
75
50
14.1
Count
100
Level
No Diff
Opposite
Same
Total
Count
103
73
29
205
Prob
0.50244
0.35610
0.14146
1.00000
N Missing
0
3 Levels
25
No Diff
Opposite
Same
5
Percentages
With whom is it easiest to make friends?
Count
Row %
Female
Male
Total
Opposite
Gender
Same
Gender
No
Difference
Total
58
42.34%
15
22.06%
73
16
11.68%
13
19.12%
29
63
45.98%
40
58.82%
103
137
100%
68
100%
205
6
Stat 301 – Lecture 33
Mosaic Plot
1.00
Same
Answer
0.75
Opposite
0.50
0.25
No Diff
0.00
Female
Male
Gender
7
Description

Almost 60% of males in the
sample say no difference
while less than 50% of
females in the sample say no
difference.
8
Description

Females in the sample are
almost twice as likely as males
in the sample to say its easiest
to make friends with the
opposite gender.
9
Stat 301 – Lecture 33
Description

Males in the sample are almost
twice as likely as females in the
sample to say its easiest to
make friends with the same
gender.
10
Comment

There appears to be some sort
of relationship between
students’ gender and their
opinion on with whom it is
easiest to make friends.
11
Inference

Can the apparent relationship
be due to chance or is there a
statistically significant
relationship?
12
Stat 301 – Lecture 33
Independence

If students’ gender and their
opinion on with whom it is
easiest to make friends were
independent the proportions for
females and males would be
similar to the total proportions.
13
Expected Percentages
With whom is it easiest to make friends?
Count
Row %
Female
Opposite
Gender
Same
Gender
No
Difference
35.61%
14.15%
50.24%
35.61%
73
35.61%
14.15%
29
14.15%
50.24%
103
50.24%
Male
Total
Total
137
100%
68
100%
205
100%14
Expected Values
With whom is it easiest to make friends?
Count
Row %
Female
Male
Total
Opposite
Gender
Same
Gender
No
Difference
Total
48.7854
35.61%
24.2146
35.61%
73
35.61%
19.3805
14.146%
9.6195
14.146%
29
14.146%
68.8341
50.244%
34.1659
50.244%
103
50.244%
137
100%
68
100%
205
100%15
Stat 301 – Lecture 33
Test Statistic
∑

χ

df = (#rows – 1)(#columns – 1)

Condition: Expected counts
should be greater than 5 (10 or
15).
16
Cell Chi-Square
With whom is it easiest to make friends?
Count
Row %
Female
Male
Opposite
Gender
Same
Gender
No
Difference
1.7405
0.5896
0.4945
3.5065
1.1880
0.9962
χ2
Prob > χ2
Total
8.515
0.0142
17
Hypotheses
Ho: Opinion is independent of
gender
 HA: Opinion is not independent
of gender

18
Stat 301 – Lecture 33
Test Statistic/P-value
χ2 = 8.515
 P-value = 0.0142
 The P-value is less than 0.05,
therefore reject the null hypothesis,
Ho.

19
Conclusion

Opinion on with whom it is
easiest to make friends and
gender are not independent.
20
Inference

For students taking introductory
statistics at a large public
university, females are more
likely than males to say it is
easiest to make friends with
someone of the opposite
gender.
21
Stat 301 – Lecture 33
Inference

For students taking introductory
statistics at a large public
university, males are more likely
than females to say it is easiest
to make friends with someone
of the same gender.
22
JMP
Gender
Female
Female
Female
Male
Male
Male
Opinion
Opposite Gender
Same Gender
No Difference
Opposite Gender
Same Gender
No Difference
Count
58
16
63
15
13
40
23
Analyze – Fit Y by X
Y, Response: Opinion
 X, Factor: Gender
 Freq: Count

24
Stat 301 – Lecture 33
Contingency Analysis of Opinion By Gender
Freq: Count
Mosaic Plot
1.00
0.90
0.80
0.70
Same Gender
Opposite Gender
0.60
0.50
0.40
0.30
0.20
No Difference
0.10
0.00
Female
Male
Gender
25
Contingency Analysis of Opinion By Gender
Freq: Count
Contingency Table
Opinion
Count
No
Opposite Same
Difference Gender
Gender
Row %
Expected
Cell Chi^2
Female
16
58
63
11.68
42.34
45.99
68.8341 48.7854 19.3805
0.5896
1.7405
0.4945
Male
13
15
40
19.12
22.06
58.82
34.1659 24.2146 9.61951
1.1880
3.5065
0.9962
103
73
29
137
68
205
Tests
N
205
DF
2
-LogLike RSquare (U)
4.4258056
0.0218
Test
ChiSquare Prob>ChiSq
8.852
0.0120*
Likelihood Ratio
8.515
0.0142*
Pearson
26
Conditions
All the expected counts are
greater than 5.
 It would be better if all the
expected counts were greater
than 10 or 15.

27