1
Chi-square (χ2)
PSY 211
4-15-08
LAST STATISTIC OF THE SEMESTER!
NEXT TIME: FOOD DAY, REVIEW, WRAP-UP
A. Introduction
1. Parametric tests: Use at least one numeric rating,
so scores can be placed in a frequency distribution
that usually has a normal shape
 Correlation/Regression: Continuous variables
only
 t-test/ANOVA: One categorical variable, one
continuous variable
2. Non-parametric tests: Do not use numerical values,
scores cannot be placed on a frequency
distribution (also sometimes used for numeric
variables that have non-normal distributions)
 Chi-square (χ2): Categorical variables only
Pronunciation note: Chi is pronounced “khi” (from
Greece), not “chai” (India), not “chee” (China)
B. Two Types of χ2
 χ2 Test for Goodness of Fit
o Involves a single categorical variable only
 χ2 Test for Independence
o Involves 2(+) categorical variables
2
C. χ2 Test for Goodness of Fit
 Generally just involves one categorical variable
 Null hypothesis specifies the proportion of the
population in each category
 Determines how well sample data conform to
proportions set forth by the null hypothesis
 Test statistic (χ2) examines whether the
proportions in the sample reliably differ from the
null hypothesis
Examples:
 Use logic or past research to guide the null
hypothesis
H0 for Gender:
Female
Male
50%
50%
H0 for Negative Childhood Emotion:
Shame
Anger
Anxiety
Sadness
25%
25%
25%
25%
 Often an equal percentage (proportion) is
chosen for each group
3
 Alternatively, null hypothesis could be based on
known proportions in a larger population
Examples:
H0 for Ethnicity:
White
non-White
75%
25%
H0 for Vegetarianism:
No
Yes
97%
3%
H0 for Religious Affiliation:
Christianity
Non-religious / Don’t care
Agnosticism
Atheism
Other
84%
10%
2%
1%
3%
 χ2 used to examine whether the proportions in a
sample reliably differ from those hypothesized
 Like F…
χ2 ranges from 0 to ∞
Is small when the null hypothesis is likely true.
Is large when the null hypothesis is rejected.
4
H0 for Gender:
Female
Male
50%
50%
N = 279
Proportion Female = 50% = .50
Hypothesized frequency Female = .50 * 279 = 139.5
Proportion Male = 50% = .50
Hypothesized frequency Male = .50 * 279 = 139.5
Chi-Square Test
Frequencies
15. Gender
Female
Male
Total
Observed N
213
66
279
Expected N
139.5
139.5
Residual
73.5
-73.5
Te st Statistics
Chi-Squarea
df
As ymp. Sig.
15. Gender
77.452
1
.000
a. 0 c ells (.0% ) have expected frequencies les s than
5. The minimum ex pec ted cell frequenc y is 139.5.
 The Observed N indicates the actual frequency
in each group
 The Expected N indicates the frequency that is
hypothesized, based on the null hypothesis.
 The Residual just indicates the Observed
frequency minus the Expected frequency
 The Test Statistics box indicates that the χ2
value is 44.17, the degrees of freedom (df) are
1, and p < .001.
The chi-square test for goodness of fit was significant,
χ2(1, N = 279) = 77.45, p < .001. The sample
included an unexpectedly high number of females.
5
Last Semester…
Chi-Square Test
Frequencies
2. Gender
Female
Male
Total
Observed N
223
103
326
Expected N
163.0
163.0
Residual
60.0
-60.0
Te st Statistics
Chi-Squarea
df
As ymp. Sig.
2. Gender
44.172
1
.000
a. 0 c ells (.0% ) have expected frequencies les s than
5. The minimum ex pec ted cell frequenc y is 163.0.
The chi-square test for goodness of fit was significant,
χ2(1, N = 326) = 44.17, p < .001. The sample
included an unexpectedly high number of females.
6
H0 for Religious Affiliation:
Christianity
Non-religious / Don’t care
Agnosticism
Atheism
Other
84%
10%
2%
1%
3%
N = 279
Hypothesized frequency Christian = .84 * 279 = 234.4
………
Hypothesized frequency Other = .03 * 279 = 8.4
Chi-Square Test
Frequencies
31. Worldview or Rel igion
Christianity
Agnos ticis m
At heis m
Don't Know / Don't Care
Ot her
Total
Observed N
204
27
10
35
3
279
Ex pec ted N
234.4
27.9
5.6
2.8
8.4
Residual
-30.4
-.9
4.4
32.2
-5. 4
Test Statistics
Chi-Square a
df
As ymp. Sig.
31. Worldview
or Religion
382.767
4
.000
a. 1 cells (20.0%) have expected frequencies less than
5. The minimum expected cell frequency is 2.8.
The chi-square test for goodness of fit was significant,
χ2(4, N = 279) = 382.77, p < .001. There were fewer
Christians than expected and more people who were
apathetic.
7
Last Semester…
Chi-Square Test
Frequencies
23. Re ligi on
Non-religious
Christianity
Agnos ticis m
At heis m
Ot her
Total
Observed N
42
231
19
16
18
326
Ex pec ted N
32.6
273.8
6.5
3.3
9.8
Residual
9.4
-42.8
12.5
12.7
8.2
Expected frequencies
based on the expected
proportions identified
previously. For
example, for Christianity
.84 x 326 = 273.84.
Te st Statistics
Chi-Squarea
df
As ymp. Sig.
23. Religion
89.997
4
.000
a. 1 c ells (20. 0%) have ex pect ed frequencies less than
5. The minimum ex pect ed c ell frequency is 3.3.
The chi-square test for goodness of fit was significant,
χ2(4, N = 326) = 90.00, p < .001. There were fewer
Christians than expected and more people than
expected in every other religious group.
8
H0 for Yearly Physical:
No
Yes
50%
50%
Chi-Square Test
Frequencies
2. Yearly Physical
No
Yes
Total
Observed N
143
136
279
Expected N
139.5
139.5
Residual
3.5
-3.5
Test Statistics
Chi-Square a
df
As ymp. Sig.
2. Yearly
Physical
.176
1
.675
a. 0 cells (.0%) have expected frequencies less than
5. The minimum expected cell frequency is 139.5.
The chi-square test for goodness of fit was not
significant, χ2(1, N = 279) = 0.18, p = .68. There was
about an equal number of people getting yearly
physicals as those who were not.
9
D. χ2 Test for Independence
 Examines the relationship between two (or
more) categorical variables to determine if they
are independent
 Two variables are said to be independent if
there is no relationship between them
 Two variables are said to be dependent if there
is a relationship between them
 Similar to the correlation coefficient, except that
instead of both variables being continuous, both
variables are categorical
o Can’t correlate Academic Major with
Favorite Barnyard Animal, but you can do
a chi-square!
 Null hypothesis: no relationship
 Alternative hypothesis: some relationship
10
Examples:
Does Gender related to Beliefs about Human Origins?
15. Ge nde r * 18. Belie f about Hum an Origins Crossta bula tion
15. Gender
Female
Male
Total
Count
Ex pec ted Count
Count
Ex pec ted Count
Count
Ex pec ted Count
18. Belief about Human
Origins
Religious
Evolutionary
Texts
Theory
124
89
117.6
95.4
30
36
36.4
29.6
154
125
154.0
125.0
Total
213
213.0
66
66.0
279
279.0
Chi-Square Tests
Pearson Chi-Square
Continuity Correction a
Likelihood Ratio
Fis her's Exact Test
Linear-by-Linear
As sociation
N of Valid Cases
Value
3.318b
2.822
3.304
3.306
df
1
1
1
1
As ymp. Sig.
(2-sided)
.069
.093
.069
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
.089
.047
.069
279
a. Computed only for a 2x2 table
b. 0 cells (.0%) have expected count less than 5. The minimum expected count is 29.
57.
Observed frequencies compared to those that are
expected, based on the null hypothesis
Sample size
Chi-square value, degrees of freedom, and p-value
11
Bar Chart
18. Belief about Human
Origins
125
Religious Texts
Evolutionary Theory
Count
100
75
50
25
0
Female
Male
15. Gender
The chi-square test for independence was nonsignificant, χ2(1, N = 279) = 3.32, p = .07. Gender
was not reliably related to beliefs about human
origins.
12
Does being a parent impact one’s political views?
13. Parent * 20. Top Political Issue Crosstabulation
13. Parent
No
Yes
Total
Count
Expected Count
Count
Expected Count
Count
Expected Count
20. Top Political Iss ue
Economy
Healthcare Foreign Policy
58
34
11
65.3
39.5
10.5
23
15
2
15.7
9.5
2.5
81
49
13
81.0
49.0
13.0
Education
76
68.5
9
16.5
85
85.0
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
15.516 a
15.831
.080
4
4
As ymp. Sig.
(2-sided)
.004
.003
1
.777
df
279
a. 1 c ells (10.0%) have ex pec ted c ount les s than 5. The
minimum expected count is 2.52.
Other
46
41.1
5
9.9
51
51.0
Total
225
225.0
54
54.0
279
279.0
13
Bar Chart
20. Top Political Issue
80
Education
Economy
Healthcare
Foreign Policy
Other
Count
60
40
20
0
No
Yes
13. Parent
The chi-square test for independence was significant,
χ2(4, N = 279) = 15.52, p = .004. Non-parents were
mainly concerned with education, whereas parents
were mainly concerned with the economy.
14
Does ethnicity relate to musical preference?
11. White * 32. Favorite Music Crosstabulation
11. White
No
Yes
Total
Count
Expected Count
Count
Expected Count
Count
Expected Count
Country
2
6.0
62
58.0
64
64.0
32. Favorite Mus ic
Rap, hip
Rock
hop, R&B
2
12
8.8
3.6
92
27
85.2
35.4
94
39
94.0
39.0
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
30.695 a
26.787
9.472
3
3
As ymp. Sig.
(2-sided)
.000
.000
1
.002
df
279
a. 1 c ells (12.5%) have ex pec ted c ount les s than 5. The
minimum expected count is 3.63.
Other
10
7.6
72
74.4
82
82.0
Total
26
26.0
253
253.0
279
279.0
15
Bar Chart
32. Favorite Music
100
Country
Rock
Rap, hip hop, R&B
Other
Count
80
60
40
20
0
No
Yes
11. White
The chi-square test for independence was significant,
χ2(3, N = 279) = 30.70, p < .001. Compared to other
ethnic groups, white people were less likely to prefer
rap, hip hop, and R&B.
16
Some Examples from Last Semester…
Is Gender related to Vegetarianism?
The sample is 68%
female, so females
are expected to
make up 68% of the
vegetarians and
68% of the nonvegetarians.
2. Gender * 10. Vegetarian Crosstabulation
2. Gender
Female
Male
Total
10. Vegetarian
No
Yes
206
17
209.3
13.7
100
3
96.7
6.3
306
20
306.0
20.0
Count
Expected Count
Count
Expected Count
Count
Expected Count
Total
223
223.0
103
103.0
326
326.0
Chi-Square Tests
Pearson Chi-Square
Continuity Correction a
Likelihood Ratio
Fis her's Exact Test
Linear-by-Linear
As sociation
N of Valid Cases
Value
2.715b
1.959
3.081
2.707
df
1
1
1
1
As ymp. Sig.
(2-sided)
.099
.162
.079
Exact Sig.
(2-sided)
Exact Sig.
(1-sided)
.136
.076
.100
326
a. Computed only for a 2x2 table
b. 0 cells (.0%) have expected count less than 5. The minimum expected count is 6.
32.
Observed frequencies compared to those that are
expected, based on the null hypothesis
Sample size
Chi-square value, degrees of freedom, and p-value
The chi-square test for independence was nonsignificant, χ2(1, N = 326) = 2.72, ns. Gender was not
reliably related to vegetarianism.
17
Does ethnicity relate to musical preference?
20. Ethnicity * 29. Music Crosstabulation
20.
Ethnicity
White
NonWhite
Total
Count
Expected
Count
Count
Expected
Count
Count
Expected
Count
29. Mus ic
Alterna
Classic
tive
Rock
82
40
Rap
16
R&B
12
Hip
Hop
25
16.1
13.4
25.1
85.1
2
3
3
1.9
1.6
18
18.0
Country
66
Other
51
Total
292
37.6
61.8
52.8
292.0
13
2
3
8
34
2.9
9.9
4.4
7.2
6.2
34.0
15
28
95
42
69
59
326
15.0
28.0
95.0
42.0
69.0
59.0
326.0
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
7.354a
7.962
.853
6
6
As ymp. Sig.
(2-sided)
.289
.241
1
.356
df
326
a. 4 c ells (28.6%) have ex pec ted c ount les s than 5. The
minimum expected count is 1.56.
The chi-square test for independence was nonsignificant, χ2(6, N = 326) = 7.35, ns. Ethnicity was
not reliably related to music preference.
White people make up
90% of the sample, so
they are expected to
make up 90% of those
who like rap, 90% of
those who like R&B,
etc.
18
Does being an athlete related to choice of hero?
4. Athlete * 22. Hero Crosstabulation
4.
Athlete
No
Yes
Total
Count
Expected
Count
Count
Expected
Count
Count
Expected
Count
22. Hero
Other
Romantic Famous Teacher/
Mom Dad Sibling Relative Friend Partner
Person
Coach Other Total
58
20
9
17
10
8
3
3
20
148
53.6 27.7
8.6
18.2
10.9
4.5
3.6
6.4
41
10
23
14
2
5
11
64.4 33.3
10.4
21.8
13.1
5.5
4.4
7.6
61
19
40
24
10
8
14
118.0 61.0
19.0
40.0
24.0
10.0
8.0
14.0
60
118
Chi-Square Te sts
Pearson Chi-Square
Lik elihood Ratio
Linear-by-Linear
As soc iation
N of Valid Cases
Value
16.937 a
17.516
.671
8
8
As ymp. Sig.
(2-sided)
.031
.025
1
.413
df
326
a. 3 c ells (16.7%) have ex pec ted c ount les s than 5. The
minimum expected count is 3.63.
The chi-square test for independence was statistically
significant, χ2(8, N = 326) = 16.94, p = .03.
Specifically, athletes were more likely than expected
to indicate that their dad was their hero.
14.5 148.0
12
178
17.5 178.0
32
326
32.0 326.0