ChiSquare Test for Independence
This lesson explains how to conduct a
chisquare test for independence.
The test is applied when you have two
categorical variables
from a single population. It is used to determine whether
there is a significant association between the two variables.
For example, in an election survey, voters might be classified
by gender (male or female) and voting preference (Democrat,
Republican, or Independent). We could use a chisquare test for
independence to determine whether gender is related to
voting preference. The
sample problem at the end of the lesson
considers this example.
When to Use ChiSquare Test for Independence
The test procedure described in this lesson is appropriate when the
following conditions are met:
 If sample data are displayed in a
contingency table,
the expected frequency count for each cell of the table is
at least 5.
This approach consists of four steps: (1) state the hypotheses,
(2) formulate an analysis plan, (3) analyze sample data, and
(4) interpret results.
State the Hypotheses
Suppose that Variable A has r levels, and
Variable B has c levels.
The
null hypothesis states that knowing the level of
Variable A does not help you predict the level of
Variable B. That is, the variables are independent.
H_{0}: Variable A and Variable B are independent.
H_{a}: Variable A and Variable B are not independent.
The
alternative hypothesis is that knowing the level of
Variable A can help you predict the level of
Variable B.
Note: Support for the alternative hypothesis suggests that the variables
are related; but the relationship is not necessarily causal, in
the sense that one variable "causes" the other.
Formulate an Analysis Plan
The analysis plan describes
how to use sample data to accept or reject the null
hypothesis. The plan should specify the following elements.
 Significance level. Often, researchers choose
significance levels
equal to
0.01, 0.05, or 0.10; but any value between 0 and
1 can be used.
 Test method. Use the
chisquare test for independence
to determine whether there is a significant relationship
between two categorical variables.
Analyze Sample Data
Using sample data, find the
degrees of freedom, expected frequencies,
test statistic, and the Pvalue associated with the test statistic.
The approach described in this section is illustrated in the
sample problem at the end of this lesson.
 Degrees of freedom. The
degrees of freedom (DF) is equal to:
DF = (r  1) * (c  1)
where r is the number of levels for one catagorical variable, and
c is the number of levels for the other categorical
variable.
 Expected frequencies. The expected frequency counts
are computed separately for each level of one categorical variable
at each level of the other categorical variable. Compute r * c
expected frequencies, according to the following formula.
E_{r,c} = (n_{r} * n_{c}) / n
where
E_{r,c} is the expected frequency count for
level r of Variable A and
level c of Variable B,
n_{r} is the total number of sample observations at
level r of Variable A,
n_{c} is the total number of sample observations at
level c of Variable B, and
n is the total sample size.
 Test statistic. The test statistic is a chisquare random variable
(Χ^{2}) defined by
the following equation.
Χ^{2} =
Σ [ (O_{r,c}  E_{r,c})^{2} / E_{r,c} ]
where
O_{r,c} is the observed frequency count at level r of
Variable A and level c of Variable B, and
E_{r,c} is the expected frequency count at level r of
Variable A and level c of Variable B.
 Pvalue. The Pvalue is the probability of observing a
sample statistic as extreme as the test statistic. Since the
test statistic is a chisquare, use the
ChiSquare Distribution Calculator
to assess the probability associated with the test statistic. Use
the degrees of freedom computed above.
Interpret Results
If the sample findings are unlikely, given
the null hypothesis, the researcher rejects the null hypothesis.
Typically, this involves comparing the Pvalue to the
significance level,
and rejecting the null hypothesis when the Pvalue is less than
the significance level.
Test Your Understanding
Problem
A public opinion poll surveyed a simple random sample of 1000 voters.
Respondents were classified by gender (male or female) and by
voting preference (Republican, Democrat, or Independent).
Results are shown in the
contingency table below.

Voting Preferences 
Row total 
Rep 
Dem 
Ind 
Male 
200 
150 
50 
400 
Female 
250 
300 
50 
600 
Column total 
450 
450 
100 
1000 
Is there a gender gap? Do the men's voting preferences differ
significantly from the women's preferences? Use a 0.05
level of significance.
Solution
The solution to this problem takes four steps:
(1) state the hypotheses, (2) formulate an analysis plan,
(3) analyze sample data, and (4) interpret results.
We work through those steps below:
 State the hypotheses. The first step is to
state the
null hypothesis and an alternative hypothesis.
H_{0}: Gender and voting preferences are independent.
H_{a}: Gender and voting preferences are not
independent.
 Formulate an analysis plan. For this analysis,
the significance level is 0.05. Using sample data, we will
conduct a
chisquare test for independence.
 Analyze sample data. Applying the chisquare
test for independence to sample data, we compute
the degrees of freedom,
the expected frequency counts, and
the chisquare test statistic. Based on the
chisquare statistic and the
degrees of freedom, we determine the
Pvalue.
DF = (r  1) * (c  1) = (2  1) * (3  1) = 2
E_{r,c} = (n_{r} * n_{c}) / n
E_{1,1} = (400 * 450) / 1000 = 180000/1000 = 180
E_{1,2} = (400 * 450) / 1000 = 180000/1000 = 180
E_{1,3} = (400 * 100) / 1000 = 40000/1000 = 40
E_{2,1} = (600 * 450) / 1000 = 270000/1000 = 270
E_{2,2} = (600 * 450) / 1000 = 270000/1000 = 270
E_{2,3} = (600 * 100) / 1000 = 60000/1000 = 60
Χ^{2} = Σ
[ (O_{r,c}  E_{r,c})^{2} / E_{r,c} ]
Χ^{2} =
(200  180)^{2}/180 + (150  180)^{2}/180 + (50  40)^{2}/40
+
(250  270)^{2}/270 + (300  270)^{2}/270 + (50  60)^{2}/60
Χ^{2} =
400/180 + 900/180 +
100/40 + 400/270 +
900/270 + 100/60
Χ^{2} =
2.22 + 5.00 + 2.50 + 1.48 + 3.33 + 1.67 = 16.2
where
DF is the degrees of freedom,
r is the number of levels of gender,
c is the number of levels of the voting preference,
n_{r} is the number of observations from level r
of gender,
n_{c} is the number of observations from level c
of voting preference,
n is the number of observations in the sample,
E_{r,c} is the expected frequency count when gender is level
r and voting preference is level c, and
O_{r,c} is the observed frequency count when gender is level
r voting preference is level c.
The Pvalue is the probability that a chisquare statistic
having 2 degrees of freedom is more extreme than 16.2.
We use the
ChiSquare Distribution Calculator
to find P(Χ^{2} > 16.2) = 0.0003.
 Interpret results. Since the Pvalue (0.0003) is
less than the significance level (0.05), we cannot accept the
null hypothesis. Thus, we conclude that there is a relationship
between gender and voting preference.
Note: If you use this approach on an exam, you may also want to mention
why this approach is appropriate. Specifically, the approach is
appropriate because the sampling method was simple random sampling, the
variables under study were categorical, and the expected frequency count
was at least 5 in each cell of the contingency table.