ChiSquare Test of Independence  Formula, Guide & Examples
A chisquare (Χ^{2}) test of independence is a nonparametric hypothesis test. You can use it to test whether two categorical variables are related to each other.
Table of contents
 What is the chisquare test of independence?
 Chisquare test of independence hypotheses
 When to use the chisquare test of independence
 How to calculate the test statistic (formula)
 How to perform the chisquare test of independence
 When to use a different test
 Practice questions
 Frequently asked questions about the chisquare test of independence
What is the chisquare test of independence?
A chisquare (Χ^{2}) test of independence is a type of Pearson’s chisquare test. Pearson’s chisquare tests are nonparametric tests for categorical variables. They’re used to determine whether your data are significantly different from what you expected.
You can use a chisquare test of independence, also known as a chisquare test of association, to determine whether two categorical variables are related. If two variables are related, the probability of one variable having a certain value is dependent on the value of the other variable.
The chisquare test of independence calculations are based on the observed frequencies, which are the numbers of observations in each combined group.
The test compares the observed frequencies to the frequencies you would expect if the two variables are unrelated. When the variables are unrelated, the observed and expected frequencies will be similar.
Contingency tables
When you want to perform a chisquare test of independence, the best way to organize your data is a type of frequency distribution table called a contingency table.
A contingency table, also known as a cross tabulation or crosstab, shows the number of observations in each combination of groups. It also usually includes row and column totals.
Household address  Intervention  Outcome 

25 Elm Street  Pamphlet  Recycles 
100 Cedar Street  Control  Recycles 
3 Maple Street  Control  Does not recycle 
123 Oak Street  Phone call  Recycles 
…  …  … 
They reorganize the data into a contingency table:
Intervention  Recycles  Does not recycle  Row totals 

Pamphlet  89  9  98 
Phone call  84  8  92 
Control  86  24  110 
Column totals  259  41  N = 300 
They also visualize their data in a bar graph:
Chisquare test of independence hypotheses
The chisquare test of independence is an inferential statistical test, meaning that it allows you to draw conclusions about a population based on a sample. Specifically, it allows you to conclude whether two variables are related in the population.
Like all hypothesis tests, the chisquare test of independence evaluates a null and alternative hypothesis. The hypotheses are two competing answers to the question “Are variable 1 and variable 2 related?”
 Null hypothesis (H_{0}): Variable 1 and variable 2 are not related in the population; The proportions of variable 1 are the same for different values of variable 2.
 Alternative hypothesis (H_{a}): Variable 1 and variable 2 are related in the population; The proportions of variable 1 are not the same for different values of variable 2.
You can use the above sentences as templates. Replace variable 1 and variable 2 with the names of your variables.
Expected values
A chisquare test of independence works by comparing the observed and the expected frequencies. The expected frequencies are such that the proportions of one variable are the same for all values of the other variable.
You can calculate the expected frequencies using the contingency table. The expected frequency for row r and column c is:
Intervention  Recycles  Does not recycle  Row totals 

Pamphlet  
Phone call  
Control  
Column totals 
The expected frequencies are such that the proportion of households who recycle is the same for all interventions:
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When to use the chisquare test of independence
The following conditions are necessary if you want to perform a chisquare goodness of fit test:
 You want to test a hypothesis about the relationship between two categorical variables (binary, nominal, or ordinal).
 The sample was randomly selected from the population.
 There are a minimum of five observations expected in each combined group.
How to calculate the test statistic (formula)
Pearson’s chisquare (Χ^{2}) is the test statistic for the chisquare test of independence:
Where
 Χ^{2} is the chisquare test statistic
 Σ is the summation operator (it means “take the sum of”)
 O is the observed frequency
 E is the expected frequency
The chisquare test statistic measures how much your observed frequencies differ from the frequencies you would expect if the two variables are unrelated. It is large when there’s a big difference between the observed and expected frequencies (O − E in the equation).
Follow these five steps to calculate the test statistic:
Step 1: Create a table
Create a table with the observed and expected frequencies in two columns.
Intervention  Outcome  Observed  Expected 

Pamphlet  Recycles  89  84.61 
Does not recycle  9  13.39  
Phone call  Recycles  84  79.43 
Does not recycle  8  12.57  
Control  Recycles  86  94.97 
Does not recycle  24  15.03 
Step 2: Calculate O − E
In a new column called “O − E”, subtract the expected frequencies from the observed frequencies.
Intervention  Outcome  Observed  Expected  O − E 

Pamphlet  Recycles  89  84.61  4.39 
Does not recycle  9  13.39  4.39  
Phone call  Recycles  84  79.43  4.57 
Does not recycle  8  12.57  4.57  
Control  Recycles  86  94.97  8.97 
Does not recycle  24  15.03  8.97 
Step 3: Calculate (O – E)^{2}
In a new column called “(O − E)^{2}”, square the values in the previous column.
Intervention  Outcome  Observed  Expected  O − E  (O − E)^{2} 

Pamphlet  Recycles  89  84.61  4.39  19.27 
Does not recycle  9  13.39  4.39  19.27  
Phone call  Recycles  84  79.43  4.57  20.88 
Does not recycle  8  12.57  4.57  20.88  
Control  Recycles  86  94.97  8.97  80.46 
Does not recycle  24  15.03  8.97  80.46 
Step 4: Calculate (O − E)^{2} / E
In a final column called “(O − E)^{2} / E”, divide the previous column by the expected frequencies.
Intervention  Outcome  Observed  Expected  O − E  (O − E)^{2}  (O − E)^{2} / E 

Pamphlet  Recycles  89  84.61  4.39  19.27  0.23 
Does not recycle  9  13.39  4.39  19.27  1.44  
Phone call  Recycles  84  79.43  4.57  20.88  0.26 
Does not recycle  8  12.57  4.57  20.88  1.66  
Control  Recycles  86  94.97  8.97  80.46  0.85 
Does not recycle  24  15.03  8.97  80.46  5.35 
Step 5: Calculate Χ^{2}
Finally, add up the values of the previous column to calculate the chisquare test statistic (Χ2).
How to perform the chisquare test of independence
If the test statistic is big enough then you should conclude that the observed frequencies are not what you’d expect if the variables are unrelated. But what counts as big enough?
We compare the test statistic to a critical value from a chisquare distribution to decide whether it’s big enough to reject the null hypothesis that the two variables are unrelated. This procedure is called the chisquare test of independence.
Follow these steps to perform a chisquare test of independence (the first two steps have already been completed for the recycling example):
Step 1: Calculate the expected frequencies
Use the contingency table to calculate the expected frequencies following the formula:
Step 2: Calculate chisquare
Use the Pearson’s chisquare formula to calculate the test statistic:
Step 3: Find the critical chisquare value
You can find the critical value in a chisquare critical value table or using statistical software. You need to known two numbers to find the critical value:
 The degrees of freedom (df): For a chisquare test of independence, the df is (number of variable 1 groups − 1) * (number of variable 2 groups − 1).
 Significance level (α): By convention, the significance level is usually .05.
Step 4: Compare the chisquare value to the critical value
Is the test statistic big enough to reject the null hypothesis? Compare it to the critical value to find out.
Step 5: Decide whether to reject the null hypothesis
 If the Χ^{2} value is greater than the critical value, then the difference between the observed and expected distributions is statistically significant (p < α).
 The data allows you to reject the null hypothesis that the variables are unrelated and provides support for the alternative hypothesis that the variables are related.
 If the Χ^{2} value is less than the critical value, then the difference between the observed and expected distributions is not statistically significant (p > α).
 The data doesn’t allow you to reject the null hypothesis that the variables are unrelated and doesn’t provide support for the alternative hypothesis that the variables are related.
Step 6: Follow up with post hoc tests (optional)
If there are more than two groups in either of the variables and you rejected the null hypothesis, you may want to investigate further with post hoc tests. A post hoc test is a followup test that you perform after your initial analysis.
Similar to a oneway ANOVA with more than two groups, a significant difference doesn’t tell you which groups’ proportions are significantly different from each other.
One post hoc approach is to compare each pair of groups using chisquare tests of independence and a Bonferroni correction. A Bonferroni correction is when you divide your original significance level (usually .05) by the number of tests you’re performing.
Chisquare test of independence  Chisquare test statistic 

Pamphlet vs. phone call  0.014 
Pamphlet vs. control  6.198 
Phone call vs. control  6.471 
 Since there are two intervention groups and two outcome groups for each test, there is (2 − 1) * (2 − 1) = 1 degree of freedom.
 There are three tests, so the significance level with a Bonferroni correction applied is α = .05 / 3 = .016.
 For a test of significance at α = .016 and df = 1, the Χ^{2} critical value is 5.803.
 The chisquare value is greater than the critical value for the pamphlet vs control and phone call vs. control tests.
Based on these findings, the city concludes that a significantly greater proportion of households recycled after receiving a pamphlet or phone call compared to the control.
There was no significant difference in proportion between the pamphlet and phone call intervention, so the city chooses the phone call intervention because it creates less paper waste.
When to use a different test
Several tests are similar to the chisquare test of independence, so it may not always be obvious which to use. The best choice will depend on your variables, your sample size, and your hypotheses.
When to use the chisquare goodness of fit test
There are two types of Pearson’s chisquare test. The chisquare test of independence is one of them, and the chisquare goodness of fit test is the other. The math is the same for both tests—the main difference is how you calculate the expected values.
You should use the chisquare goodness of fit test when you have one categorical variable and you want to test a hypothesis about its distribution.
When to use Fisher’s exact test
If you have a small sample size (N < 100), Fisher’s exact test is a better choice. You should especially opt for Fisher’s exact test when your data doesn’t meet the condition of a minimum of five observations expected in each combined group.
When to use McNemar’s test
You should use McNemar’s test when you have a closelyrelated pair of categorical variables that each have two groups. It allows you to test whether the proportions of the variables are equal. This test is most often used to compare before and after observations of the same individuals.
When to use a G test
A G test and a chisquare test give approximately the same results. G tests can accommodate more complex experimental designs than chisquare tests. However, the tests are usually interchangeable and the choice is mostly a matter of personal preference.
One reason to prefer chisquare tests is that they’re more familiar to researchers in most fields.
Practice questions
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Frequently asked questions about the chisquare test of independence
 How do I perform a chisquare test of independence in Excel?

You can use the CHISQ.TEST() function to perform a chisquare test of independence in Excel. It takes two arguments, CHISQ.TEST(observed_range, expected_range), and returns the p value.
 How do I perform a chisquare test of independence in R?

You can use the chisq.test() function to perform a chisquare test of independence in R. Give the contingency table as a matrix for the “x” argument. For example:
m = matrix(data = c(89, 84, 86, 9, 8, 24), nrow = 3, ncol = 2)
chisq.test(x = m)
 What are the two main types of chisquare tests?

The two main chisquare tests are the chisquare goodness of fit test and the chisquare test of independence.
 What properties does the chisquare distribution have?

A chisquare distribution is a continuous probability distribution. The shape of a chisquare distribution depends on its degrees of freedom, k. The mean of a chisquare distribution is equal to its degrees of freedom (k) and the variance is 2k. The range is 0 to ∞.
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