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Chi-Square Independence Test – What and Why?

Chi-Square Independence Test - What Is It?

The chi-square independence test is a procedure for testing if two categorical variables are related in some population. Example: a scientist wants to know if education level and marital status are related for all people in some country. He collects data on a simple random sample of n = 300 people, part of which are shown below.

Chi-Square Test - Raw Data View

Chi-Square Test - Observed Frequencies

A good first step for these data is inspecting the contingency table of marital status by education. Such a table -shown below- displays the frequency distribution of marital status for each education category separately. So let's take a look at it.

Chi-Square Test - Contingency Table

The numbers in this table are known as the observed frequencies. They tell us an awful lot about our data. For instance,

Chi-Square Test - Column Percentages

Although our contingency table is a great starting point, it doesn't really show us if education level and marital status are related. This question is answered more easily from a slightly different table as shown below.

Chi-Square Test - Column Percentages

This table shows -for each education level separately- the percentages of respondents that fall into each marital status category. Before reading on, take a careful look at this table and tell me is marital status related to education level and -if so- how? If we inspect the first row, we see that 46% of respondents with middle school never married. If we move rightwards (towards higher education levels), we see this percentage decrease: only 18% of respondents with a PhD degree never married (top right cell).
Reversely, note that 64% of PhD respondents are married (second row). If we move towards the lower education levels (leftwards), we see this percentage decrease to 31% for respondents having just middle school.
In short, more highly educated respondents seem to marry more often than less educated respondents.

Chi-Square Test - Stacked Bar Chart

Our last table shows a relation between marital status and education. This becomes much clearer by visualizing this table as a stacked bar chart, shown below.

Chi-Square Independence Test - Stacked Bar Chart Showing Dependence

If we move from top to bottom (highest to lowest education) in this chart, we see the dark blue bar (never married) increase. Marital status is clearly associated with education level: the lower someone’s education, the smaller the chance he’s married. That is: education “says something” about marital status (and reversely) in our sample. So what about the population?

Chi-Square Test - Null Hypothesis

The null hypothesis for a chi-square independence test is that two categorical variables are independent in some population. Now, marital status and education are related -thus not independent- in our sample. However, we can't conclude that this holds for our entire population. The basic problem is that sample outcomes usually differ from populations.
If marital status and education are perfectly independent in our population, we may still see some relation in our sample by mere chance. However, a strong relation in a large sample is extremely unlikely and hence refutes our null hypothesis. In this case we'll conclude that the variables weren't unrelated in our population anyway.
So exactly how strong is the relation in our sample? And what's the probability -or significance level- of finding it if the variables are (perfectly) independent in the entire population?

Chi-Square Test - Statistical Independence

Before we continue, let's first make sure we understand what “independence” really means in the first place. Practically speaking, statistical independence means that the frequency distribution of a variable is the same for all levels of some other variable. Uh... say again? Well, what if we had found the chart below?

Chi-Square Independence Test - Stacked Bar Chart Showing Statistical Independence

What does education “say about” marital status? Absolutely nothing! Why? Because the frequency distributions of marital status are identical over education levels: no matter the education level, the probability of being married is 50% and the probability of never being married is 30%. In this chart, education and marital status are perfectly independent. The hypothesis of independence tells us which frequencies we should have found in our sample: the expected frequencies.

Expected Frequencies

Expected frequencies are the frequencies we expect in our sample if the null hypothesis holds. If education and marital status are independent in our population, then we expect this in our sample too. This implies the contingency table -holding expected frequencies- shown below.

Chi-Square Test - Expected Frequencies

These expected frequencies are calculated as $$eij = \frac{oi.\,o.j}{N}$$

So for our first cell, that'll be $$eij = \frac{39 \cdot 90}{300} = 11.7$$
and so on. But let's not bother too much as our software will take care of all this.
Note that many frequencies are non integers, for instance 11.7 respondents with middle school who never married. Although there's no such thing as “11.7 respondents” in the real world, such non integer frequencies are just fine mathematically. So at this point, we've 2 contingency tables:

Insofar as these tables are different, our data deviate more from independence. If they deviate a lot, we no longer believe our variables are independent and -thus- related. So how different are these 2 tables? We basically add up the differences for each of our (5 * 4 =) 20 cells, resulting in a single number that indicates how much our data differ from our hypothesis: the χ2 (pronounce “chi-square”) test statistic.

Test Statistic

The chi-square test statistic is calculated as
$$\chi^2 = \Sigma{\frac{(oij - eij)^2}{eij}}$$
so for our data $$\chi^2 = \frac{(18 - 11.7)^2}{11.7} + \frac{(36 - 27)^2}{27} + ... + \frac{(6 - 5.4)^2}{5.4} = 23.57$$

Again, our statistical software will take great care of all this. However, if you really want to see all calculations, take a look at this GoogleSheet, a screenshot of which is shown below.

Chi-Square Test Statistic in GoogleSheet

So χ2 = 23.57 in our sample. This number summarizes the difference between our data and our independence hypothesis. Is 23.57 a large value? What's the probability of finding this? Well, we can calculate it from its sampling distribution but this requires a couple of assumptions.

Chi-Square Test Assumptions

The assumptions for a chi-square independence test are

If these assumptions hold, our χ2 test statistic follows a χ2 distribution. It's this distribution that tells us the probability of finding χ2 = 23.57.

Chi-Square Test - Degrees of Freedom

We'll get the significance level we're after from the chi-square distribution if we give it 2 numbers:

The degrees of freedom is basically a number that determines the exact shape of our distribution. It's calculated as $$df = (i - 1) \cdot (j - 1)$$

so in our example $$df = (5 - 1) \cdot (4 - 1) = 12.$$
And with df = 12, the probability of finding χ2 ≥ 23.57 ≈ 0.023.We simply look this up in SPSS or other appropriate software. This is our 1-tailed significance. It basically means, there's a 0.023 (or 2.3%) chance of finding this assocation in our sample if it is zero in our population.

Chi-Square Distribution with 1-Tailed P-Value

Since this is a small chance, we no longer believe our null hypothesis of our variables being independent in our population. Conclusion: marital status and education are related. Now, keep in mind that our p-value of 0.023 only tells us that the assocation between our variables is probably not zero. It doesn't say anything about the strength of this association: the effect size.

Effect Size

For the effect size of a chi-square independence test, consult the appropriate association measure. If at least one nominal variable is involved, that'll usually be Cramér’s V (a sort of Pearson correlation for categorical variables). In our example Cramér’s V = 0.162. Since Cramér’s V takes on values between 0 and 1, 0.162 indicates a very weak association. If both variables had been ordinal, Kendall’s tau or a Spearman correlation would have been suitable as well.


For reporting our results in APA style, we may write something like “An association between education and marital status was observed, χ2(12) = 23.57, p = 0.023.”

Chi-Square Independence Test - Software

You can run a chi-square independence test in Excel or Google Sheets but you probably want to use a more user friendly package such as

The figure below shows the output for our example generated by SPSS.

Chi-Square Test - SPSS Output

For a full tutorial (using a different example), see SPSS Chi-Square Independence Test.

Thanks for reading!

Previous tutorial: What is a Dichotomous Variable?

Next tutorial: SPSS Chi-Square Independence Test Tutorial

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