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# What is Chi-Square Test? Formula, Types, and Examples

The chi-square test is a statistical gem that uncovers associations or dependencies between categorical variables. In this blog, we’ll get to know about the chi-square test, its formula, types, examples, and properties, along with its limitations.

## Table of Contents

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## What is a Chi-Square Test?

A chi-square test is a statistical method used to determine if there’s a significant association between categorical variables. It compares observed data with expected data to see if the differences are statistically significant or simply due to chance. We use this test in various fields like biology, social sciences, and business to analyze categorical data and determine if there’s a relationship between variables.

The formula involves calculating the differences between observed and expected values squared divided by the expected values and summed across all categories. If the calculated chi-square value exceeds a critical value from a chi-square distribution table for a given significance level, it suggests a significant relationship between the variables.

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## Formula for the Chi-Square Test

The formula for the chi-square test statistic depends on the type of data analyzed. Below is the formula for the chi-square test:

## What Do Chi-Square Statistics Tell You?

Chi-square statistics are used to determine if there’s a significant association between categorical variables. It helps assess whether the observed distribution of categorical data differs from the expected distribution, providing insight into whether variables are independent or related. It’s a powerful tool for hypothesis testing and determining the goodness-of-fit of categorical data.

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### Types of Chi-Square Test

There are mainly two types of chi-square tests. Both tests use the chi-square statistic to evaluate the discrepancy between observed and expected frequencies. The calculated chi-square value is compared to a critical value from a chi-square distribution to determine statistical significance. If the calculated value is greater than the critical value, we reject the null hypothesis and conclude that there’s a significant difference or association between the variables being tested.

#### Goodness-of-Fit Test

• This test is used to determine if sample data fits a certain distribution or theoretical expectation.
• It compares observed frequencies in different categories with the frequencies we would expect if the variables were independent.
• For example, if you were rolling a fair six-sided die 60 times and wanted to test whether the frequencies of each number rolled match the expected frequencies (1/6 for each number), you could use a goodness-of-fit chi-square test.

#### Test of Independence

• This test examines whether there is a relationship between two categorical variables.
• It involves creating a contingency table that displays the frequency distributions of the two categorical variables.
• The test determines whether the variables are independent or if changes in one variable are associated with changes in the other.
• For instance, in a survey, you might want to analyze whether there’s a relationship between gender and voting preference by creating a contingency table of these two variables and conducting a chi-square test of independence to see if there’s a significant association between them.

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## How to Perform a Chi-Square Test

Performing a chi-square test involves several steps. Here’s a simplified guide:

1. Formulate Hypotheses: Define your null hypothesis (usually stating no association between variables) and alternative hypothesis.
1. Create a Contingency Table: To analyze the variables, organize your data into a contingency table that displays the frequencies or counts of each category.
1. Calculate Expected Frequencies: Compute the expected frequencies for each cell in the table based on the null hypothesis.
1. Compute Chi-Square Statistic: Use the formula to calculate the chi-square statistic by comparing observed and expected frequencies.
1. Determine Degrees of Freedom: Calculate degrees of freedom using the formula (df = (rows − 1) ✕ (columns − 1)).
1. Find Critical Value or P-Value: Based on the degrees of freedom and chosen significance level, find the critical value from the chi-square distribution table or compute the p-value.
1. Draw Conclusion: Compare the calculated chi-square statistic with the critical value or p-value. If the calculated chi-square value is greater than the critical value or if the p-value is less than the alpha level, reject the null hypothesis.

### Example of a Chi-Square Test

Suppose we surveyed 200 adults and 150 children about their favorite ice cream flavors. The results are as follows:

Step-by-Step Calculation:

Step 1: Set Hypotheses

Imagine we’re trying to figure out if there’s a link between age group and preferred ice cream flavor. The null hypothesis suggests there’s no connection, while the alternative hypothesis proposes there is a link.

Null Hypothesis (H0): There is no association between age group and favorite ice cream flavor.

Alternative Hypothesis (H1): There is an association between age group and favorite ice cream flavor.

Step 2: Contingency Table

Now, let’s organize our data. We create a table with age groups (maybe “Kids,” “Teens,” and “Adults”) on one side and ice cream flavors (like “Chocolate,” “Vanilla,” and “Strawberry”) on the other. In each cell, we count how many people fall into both the age group and flavor categories.

Step 3: Calculate Expected Frequencies

We calculate what we’d expect in each cell if the age group and ice cream flavor were independent. This gives us a baseline to compare against our actual observations.

Step 4: Compute Chi-Square Statistic

Using a formula, we find the chi-square statistic by comparing what we observed in our table to what we expected. Use the formula to calculate the chi-square statistic by comparing observed and expected frequencies.

Step 5: Degrees of Freedom

The degrees of freedom helps us interpret our chi-square value. For our ice cream example, df = (number of age groups −1) ✕ (number of flavors − 1).

Step 6: Find Critical Value or P-Value

Based on our degrees of freedom and a chosen significance level (let’s say 0.05), we check a chi-square distribution table to find the critical value or calculate the p-value. This tells us how likely our results are due to chance.

Step 7: Conclusion

Compare the calculated chi-square value with the critical value, or p-value, to determine if you reject the null hypothesis.

Performing these calculations (with software or manually) will yield the chi-square statistic. Then, comparing it to the critical value or p-value would determine if there’s a significant association between age group and ice cream flavor preference.

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## When Should You Use the Chi-Square Test?

The chi-square test serves as a robust tool in statistical analysis, particularly when exploring associations or dependencies between categorical variables. It finds application across diverse fields and scenarios.

This test is invaluable when you seek to understand relationships between variables without assuming a cause-and-effect link. For example, in the social sciences, it helps to examine if there’s an association between political affiliation and voting behavior.

In quality control, it helps determine if observed defects align with expected distributions across manufacturing lines. Additionally, it’s pivotal in genetics to assess whether observed genetic frequencies match expected patterns.

Essentially, the chi-square test shines whenever you’re investigating the relationships, distributions, or disparities within categorical data sets, offering insights crucial for decision-making in various domains.

## Properties of the Chi-Square Test

The chi-square test is a statistical test to determine whether there is a significant association between categorical variables. Here are some key properties and characteristics of the chi-square test:

• Assumption: It assumes that the data are drawn from a random sample and that the expected frequency for each cell in a contingency table is not too small (typically, no more than 20% of cells should have an expected frequency below 5).
• Test Statistic: The chi-square statistic (chi^2) is calculated by comparing observed and expected frequencies in a contingency table.
• Degrees of Freedom: In a chi-square test, degrees of freedom are calculated based on the number of categories in each variable. For a contingency table with r rows and c columns, the degrees of freedom are (r − 1) ✕ (c − 1).
• Interpretation: The chi-square test produces a p-value that indicates the probability of observing the data if the variables are independent. A small p-value (typically less than 0.05) suggests that there is a significant association between the variables.
• Effect Size: Cramer’s V or Phi coefficient is often used as a measure of effect size for chi-square tests, indicating the strength of association between variables.
• Non-parametric Test: The chi-square test is a non-parametric test, meaning it doesn’t make assumptions about the underlying distribution of the data.

## Limitations of the Chi-Square Test

• The chi-square test assumes categorical data and independence of observations, potentially leading to unreliable results if assumptions are violated.
• Small sample sizes may not yield conclusive or accurate results, affecting the reliability of the test.
• Low expected cell frequencies (typically less than 5) can compromise the test’s validity, requiring alternative methods.
• It shows associations but doesn’t establish causation between variables.
• The results might vary based on how categories are defined or grouped for analysis.
• The chi-square test is not suitable for continuous data unless appropriately categorized.
• Conducting multiple tests on the same data increases the chance of finding significant results by chance.
• In cases of high-dimensional tables with small samples, the test’s reliability diminishes.
• Different ways of grouping data can lead to varied conclusions, impacting the test’s outcome.

## Conclusion

In conclusion, the chi-square test stands as a powerful statistical tool widely used across various fields to assess the association between categorical variables. Its flexibility, simplicity, and ability to handle large datasets make it indispensable in research, from biology to social sciences and beyond. By providing a robust method to evaluate observed versus expected data, the chi-square test empowers researchers to draw meaningful conclusions, identify patterns, and make informed decisions based on statistical significance. Understanding its applications and limitations equips us with the means to delve deeper into data analysis, aiding in the pursuit of accurate interpretations and discoveries in diverse fields.

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## FAQs

When should I use a chi-square test?

Use it when you want to analyze categorical data to determine if there is a relationship between two variables or if the distribution of categorical variables differs from what would be expected by chance.

What are the types of chi-square tests?

There are two main types:

• The chi-square test of independence tests whether two categorical variables are independent of each other.
• The chi-square goodness-of-fit test checks whether the observed categorical data matches the expected data.
How is the chi-square test calculated?

The test statistic is calculated by comparing the observed frequencies of categorical data with the frequencies that would be expected if the variables were independent.

What are the assumptions of the chi-square test?
• The data is categorical.
• The observations are independent.
• The sample size is large enough for the test to be valid (typically, each expected cell count should be at least 5).
What does the chi-square test result mean?

The test generates a p-value. A small p-value (usually < 0.05) suggests that there is a significant relationship between the variables.

Can the chi-square test be used for large datasets?

Yes, chi-square tests can be used for large datasets, but if the sample size is very large, small differences might be flagged as significant due to the test’s sensitivity.

## About the Author

Principal Data Scientist

Meet Akash, a Principal Data Scientist who worked as a Supply Chain professional with expertise in demand planning, inventory management, and network optimization. With a master’s degree from IIT Kanpur, his areas of interest include machine learning and operations research.