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

• What is a Chi-Square Test?
• Formula for the Chi-Square Test
• Types of Chi-Square Test
• How to Perform a Chi-Square Test
  • Example of a Chi-Square Test
• When Should You Use the Chi-Square Test?
• Properties of the Chi-Square Test
• Limitations of the Chi-Square Test
• Conclusion
• FAQs

What is a Chi-Square Test?

The Chi-Square test is a statistical hypothesis test used in the analysis of contingency tables to determine whether there is a significant association between two categorical variables. It is widely applied in data analysis when working with observations from a random set of variables.

A Chi-Square (χ²) statistic is a test that measures how expected values compare to observed values. Since this test is used for categorical data, it does not require the data to follow a normal distribution. Therefore, the Chi-Square test is considered a non-parametric (distribution-free) tool in statistics.

This test assists in establishing whether differences between categories in a data set are due to random chance or a significant relationship. It is widely applied in medical research, social sciences, and marketing to examine survey responses, customer tastes, and experimental results.

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:

Types of Chi-Square Test

Chi-Square Goodness of Fit Test

• This test verifies if an observed frequency distribution is equal to an expected distribution.
• Applied to find out if a sample is representative of a population.
• Example: Testing whether customer choices of various product colors are equal to expected proportions of sales.

Chi-Square Test of Independence (or Differences)

• This test verifies if two categorical variables are independent.
• Applied to examine contingency tables.
• Example: Verifying if gender and voting preference are independent in survey data.

How to Perform a Chi-Square Test

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

To conduct a Chi-Square test, follow these steps:

Step 1: Formulate the Hypotheses

• Null Hypothesis (H₀): Assumes no relationship between the categorical variables.
• Alternative Hypothesis (H_A): Assumes a significant relationship exists.

Step 2: Create a Table of Observed and Expected Frequencies

Calculate the expected frequencies using:

E = (Row Total×Column Total)​/Grand Total

Each variable must have two or more categories for the test to be valid.

Step 3: Calculate the Chi-Square Statistic

Use the formula: (χ²) = Σ [(O – E)² / E]

For each observed number in the data, subtract the corresponding expected value, square the result, and divide by the expected value.

Step 4: Compare with the Critical Value

• Determine degrees of freedom: df = (Rows−1)×(Columns−1)
• Find the critical value from the Chi-Square table based on the significance level (usually 0.05).
• If χ² is greater than the critical value, reject H₀; otherwise, fail to reject H₀.

This process helps determine if two categorical variables are independent or significantly related.

Organize the observed frequencies from your dataset.

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 help 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.

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 follows the Chi-Square distribution, which has the following key properties:

• Mean: The mean of the distribution is equal to the degrees of freedom (df).
• Variance: The variance is twice the number of degrees of freedom: Variance = 2 × df
• Shape of the Distribution:
  • The Chi-Square distribution starts at zero (χ² ≥ 0) and is positively skewed.
  • As the degrees of freedom increase, the distribution approaches a normal shape.
• Hypothesis Testing: It is primarily used in hypothesis testing to determine whether two categorical variables are related or if differences occur due to chance.
• Non-Parametric Test: The Chi-Square test is non-parametric, meaning it makes no assumptions about the data’s underlying distribution.

These properties make the Chi-Square test a powerful tool for analyzing categorical data.

Limitations of the Chi-Square Test

While the Chi-Square test is a useful statistical tool, it has several limitations:

• Limited analysis in contingency tables – The test only partially completes the analysis and does not provide detailed insights into the strength or direction of relationships.
• Cannot establish causation – It only tests for associations and cannot determine if one variable causes changes in another.
• Requires independent observations – All participants or data points must be independent; otherwise, the results may be invalid.
• Only works with categorical data – The Chi-Square test cannot be used for numerical or continuous data.
• Sample size sensitivity – It is extremely sensitive to sample size; large samples may show statistically significant results even if the effect is weak.
• Expected frequency condition – No analyzed category should have an expected count less than one, and at least 80% of categories should have counts above five.

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. If you want to learn more about this technology, then check out our Comprehensive Data Science Course.

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