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Are Preferences Really That Different? Unpacking the Chi-Square Test

Michael Lee, MBA
3 days ago
3 min read

The Chi-Square Test: Unraveling Relationships Between Categorical Variables

The Story Behind the Test

The Chi-Square test has deep historical roots, originally developed by Karl Pearson in 1900. It became a foundational statistical method for examining categorical data, answering crucial questions about whether two variables are related. It has since been used in fields ranging from genetics (determining inherited traits) to marketing (analyzing customer preferences) and beyond.

What Is the Objective?

The Chi-Square test helps determine if there is a statistical relationship between two categorical variables. It tells us whether observed differences between groups are real or simply due to random chance.

For example, if a beverage company wants to know whether age groups prefer different types of drinks, the Chi-Square test helps evaluate if the difference in preferences is statistically significant.

✅ When to Use Chi-Square: When both variables are categorical (e.g., gender, region, product type)When comparing frequencies or counts

🚫 When Not to Use Chi-Square: When data is numerical (use correlation or regression instead)When expected cell values are below 5 (consider combining categories or using Fisher’s Exact Test)

How It Works

Step 1: Define Hypotheses

Null Hypothesis (H₀): There is no relationship between age groups and drink preferences.
Alternative Hypothesis (H₁): There is a significant relationship between age groups and drink preferences.

Step 2: Collect Data and Create a Contingency Table

A survey records the drink preferences (Juice, Soda, Tea) among different age groups (Children, Adults, Older Adults). The observed values are shown below:

	Juice	Soda	Tea	Total
Children	30	10	10	50
Adults	20	25	30	75
Older Adults	10	15	50	75
Total	60	50	90	200

Step 3: Calculate the Expected Values

The expected value for each cell is computed using: E = (Row Total x Column Total)/ (Grand Total)

The complete expected table is:

	Juice	Soda	Tea	Total
Children	15	12.5	22.5	50
Adults	22.5	18.75	33.75	75
Older Adults	22.5	18.75	33.75	75
Total	60	50	90	200

Step 4: Compute the Chi-Square Statistic

Plugging in the values:

Step 5: Determine Statistical Significance

Degrees of freedom (df) = (number of rows - 1) × (number of columns - 1)
With 3 age groups and 3 drink types:
The critical Chi-Square value at df = 4 and α = 0.05 is 9.49.
Since our Chi-Square value is 40.5, which is much greater than 9.49, the p-value is < 0.001.

Interpreting the Results

Since our Chi-Square value exceeds the critical value and the p-value is less than 0.05, we conclude that there is a statistically significant relationship between age groups and drink preferences. Age does influence what people drink, and this is not likely due to random chance.

Real-World Applications

1. Marketing & Customer Preferences

A beverage company can use this test to tailor their marketing strategies by understanding which drinks appeal to different age groups.

2. Healthcare & Dietary Trends

Nutritionists can analyze whether dietary choices (e.g., plant-based diets vs. meat-based diets) differ significantly across age groups.

3. Education & Learning Preferences

Schools can study whether students of different age groups prefer different teaching methods (e.g., visual, hands-on, or auditory learning).

Final Thoughts

The Chi-Square test is an essential tool for identifying relationships in categorical data. It provides a foundation for decision-making in business, healthcare, social sciences, and beyond.

If you want to gain hands-on experience with hypothesis testing and other powerful analytical techniques, our 2-day course, Problem Solving Using Data Analytics, provides practical applications and real-world exercises. For those curious about how Generative AI can enhance statistical testing, our Data Analytics in the Age of AI course explores AI-driven analytics and automation.