Chi-Square Test: A Comprehensive Guide

Introduction

The Chi-Square Test is a non-parametric statistical test used to determine whether there is a significant association between two categorical variables. It is widely applied in fields like data science, business analytics, medical research, marketing, and social sciences to analyze frequency data.

✅ Why Use the Chi-Square Test?

• Helps determine if two categorical variables are independent or related.
• Useful in analyzing survey results, market research, and customer behavior.
• Works on count data (not numerical or continuous data).

📌 Key Questions Chi-Square Can Answer:
✔ Does gender affect product preference?
✔ Is there a relationship between smoking habits and lung disease?
✔ Does education level influence voting patterns?

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1. Types of Chi-Square Tests

Chi-Square tests are divided into two main types:

A. Chi-Square Goodness-of-Fit Test

✅ Purpose: Determines whether the observed frequency distribution fits an expected distribution.
✅ Used when comparing one categorical variable against a known expected distribution.

📌 Example:

• A company claims that their customer complaints are evenly distributed across four quarters of the year.
• We collect real data on complaints per quarter and compare it to the expected uniform distribution.

✔ Hypothesis:

• Null Hypothesis (H0H_0): The observed distribution matches the expected distribution.
• Alternative Hypothesis (HAH_A): The observed distribution is different from the expected distribution.

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B. Chi-Square Test for Independence

✅ Purpose: Determines if two categorical variables are independent or associated.
✅ Used when we want to analyze relationships between two categorical factors.

📌 Example:

• Examining if gender (Male, Female) affects voting preference (Party A, Party B, Party C).
• Analyzing if customer age group is related to product purchase preference.

✔ Hypothesis:

• Null Hypothesis (H0H_0): The two variables are independent (no association).
• Alternative Hypothesis (HAH_A): The two variables are dependent (have an association).

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2. Assumptions of Chi-Square Test

For a valid Chi-Square test, these conditions must be met:

✅ 1. Data Must Be Categorical – Both variables should be categorical, not numerical.
✅ 2. Observations Must Be Independent – Each observation should be unique and unrelated to others.
✅ 3. Expected Frequency ≥ 5 – Each category should have an expected count of at least 5 for accurate results.
✅ 4. Sufficient Sample Size – A larger sample size increases the reliability of results.

📌 What if assumptions are violated?

• Use Fisher’s Exact Test for small sample sizes.
• Combine low-frequency categories to increase expected counts.

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3. Chi-Square Formula and Calculation

A. Understanding the Chi-Square Formula

χ2=∑(O−E)2E\chi^2 = \sum \frac{(O – E)^2}{E}

Where:
✔ OO = Observed frequency (actual counts from data)
✔ EE = Expected frequency (theoretical counts under H0H_0)

📌 Higher χ2\chi^2 Value → More Likely Variables Are Related.

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B. Steps in Conducting a Chi-Square Test

✅ Step 1: Formulate Hypotheses

• Null Hypothesis (H0H_0): No relationship exists between the variables.
• Alternative Hypothesis (HAH_A): The variables are related.

✅ Step 2: Create a Contingency Table

• Organize observed frequencies into a table format.

✅ Step 3: Compute Expected Frequencies E=Row Total×Column TotalGrand TotalE = \frac{\text{Row Total} \times \text{Column Total}}{\text{Grand Total}}

✅ Step 4: Calculate Chi-Square Statistic χ2=∑(O−E)2E\chi^2 = \sum \frac{(O – E)^2}{E}

✅ Step 5: Compare with Critical Value

• Find the critical χ2\chi^2 value based on degrees of freedom and significance level (α\alpha).
• If χ2\chi^2 > critical value, reject H0H_0 (variables are related).
• If χ2\chi^2 ≤ critical value, fail to reject H0H_0 (variables are independent).

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4. Python Implementation of Chi-Square Test

Example 1: Chi-Square Test for Independence

import pandas as pd
import scipy.stats as stats

# Creating a contingency table for Gender vs Product Preference
data = [[30, 10],   # Male
        [20, 40]]   # Female

# Perform Chi-Square Test
chi2_stat, p_value, dof, expected = stats.chi2_contingency(data)

# Print results
print(f"Chi-Square Statistic: {chi2_stat}")
print(f"P-Value: {p_value}")
print(f"Degrees of Freedom: {dof}")
print("Expected Frequencies:\n", expected)

# Decision
alpha = 0.05
if p_value < alpha:
    print("Reject Null Hypothesis: The variables are related.")
else:
    print("Fail to Reject Null Hypothesis: No significant association.")

📌 Interpretation:

• If p-value < 0.05, there is a statistically significant relationship between gender and product preference.
• If p-value > 0.05, the two variables are independent.

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5. Post-Hoc Analysis for Chi-Square

If a significant association is found, we use post-hoc tests to analyze which categories differ.

📌 Common Methods:

• Standardized Residuals – Identify which cells contribute most to significance.
• Pairwise Chi-Square Tests – Compare specific group pairs.
• Bonferroni Correction – Adjusts for multiple comparisons.

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6. Real-World Applications of Chi-Square Test

📌 Healthcare – Examining the relationship between smoking and lung disease.
📌 Marketing – Analyzing customer purchase behavior based on demographics.
📌 Education – Studying whether gender influences subject choice.
📌 Politics – Investigating voting preferences across different age groups.

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