Chi-Squared Test

The Chi-Squared test (pronounced “kai-squared”) is a statistical hypothesis test used to determine whether there is a significant relationship between categorical variables or whether observed frequencies differ from expected frequencies.

It’s one of the most widely used non-parametric tests for discrete (count) data.

In simple terms:

“The Chi-Squared test checks whether what you observed is different from what you expected — for example, are two categorical variables related, or are differences just due to chance?”

When to Use It

• Data are categorical - e.g., gender, colour, product type, etc
• You have counts/frequencies - Observed vs expected
• Expected frequency per cell ≥ 5 - (for accuracy)

When Not to Use It

• Data are continuous - Use other tests (t, ANOVA, etc.)
• Small sample (< 5 per cell) - Use Fisher’s exact test

Types of Chi-Squared Tests

1. Goodness-of-Fit Test - Tests whether observed frequencies match expected proportions eg. “Are dice fair?”
2. Test of Independence - Tests whether two categorical variables are related eg. "Is gender related to buying preference?”
3. Test of Homogeneity - Tests whether distributions are the same across populations eg. “Do different stores sell similar proportions of products?”

1. Goodness-of-Fit Test

Hypotheses

• H₀ (Null) - Observed frequencies match expected frequencies
• H₁ (Alt) - Observed frequencies differ from expected frequencies

Formula:

Where:

• O_i​ = observed frequency
• E_i = expected frequency

Degrees of Freedom (df): df = k − 1

where k = number of categories

Example: Are dice fair?

You roll a die 60 times and get:

Expected (E) = 10 each (since fair die → equal chance)

Python Example

import numpy as npfrom scipy.stats import chisquare​observed = np.array([8, 9, 10, 12, 11, 10])expected = np.array([10, 10, 10, 10, 10, 10])​chi2, p = chisquare(observed, expected)print(f"Chi-Squared Statistic: {chi2:.3f}")print(f"P-value: {p:.4f}")​if p > 0.05:    print("No significant difference — die appears fair.")else:    print("Significant difference — die may be biased.")

Interpretation:

• p > 0.05 → data fit the expected distribution (no bias)
• p ≤ 0.05 → observed frequencies differ significantly (possible bias)

2. Chi-Squared Test of Independence

Used with a contingency table — tests if two categorical variables are related or independent.

Hypotheses

• H₀ (Null) - The variables are independent
• H₁ (Alt) - The variables are dependent (associated)

Example: Gender vs Purchase Preference

Python Example

import numpy as npimport pandas as pdfrom scipy.stats import chi2_contingency​# Contingency tabledata = np.array([[40, 20],                 [30, 30]])​chi2, p, dof, expected = chi2_contingency(data)​print(f"Chi-Squared Statistic: {chi2:.3f}")print(f"Degrees of Freedom: {dof}")print(f"P-value: {p:.4f}")print("Expected Frequencies:\n", expected)​if p < 0.05:    print("Reject H₀ — variables are dependent (related).")else:    print("Fail to reject H₀ — variables are independent.")

Interpretation:

• If p < 0.05 → significant relationship (e.g., gender affects buying)
• If p > 0.05 → no evidence of relationship (variables independent)

3. Chi-Squared Test of Homogeneity

Similar to the independence test but used to compare distributions across different populations (e.g., region A vs region B sales distribution).

Same formula, same interpretation — only the data come from different samples rather than one sample classified by two variables.

Degrees of Freedom (df):

For independence or homogeneity:

where:

• r = number of rows (categories of variable 1)
• c = number of columns (categories of variable 2)

Visualisation Example

import pandas as pdimport seaborn as snsimport matplotlib.pyplot as plt​df = pd.DataFrame({    'Gender': ['Male'] * 60 + ['Female'] * 60,    'Purchase': ['Buy'] * 40 + ['Don’t Buy'] * 20 + ['Buy'] * 30 + ['Don’t Buy'] * 30})​sns.countplot(data=df, x='Gender', hue='Purchase', palette='Set2')plt.title("Gender vs Purchase Preference")plt.show()

The visual helps show whether proportions differ across categories.

Assumptions

• Data type - Frequencies/counts (not percentages)
• Independence - Observations must be independent
• Expected frequencies - ≥ 5 per cell for valid χ² approximation
• Sample size - Reasonably large

Python code

import pingouin as pgimport pandas as pd​# Example raw datadf = pd.DataFrame({    'fbs': [0, 1, 1, 0, 0, 1, 0, 1, 1, 0],   # Fasting blood sugar    'target': [1, 1, 0, 0, 1, 0, 1, 1, 0, 0] # Heart disease})​# Run chi-squared test on raw categorical columnsexpected, observed, stats = pg.chi2_independence(data=df, x='fbs', y='target')​print("Observed Frequencies:\n", observed)print("\nExpected Frequencies:\n", expected)print("\nTest Results:\n", stats)

Outputs:

Observed Frequencies: target    0    1fbs             0       2.5  2.51       2.5  2.5​Expected Frequencies: target    0    1fbs             0       2.5  2.51       2.5  2.5​Test Results:                  test    lambda  chi2  dof  pval  cramer  power0             pearson  1.000000   0.0  1.0   1.0     0.0   0.051        cressie-read  0.666667   0.0  1.0   1.0     0.0   0.052      log-likelihood  0.000000   0.0  1.0   1.0     0.0   0.053       freeman-tukey -0.500000   0.0  1.0   1.0     0.0   0.054  mod-log-likelihood -1.000000   0.0  1.0   1.0     0.0   0.055              neyman -2.000000   0.0  1.0   1.0     0.0   0.05