Chi-Square Tests

Chi-square tests are used for categorical data, not numeric measurements.

They help answer two major questions:

1. Do observed frequencies match expected frequencies? - Goodness of Fit Test
2. Are two categorical variables associated? - Test of Independence (Contingency Table)

Chi-Square Goodness of Fit Test

“Does my observed distribution match an expected distribution?”

Use when:

• You have one categorical variable
• You want to compare observed counts to expected counts
• Example: Dice rolls, survey choices, defect types, etc.

Example

A six-sided die was rolled 60 times. Here are the counts:

observed = np.array([8, 12, 10, 11, 9, 10])​# perfectly fair dieexpected = np.array([10, 10, 10, 10, 10, 10])  

Run the test:

chi2, p = stats.chisquare(f_obs=observed, f_exp=expected)print(chi2, p)

Interpretation

• p < 0.05 → observed distribution ≠ expected distribution
• p ≥ 0.05 → no evidence of difference

If expected = uniform distribution

You can skip f_exp:

chi2, p = stats.chisquare(observed)​

Chi-Square Test of Independence

“Are these two categorical variables related?”

Use when:

• You have two categorical variables
• You want to test whether they are associated
• Example: Gender vs purchase decision, education vs voting preference, etc.

Requires a contingency table (cross-tabulation).

Test of Independence — Example (Manual Table)

Consider data on whether people bought a product:

Represent this as:

table = np.array([    [30, 10],    [20, 40]])

Run the test

chi2, p, dof, expected = stats.chi2_contingency(table)print(chi2, p)print("Expected frequencies:\n", expected)

Interpretation

• p < 0.05 → variables are associated (dependent)
• p ≥ 0.05 → variables are not associated (independent)

Expected frequencies tell what we would expect if variables were independent.

Test of Independence — Example with Pandas

With real datasets, you usually start with a DataFrame:

df = pd.DataFrame({    "gender": ["M","M","F","F","F","M","F"],    "bought": ["yes","no","yes","no","no","yes","no"]})

Create the contingency table

table = pd.crosstab(df['gender'], df['bought'])print(table)​

Run the test

chi2, p, dof, expected = stats.chi2_contingency(table)print(chi2, p)​

Requirements & Assumptions

• Counts, not proportions - Chi-square tests require frequency counts, not percentages.
• Observations must be independent - Each person/item is counted once.
• Expected frequency rule - At least 80% of expected counts should be ≥ 5.

If not, use Fisher’s Exact Test (SciPy supports it for 2×2 tables):

oddsratio, p = stats.fisher_exact(table)​

When to Use Fisher’s Exact Test

Use instead of chi-square when:

• Sample size is small (< 40)
• Expected counts < 5 in any cell

Example

table = np.array([    [3, 1],    [1, 3]])​oddsratio, p = stats.fisher_exact(table)print(p)

Post-hoc Testing for Chi-Square

The chi-square test only tells if any association exists.

It does not tell:

• Which categories differ
• Where the difference occurs

To dig deeper:

• Examine standardised residuals
• Perform pairwise chi-square tests with Bonferroni correction

Standardised Residuals

residuals = (table - expected) / np.sqrt(expected)print(residuals)

Cells with large |residual| (> ~2) indicate areas of significant difference.

Effect Sizes for Chi-Square Tests

Cramér’s V (common effect size)

def cramers_v(table):    chi2, p, dof, expected = stats.chi2_contingency(table)    n = table.sum()    k = min(table.shape) - 1    return np.sqrt(chi2 / (n * k))​print(cramers_v(table))​

Interpretation (Cohen’s guidelines):

Choosing Between Chi-Square and Other Tests