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

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

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

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

Outputs: