Independence vs. Homogeneity

You’ll be able to

Distinguish a test of independence from a test of homogeneity
Match the test to the sampling design
State appropriate hypotheses for each

Same arithmetic, different design

The chi-square test for independence and the test for homogeneity use the identical statistic, expected-count formula, and degrees of freedom — what differs is the study design and the hypotheses. The distinction is about how the data were collected, so read the sampling scheme carefully.

How to tell them apart

Independence: one random sample is taken and two categorical variables are recorded on each individual; H₀ says those two variables are independent. Homogeneity: several separate samples (or groups/populations, or treatment groups in an experiment) are compared on one categorical variable; H₀ says the distribution of that variable is the same across all the groups.

Choosing the test

one sample, two variables → INDEPENDENCE · multiple samples/groups, one variable → HOMOGENEITY

Both use χ² = Σ(O−E)²/E with df = (r−1)(c−1). Only the design and the wording of the hypotheses change.

Worked example

Design A: 500 randomly chosen voters are each classified by party and by opinion on a law. Design B: 200 Republicans, 200 Democrats, and 200 Independents are separately sampled and each classified by opinion. Which chi-square test fits each?

1.Design A: a single sample of 500 with two variables (party and opinion) recorded on each person.
2.One sample, two variables → test for independence; H₀: party and opinion are independent.
3.Design B: three separate samples, one drawn from each party, compared on a single variable (opinion).
4.Multiple samples, one variable → test for homogeneity; H₀: the distribution of opinion is the same across the three parties.

Answer: Design A is a test for independence (one sample, two variables recorded). Design B is a test for homogeneity (three separate samples compared on one variable). The calculations are identical; only the design and hypotheses differ.

Checkpoint

A researcher takes separate random samples of 150 first-years, 150 sophomores, and 150 juniors and records each student’s favorite class format. Which chi-square test is appropriate?

On the exam

Decide independence vs. homogeneity by counting samples: one random sample with two variables → independence; several samples/groups compared on one variable → homogeneity. State the matching hypotheses — "independent" vs. "same distribution across groups" — to earn the setup point.

Checkpoint

For a chi-square test of homogeneity, the null hypothesis is best stated as:

Answer the 2 checkpoints as you read.