Chi-square (χ2) Distribution — Tests, Formula, Table [Worked Examples]

by Jack Bodeley on September 28, 2021

The chi-square (χ2) test is a test of discrepancies between observed frequencies and a corresponding set of expected frequencies derived from a null hypothesis.

Chi-square (χ2) Tests — Goodness of Fit v Test of Independence

One-variable Goodness of fit χ2 Tests

When data is distributed along a single qualitative variable, the one-variable χ2 test evaluates these discrepancies as a "goodness of fit" test.

A one-variable χ2 test evaluates whether observed frequencies for a single qualitative variable are adequately described by hypothesized or expected frequencies.

One-variable χ2 tests are designed to evaluate the adequacy with which observed frequencies are described by hypothesized or expected frequencies i.e. goodness of fit.

Observations are classified in only one way.

Two-variables Test of independence χ2 Tests

When data are cross-classified along two qualitative variables, the two-variable χ2 test evaluates these discrepancies as a "test of independence" or lack of predictability between the two qualitative variables.

Observations are classified in two ways, i.e. cross-classified according to two qualitative variables.


$$\chi^{2}= \sum \frac{(observed\ frequency - expected\ frequency)^2}{expected\ frequency}$$

Steps in χ2 Hypothesis Testing

#1 — Determine the Statistical Hypothesis (Null and ALternate)

#2 — Determine the Decision Rule

#3 — Calculate χ2

#4 — Make a Decision

#5 — Make an Interpretation