χ2-tests

Further Maths · Further Probability & Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A χ²-test compares the frequencies you observed (O) with the frequencies a hypothesis says you should expect (E). If the hypothesis is true, O and E should be close and the test statistic is small. A large statistic is evidence against the hypothesis, so the test always uses the upper tail.

In a goodness of fit test the hypothesis is that the data follow a stated distribution, such as a fair die, a binomial or a Poisson. In a test for independence the data are in a contingency table and the hypothesis is that the two factors are not associated.

The critical value depends on the degrees of freedom. Every expected frequency must be at least 5, so small classes are combined first.

Rules to remember

Test statistic: χ² = Σ(O − E)²/E, summed over all classes or cells; the totals of O and E must be equal
Goodness of fit: E = total frequency × probability of the class under H₀; the last class is usually 'this value or more', so the probabilities add to 1
Goodness of fit: ν = (number of classes after combining) − 1 − (number of parameters estimated from the data)
Contingency table: E = (row total × column total) ÷ grand total, and ν = (rows − 1)(columns − 1)
If any E is less than 5, combine neighbouring classes (or rows or columns) before working out χ² and ν
H₀: the distribution fits (or the factors are independent). Reject H₀ if χ² is greater than the critical value

Common mistake

Students count the degrees of freedom before combining classes, or forget to subtract 1 for each parameter estimated from the data (for example a Poisson mean worked out from the sample). Combine first, then count the classes that are left.

Worked example

A model predicts that three types of offspring occur in the ratio 1 : 2 : 1. In a sample of 80 offspring the observed frequencies are 14, 46 and 20. Test at the 5% significance level whether the model fits. The critical value for ν = 2 is 5.991.

Expected frequencies: 80 × 1/4 = 20, 80 × 2/4 = 40, 80 × 1/4 = 20.
χ² = (14 − 20)²/20 + (46 − 40)²/40 + (20 − 20)²/20 = 1.8 + 0.9 + 0 = 2.7.
ν = 3 − 1 = 2 (no parameters estimated). 2.7 < 5.991, so the result is not in the critical region.

Answer: χ² = 2.7; do not reject H₀. The data are consistent with the ratio 1 : 2 : 1.

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