Math · Introductory statistics · Concept

The chi-square goodness-of-fit test

A goodness-of-fit test asks whether observed counts in categories match a claimed distribution, such as a fair die or a 3:1 genetic ratio. Each category contributes (observed − expected)²/expected, and the total χ² follows a chi-square distribution with k − 1 degrees of freedom when the model is right. A small p-value means the counts fit the model poorly.

Expected counts

Multiply the total count by each category’s claimed proportion. Expected counts need not be whole numbers.

E_{i} = n p_{i}

The statistic

Each term measures one category’s mismatch relative to its expected size. χ² is never negative, and it is 0 only when every count matches exactly.

χ^{2} = \sum \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Degrees of freedom and the p-value

With k categories, df = k − 1: once k − 1 counts are known, the total fixes the last. The p-value is the right-tail area beyond χ² under the chi-square curve.

Conditions

The counts must come from independent observations, and every expected count should be at least 5 for the chi-square approximation to hold; combine sparse categories if needed.

What a result means

A small p-value says the model does not fit. A large p-value says the data are consistent with the model, not that the model is proven.

Common mistakes

Using percentages or proportions instead of counts.
Dividing by the observed count instead of the expected count.
Using k instead of k − 1 degrees of freedom.
Running the test with expected counts below 5.

Key terms

Chi-square goodness-of-fit test: A test of whether observed counts in categories match the counts a model expects, using χ² = Σ(observed − expected)²/expected. It needs expected counts that aren’t too small, usually at least 5 each.
Chi-square distribution: A right-skewed distribution of nonnegative values, set by its degrees of freedom. It is used for chi-square tests on counts and for inference about variances.
Expected count: The count a model predicts on average, such as total × category probability. It doesn’t have to be a whole number, and a real sample will usually differ from it.
Degrees of freedom: The number of values free to vary once estimates have been fixed, such as n − 1 for one sample’s standard deviation. It sets the shape of the t, chi-square and F distributions.
p-value: The probability, assuming the null hypothesis is true, of a result at least as extreme as the one observed. A small p-value is evidence against the null; it is not the probability that the null is true.
Null hypothesis: H₀, the claim a test assumes so it can compute probabilities, usually “no effect” or a specific value such as μ = 50. Failing to reject it doesn’t prove it true.

Work through an example

A die is rolled 120 times, giving 25 ones, 17 twos, 15 threes, 23 fours, 24 fives and 16 sixes. Is this consistent with a fair die at α = 0.05?

Test whether a die is fair →

Test a 3:1 genetic ratio →

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