Chi-square goodness-of-fit test

Tests whether the counts in k categories are consistent with a stated set of proportions.

X^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i},\qquad E_i = N p_i,\qquad df = k - 1 - m

$O_i$: Observed count in category i.
$E_i$: Expected count under H₀.
$p_i$: Proportion for category i stated by H₀.
$N$: Total count.
$m$: Number of parameters estimated from the data (e.g. 1 for a fitted Poisson mean; 0 when all p_i are given).

When to use

One categorical variable and a hypothesised distribution: is a die fair, do calls spread evenly over weekdays, do counts follow a fitted Poisson model? The p-value is the upper tail of $\chi^2_{df}$ .

Assumptions

Independent observations, each counted in exactly one category.
Counts, not percentages or means.
Expected counts large enough: all $E_i \ge 5$ is the classic rule (Cochran's: no more than 20% below 5, none below 1). Merge sparse categories or simulate otherwise.

How to interpret

A small p-value says the observed distribution differs from the hypothesised one; look at the contributions $(O-E)^2/E$ or residuals $(O-E)/\sqrt E$ to see where. A large p-value means the data are consistent with the model, not that the model is right.

Common mistakes

Running the test on percentages (the statistic depends on the counts).
Forgetting to subtract estimated parameters from the df.
Checking observed counts instead of expected counts against the "≥ 5" rule.

Example

60 rolls of a die: 8, 12, 9, 15, 10, 6. Each $E_i = 10$ . Contributions 0.4, 0.4, 0.1, 2.5, 0, 1.6, so $X^2 = 5.0$ on 5 df, $p = 0.42$ : no evidence that the die is unfair (60 rolls also have little power to detect a modest bias).