Last updated: 2026-06-23

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test asks a single question: do a set of observed category counts match the counts a hypothesis predicts? It is the standard tool for comparing one categorical sample against a theoretical distribution — checking whether a die is fair, whether genetic crosses follow a 9:3:3:1 ratio, or whether the colors in a bag of candy appear in the advertised proportions. You enter the counts you actually saw in each category and the counts expected under the null hypothesis, and the test returns a statistic, its degrees of freedom, and a p-value.

The statistic

The test summarizes the gap between observation and expectation with the Pearson statistic

$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}$

where $O_i$ is the observed count in category $i$, $E_i$ is the expected count, and $k$ is the number of categories. Each term measures how far one category strayed from its prediction. The deviation is squared so that surpluses and shortfalls both count as evidence against the hypothesis rather than cancelling out, and so that larger absolute gaps weigh more heavily. Dividing by $E_i$ normalizes each squared deviation: a miss of ten counts is a serious discrepancy when only five were expected but trivial when five thousand were expected, and dividing by the expected count puts every category on a comparable scale. Summing across categories gives a single number that grows as the data drift away from the hypothesis. When the observed counts equal the expected counts exactly, every term is zero and $\chi^2 = 0$.

Degrees of freedom and the p-value

The statistic is compared against a chi-square distribution whose shape is fixed by its degrees of freedom. For a simple goodness-of-fit test,

$\text{df} = k - 1$

One degree of freedom is spent because the category counts must add up to a known total: once the first k − 1 counts are fixed, the last is determined, so only k − 1 of them are free to vary. (If the expected distribution is itself estimated from the data using extra parameters, subtract one more degree of freedom per estimated parameter.)

The p-value is the upper-tail probability — the chance of seeing a statistic at least as large as the one observed if the null hypothesis were true. Goodness-of-fit is inherently a one-sided, upper-tail test: only large values of $\chi^2$ signal a poor fit, while small values mean the data sit close to the prediction. You compare the p-value against a significance level $\alpha$ (commonly five percent). When $p < \alpha$, the discrepancy is larger than sampling noise comfortably explains, and you reject the hypothesis that the data follow the expected distribution. When $p \ge \alpha$, the data are consistent with the hypothesis — which is not the same as proving it true.

Worked example

Suppose a store expects its three checkout lanes to handle traffic in a 40:40:20 split across 100 customers, and one afternoon it records 50, 30, and 20 customers. The expected counts are 40, 40, and 20. Then

With three categories the test has two degrees of freedom. The upper-tail p-value for $\chi^2 = 5$ at two degrees of freedom is about $0.082 \approx p$. Since $p \gt 0.05$, the observed split is consistent with the expected 40:40:20 distribution at the 5% level: the apparent imbalance between the first two lanes is within ordinary sampling variation.

Assumptions

The chi-square approximation rests on a few conditions. Expected counts should be reasonably large — a common rule of thumb requires every expected count to be at least 5; when several are smaller, combine sparse categories or use an exact test instead. Observations must be independent, each unit falling into exactly one category. And the test works on raw frequencies, not percentages or proportions: if a hypothesis is stated as proportions, multiply each by the total sample size to get expected counts before running the test, and confirm that the expected counts sum to the same total as the observed counts.

For comparing the means of two groups rather than categorical frequencies, see the Two-Sample t-Test Calculator (Welch); for turning a z-statistic into a tail probability, see P-Value from Z-Score Calculator.