Probability · interactive laboratory
Are the observed frequencies close enough to those predicted by a hypothesis? The chi-square test measures the discrepancies, but its conclusion depends on precise conditions.
The null hypothesis H₀ assigns a probability to each category before the sample is observed. H₁ says the distribution does not follow those probabilities. For each category we calculate the expected frequency E = N × p, then add the weighted discrepancies.
χ² = Σ (O − E)² / E
Observations must be independent, categories mutually exclusive and exhaustive, and every expected frequency at least 5. If a category is too small, combine it with an adjacent category before testing.
Here the theoretical probabilities are fixed independently of the data. In the Poisson example, λ = 2 is specified beforehand, not estimated from the sample. If you estimate parameters from the same data, the degrees of freedom change and this p-value calculation does not apply directly.
One line per category: name; observed count; theoretical probability. You can use 1/6 or 0.166667. The probabilities must sum to 1; small rounding differences are allowed.
For every category, E = N × p. Its contribution to χ² is (O − E)² / E; the sum of these contributions is the test statistic.
Failing to reject H₀ does not prove it is true. The p-value is not the probability that H₀ is true.
| Category | Observed O | Probability p | Expected E | Contribution |
|---|
With k categories and no parameters estimated from the sample: degrees of freedom = k − 1.
Further reading: NIST: chi-square goodness-of-fit test.