Qui-Quadrado (Goodness-of-Fit)

Chi-square goodness-of-fit test

Pearson's chi-square goodness-of-fit test, from 1900, asks whether an observed categorical distribution lines up with a theoretical one: χ² = Σ (O − E)² / E, where O is the observed and E the expected frequency in each category. The degrees of freedom come from df = k − 1 − p, with k categories and p parameters estimated from the data. You reject H₀ (that the data fits the model) once χ² climbs past the critical value χ²_crit in the chi-square table. Picture a die rolled 600 times: if it is fair, E = 100 per face, and a large χ² is a sign the die is loaded. As a working rule, keep each E ≥ 5 so the approximation holds up.

Applications

It turns up wherever categorical data needs checking: genetics, where you test Mendelian segregation ratios such as 9:3:3:1; market research and survey responses; A/B tests comparing proportions across several variants; validating pseudo-random number generators; and confirming that a sample reflects the distribution recorded in a population census.

FAQ

Goodness-of-fit or test of independence? Goodness-of-fit pits a single variable against a theoretical distribution. The test of independence compares two variables using a contingency table.

What if some expected counts are below 5? The chi-square approximation starts to break down. Merge the small categories together, or switch to Fisher's exact test or a Monte Carlo p-value.

How are degrees of freedom counted? Begin at k − 1, since one constraint already fixes the totals to match. Then drop one more df for every parameter you estimate from the sample, such as a Poisson mean.