Chi-Square Test: Observed vs Expected Frequencies

This simulator makes the chi-square goodness-of-fit test tangible: pick a number of categories and a sample size, choose how "loaded" the true generator secretly is, then roll a random sample and watch three bars rise for every category — the expected count under fairness, the actual observed count, and a glowing amber rod showing exactly how much that category contributes to the total χ² statistic. Live readouts track the χ² statistic, degrees of freedom, an approximate p-value, and the α = 0.05 critical value, with a clear reject/fail-to-reject verdict — the same statistical machinery used to test whether a die is fair, a survey matches an expected split, or an A/B test's category counts diverge from chance.