Chi-Square Goodness of Fit

The chi-square test

Tests whether observed frequencies match expected frequencies.

$\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}$

Step-by-step

Observed: {18, 22, 16, 20, 24, 20}. Total = 120. If fair: Expected = 20 each.

Step 1 — Compute $\chi^2$:

$\chi^2 = \frac{(18-20)^2}{20} + \frac{(22-20)^2}{20} + \frac{(16-20)^2}{20} + \frac{0}{20} + \frac{(24-20)^2}{20} + \frac{0}{20}$

$= 0.2 + 0.2 + 0.8 + 0 + 0.8 + 0 = 2.0$

Step 2 — Degrees of freedom: $df = 6 - 1 = 5$.

Step 3 — Critical value at $\alpha = 0.05$: $\chi^2_{\text{crit}} = 11.07$.

Step 4 — Decision: $\chi^2 = 2.0 < 11.07$. Fail to reject $H_0$. No evidence the die is unfair.

Try it in the visualization

Bar chart shows observed vs expected side by side. The chi-square value and p-value update as you change observed counts.