Chi-Square Test Calculator

Intro: Computes chi-square statistics for goodness-of-fit and independence, including degrees of freedom, p-value, and conclusion.

Worked example

• A die is rolled 60 times with observed counts: 8, 10, 9, 11, 12, 10. Test if the die is fair at α = 0.05.



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• We are testing whether the die is fair, so we use a chi-square goodness-of-fit test with equal expected probabilities.



• Total rolls are $N = 8+10+9+11+12+10 = 60$.



• If the die is fair, the probability for each face is $1/6$, so the expected count for each face is $E_i = 60/6 = 10$.



• The chi-square formula is $\chi^2 = \sum \dfrac{(O_i - E_i)^2}{E_i}$ where $O_i$ are observed counts and $E_i$ expected counts.



• Compute contributions for each face: 1: $(8-10)^2/10 = 4/10 = 0.4$ 2: $(10-10)^2/10 = 0$ 3: $(9-10)^2/10 = 1/10 = 0.1$



• Continue: 4: $(11-10)^2/10 = 1/10 = 0.1$ 5: $(12-10)^2/10 = 4/10 = 0.4$ 6: $(10-10)^2/10 = 0$.



• Add them: $\chi^2 = 0.4+0+0.1+0.1+0.4+0 = 1.0$.



• Degrees of freedom are $df = k-1 = 6-1 = 5$ (k = number of categories).



• Using a chi-square distribution with df = 5, the p-value for $\chi^2 = 1.0$ is large (around $p \approx 0.96$).



• Since $p > 0.05$, we fail to reject $H_0$ and conclude there is no evidence that the die is unfair.



• Answer: $\boxed{\chi^2 = 1.0,\; df=5,\; p \approx 0.96 \Rightarrow \text{fail to reject }H_0.}$

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How this calculator works

• Type or paste your function (LaTeX like \sin, \ln works too).
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Related

• Hardy–Weinberg Equilibrium
• Variance & Standard Deviation