Chi-Square Test: Observed vs Expected Frequencies

Sample Generator

Categories (k) 6

Sample size (N) 300

True-generator bias 0%

Test Results

Fail to reject H₀

χ² statistic

0.00

Degrees of freedom

p-value

1.000

Critical χ² (α=0.05)

11.07

How it works

The chi-square goodness-of-fit test checks whether observed category counts are consistent with an expected (null-hypothesis) distribution — here the uniform distribution E_i = N/k, i.e. "the generator is fair":

χ² = Σᵢ (Oᵢ − Eᵢ)² / Eᵢ
df = k − 1

Each category contributes (Oᵢ − Eᵢ)²/Eᵢ to the total — the amber rod behind each bar pair shows that contribution directly, so you can see which category is driving the statistic.

The p-value is the probability of seeing a χ² this large (or larger) if H₀ (fairness) were actually true. It is computed from the χ² sampling distribution via the Wilson–Hilferty cube-root normal approximation, which is accurate to a few thousandths for df ≥ 2:

z = ((χ²/df)^(1/3) − (1 − 2/(9·df))) / √(2/(9·df))
p ≈ 1 − Φ(z)

k slider — number of categories (e.g. faces of a die), 3–8.
N slider — how many random draws make up one sample; larger N gives the test more power to detect a small bias.
Bias slider — how far the true generating probabilities deviate from uniform (0% = perfectly fair, 100% = heavily loaded toward category 1). This is what the test is trying to detect — it never sees the bias value directly, only the resulting counts.
Roll New Sample — draws N fresh random observations from the true (possibly biased) distribution and recomputes everything.
If χ² exceeds the critical value at α = 0.05 (equivalently p < 0.05), the test rejects H₀ — the deviation is unlikely to be due to chance alone.

Real-world relevance: this exact test is used to check whether a die or coin is fair, whether survey responses match an expected demographic split, or whether an A/B-test's category counts (clicks vs. no-clicks across variants) diverge from what random chance would produce.

Expected (E) Observed (O) (O−E)²/E contribution

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