Chi-Square Test: Distribution & Category Rose (2D)

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

This is a 2D-native reading of the same chi-square goodness-of-fit test: instead of stacking 3D bars, the categories are laid out as a polar rose — one petal per category, petal length = observed count, with a dashed reference ring at the expected (uniform) count and an amber outer tick showing that category's (Oᵢ − Eᵢ)²/Eᵢ contribution.

Below it is the actual χ² sampling distribution curve for the current degrees of freedom — something the 3D bar view never shows directly. The shaded tail is the p-value region (P(X ≥ observed χ²) under H₀), the dashed red line marks the α = 0.05 critical value, and the solid marker is where your rolled sample landed.

χ² = Σᵢ (Oᵢ − Eᵢ)² / Eᵢ,  df = k − 1
f(x; df) = x^(df/2−1) e^(−x/2) / (2^(df/2) Γ(df/2))

The p-value readout uses the Wilson–Hilferty cube-root normal approximation (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 (petals in the rose), 3–8.
N slider — how many random draws make up one sample; larger N gives the test more power.
Bias slider — how far the true generating probabilities deviate from uniform. The test never sees this directly, only the resulting counts.
Roll New Sample — draws N fresh observations and recomputes the rose, the curve marker, and every statistic.
If the χ² marker falls to the right of the red critical line (equivalently p < 0.05), the test rejects H₀.

Expected ring (E) Observed petal (O) (O−E)²/E tick