Statistics Tools

Comprehensive Chi-Square Calculator

Compute χ² statistics, p-values, critical values, and visualize the chi-square distribution

Inputs

Choose a chi-square use-case and enter your data

This also determines how df is computed

For GOF: k−1−m; for independence: (r−1)(c−1)

Test statistic / evaluation point (x ≥ 0)

Most χ² tests use the right tail

Used for critical values and conclusion (default 0.05)

Probability & Critical Values

Optional: ranges, percentiles, and critical χ² values

Leave blank for open-ended

Leave blank for open-ended

Computes x such that P(X ≤ x) = percentile

For equal-tail critical values: α = 1 − CL/100

Quick Scenarios

Summary

Inputs, derived statistics, and interpretation

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Charts

PDF, CDF, and tail areas for the chi-square distribution

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Chi-Square Table & Critical Values

Common right-tail critical values and quantiles by df

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Reference

Formulas, interpretation, and usage notes

Chi-Square Distribution
PDF: f(x)=\(\frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2}\)
CDF: F(x)=P\(\frac{k}{2},\frac{x}{2}\) (regularized gamma)
Goodness-of-Fit / Independence Test Statistic
χ² = Σ (O − E)² / E
Variance Test (Normal)
χ² = (n − 1)s² / σ₀²,   df = n − 1

How to Use

Pick a mode

Choose the calculation type that matches your problem.

  • Given χ² & df: get p-values, percentiles, and critical values
  • GOF: compare observed vs expected counts (k categories)
  • Independence: paste an r×c table of observed counts
  • Variance: test or build a CI for variance under normality
Interpret p-values

Most χ² tests are right-tailed: large χ² indicates poor fit / dependence.

  • Right-tail p = P(X ≥ χ² | df)
  • If p < α: reject H₀
  • If p ≥ α: fail to reject H₀
Assumptions & cautions

Check conditions before relying on asymptotic χ² approximations.

  • Counts should be independent observations
  • Expected counts should generally not be too small
  • For variance mode, data should be approximately normal
  • Two-tailed p uses 2×min(tails) (common, but distribution is skewed)