What is the chi-square distribution?

The chi-square distribution with df degrees of freedom is the distribution of a sum of df independent squared standard normal random variables. It is a special case of the Gamma distribution with shape k = df/2 and scale θ = 2.

What is the p-value in a chi-square test?

The p-value is the right-tail probability P(χ² ≥ x), i.e. the probability of observing a test statistic at least as extreme as x under the null hypothesis. A p-value below your significance level α (commonly 0.05) leads to rejecting the null.

What is a critical value?

The critical value is the chi-square statistic threshold for a given significance level α such that P(χ² ≥ critical value) = α. If your test statistic exceeds the critical value, you reject the null hypothesis at level α.

How are degrees of freedom determined?

Degrees of freedom depend on the test. For a goodness-of-fit test with k categories, df = k − 1 (minus additional estimated parameters). For a two-way contingency table with r rows and c columns, df = (r − 1)(c − 1).

How is the CDF computed?

The CDF uses the regularized lower incomplete gamma function: P(df/2, x/2). This calculator evaluates it via a convergent series expansion for small x and a continued-fraction method for large x, giving accuracy to approximately 10 significant figures.

How is the critical value (inverse CDF) found?

The inverse CDF is found by bisection search: the algorithm brackets the target cumulative probability and narrows the interval until the result is accurate to 8 significant figures.

Chi-Square Distribution Calculator

Name: Chi-Square Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter a chi-square statistic and degrees of freedom to get the PDF, CDF (p-value), and critical value at any significance level.

Parameters

Chi-Square Statistic (χ²) (≥ 0)

The observed chi-square test statistic value.

Degrees of Freedom (df) (≥ 1)

Positive integer; typically (categories − 1) for goodness-of-fit.

Critical Value Finder

Significance Level α (0 < α < 1)

Right-tail probability; common values: 0.10, 0.05, 0.01.

Quick Presets

Results at χ² = x

Enter parameters and press Calculate.

Distribution Statistics

Enter parameters and press Calculate.

Key Formulas

PDF:: f(x) = x^df/2−1 e^−x/2 / (2^df/2 Γ(df/2))
CDF:: F(x) = P(df/2, x/2) — regularized incomplete gamma
p-value:: 1 − F(x) = P(χ² ≥ x)
Mean:: df
Variance:: 2 · df
Mode:: max(df − 2, 0)

Summary

Enter a chi-square statistic and degrees of freedom to get the PDF, CDF (p-value), and critical value at any significance level.

How it works

Enter the chi-square statistic (χ² value) — must be ≥ 0.
Enter the degrees of freedom (df) — a positive integer, typically n−1 for goodness-of-fit tests.
Press Calculate to get the PDF and CDF at your statistic.
The p-value shown is P(χ² ≥ x) — the right-tail probability used in most hypothesis tests.
Enter a significance level α (e.g. 0.05) to find the critical value x such that P(χ² ≥ x) = α.
Use the preset buttons to load common textbook examples instantly.

Use cases

Chi-square goodness-of-fit tests to check if observed counts match expected distributions.
Chi-square test of independence for contingency tables.
Confidence intervals for a population variance or standard deviation.
Likelihood-ratio tests in regression and model comparison.
Quality control: checking whether process variance exceeds a specification limit.
Genomics: testing Hardy-Weinberg equilibrium in allele frequency data.
Statistics coursework: verifying p-values and critical values from textbook problems.
Machine learning: feature selection using chi-square statistics for categorical data.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu