Continuous probability distribution

Chi-squared (χ²) distribution

X ~ χ²(k)

The sum of k squared independent standard-normal variables. It is the distribution behind variance estimation and the chi-square tests of goodness-of-fit and independence.

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Key facts

Notation: X ~ χ²(k)
Type: continuous
Parameters: k — degrees of freedom
Support: [0, ∞)
PDF: f(x) = 1 / (2^(k/2) Γ(k/2)) · x^(k/2 − 1) e^(−x/2)
Mean: k
Variance: 2k

When to use it

Chi-square test of independence
Goodness-of-fit tests
Confidence intervals for a variance

Related distributions

Standard Normal (Z)F (Fisher–Snedecor)

Frequently asked questions

What is the Chi-squared (χ²) distribution?

The sum of k squared independent standard-normal variables. It is the distribution behind variance estimation and the chi-square tests of goodness-of-fit and independence.

What are the mean and variance of the Chi-squared (χ²) distribution?

For X ~ χ²(k), the mean is k and the variance is 2k.

When is the Chi-squared (χ²) distribution used?

It is commonly used for: Chi-square test of independence; Goodness-of-fit tests; Confidence intervals for a variance.

All distribution guides

Bernoulli Binomial Poisson Discrete Uniform Continuous Uniform Normal (Gaussian)Standard Normal (Z)Exponential Chi-squared (χ²)F (Fisher–Snedecor)

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Learn the concept

Degrees of freedom, explained