chi-squared distribution

Key facts

Notation

χ 2 ( k ) {\displaystyle \chi ^{2}(k)\;} or χ k 2 {\displaystyle \chi _{k}^{2}\!}

Parameters

k ∈ N ∗ {\displaystyle k\in \mathbb {N} ^{*}~~} (known as "degrees of freedom")

Support

x ∈ ( 0 , + ∞ ) {\displaystyle x\in (0,+\infty )\;}

Pdf

1 2 k / 2 Γ ( k / 2 ) x ( k / 2 ) − 1 e − x / 2 {\displaystyle {\frac {1}{2^{k/2}\Gamma (k/2)}}\;x^{(k/2)-1}e^{-x/2}\;}

Cdf

1 Γ ( k / 2 ) γ ( k 2 , x 2 ) {\displaystyle {\frac {1}{\Gamma (k/2)}}\;\gamma {\left({\frac {k}{2}},\,{\frac {x}{2}}\right)}\;}

Mean

k {\displaystyle k}

Median

≈ k ( 1 − 2 9 k ) 3 {\displaystyle \approx k{\bigg (}1-{\frac {2}{9k}}{\bigg )}^{3}\;}

Mode

max ( k − 2 , 0 ) {\displaystyle \max(k-2,0)\;}

Variance

2 k {\displaystyle 2k\;}

Skewness

8 / k {\textstyle {\sqrt {8/k}}\,}

Excess kurtosis

12 k {\displaystyle {\frac {12}{k}}}

Entropy

k 2 + log ⁡ ( 2 Γ ( k 2 ) ) + ( 1 − k 2 ) ψ ( k 2 ) {\displaystyle {\begin{aligned}{\frac {k}{2}}&+\log \left(2\Gamma {\left({\frac {k}{2}}\right)}\right)\\&\!+\left(1-{\frac {k}{2}}\right)\psi {\left({\frac {k}{2}}\right)}\end{aligned}}}

Mgf

( 1 − 2 t ) − k / 2 {\displaystyle (1-2t)^{-k/2}} for t < 1 2 {\displaystyle t<{\tfrac {1}{2}}\;}

Cf

( 1 − 2 i t ) − k / 2 {\displaystyle (1-2it)^{-k/2}}

Pgf

( 1 − 2 ln ⁡ t ) − k / 2 {\displaystyle (1-2\ln t)^{-k/2}} for 0 < t < e {\displaystyle 0<t<{\sqrt {e}}\;}

via Wikipedia infobox

• [Pearson's chi-squared test].Tidsskrift for den Norske laegeforening : tidsskrift for praktisk medicin, ny raekke · 2019
• Sequential rerandomization.Biometrika · 2018
• Chi-squared test.Journal of clinical nursing · 2000
• A parametric meta-analysis.Statistics in medicine · 2019
• Uncertainty-Aware Spectral Visualization.IEEE transactions on visualization and computer graphics · 2025

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