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Chi-Square Distribution

NIST/SEMATECH Section 1.3.6.6.6 Chi-Square Distribution

PDF

Chi2(k=5) PDF 0 2 4 6 8 10 12 14 16 x 0 0.05 0.1 0.15 f(x)

CDF

Chi2(k=5) CDF 0 2 4 6 8 10 12 14 16 x 0 0.2 0.4 0.6 0.8 1 F(x)

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PDF

CDF

Formulas

Probability Density Function

f(x)=xk/2−1 e−x/22k/2 Γ(k/2),x>0f(x) = \frac{x^{k/2-1}\,e^{-x/2}}{2^{k/2}\,\Gamma(k/2)}, \quad x > 0

Cumulative Distribution Function

F(x)=γ(k/2, x/2)Γ(k/2)F(x) = \frac{\gamma(k/2,\, x/2)}{\Gamma(k/2)}

Properties

Mean

kk

Variance

2k2k

Overview

The chi-square distribution with kk degrees of freedom is the distribution of a sum of squares of kk independent standard normal random variables. It is fundamental to hypothesis testing and confidence interval estimation.

Related Distributions

Normal Distribution Gamma Distribution F-Distribution