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Chi-Squared Distribution (χ²₍k₎) ○◂｜Definition｜1st｜20260608000813-00-⌔

Chi-squared distribution - Wikipedia

Chi-squared distribution

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In probability theory and statistics, the ${\chi }^2$-distribution with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.¹

The chi-squared distribution ${\chi }_k^2$ is a special case of the gamma distribution and the univariate Wishart distribution. Specifically if $X\sim {\chi }_k^2$ then $X\sim \text{Gamma}(\alpha =\frac{k}{2},\theta =2)$ (where $\alpha$ is the shape parameter and $\theta$ the scale parameter of the gamma distribution) and $X\sim {\text{W}}_1(1,k)$.

The scaled chi-squared distribution $s^2{\chi }_k^2$ is a reparametrization of the gamma distribution and the univariate Wishart distribution. Specifically if $X\sim s^2{\chi }_k^2$ then $X\sim \text{Gamma}(\alpha =\frac{k}{2},\theta =2s^2)$ and $X\sim {\text{W}}_1(s^2,k)$.

The chi-squared distribution is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.²³⁴⁵ This distribution is sometimes called the central chi-squared distribution, a special case of the more general noncentral chi-squared distribution.⁶

The chi-squared distribution is used in the common chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in finding the confidence interval for estimating the population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman’s analysis of variance by ranks.

Printed 2026-06-28.

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Footnotes

1. Weisstein, Eric W. “Chi-Squared Distribution”. mathworld.wolfram.com. Retrieved 2024-10-11. ↩

2. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. “Chapter 26”. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 940. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. ↩

3. NIST (2006). Engineering Statistics Handbook – Chi-Squared Distribution ↩

4. Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). “Chi-Square Distributions including Chi and Rayleigh”. Continuous Univariate Distributions. Vol. 1 (Second ed.). John Wiley and Sons. pp. 415–493. ISBN 978-0-471-58495-7. ↩

5. Mood, Alexander; Graybill, Franklin A.; Boes, Duane C. (1974). Introduction to the Theory of Statistics (Third ed.). McGraw-Hill. pp. 241–246. ISBN 978-0-07-042864-5. ↩

6. “The Chi-Squared Distribution” (PDF). University of Regina. ↩

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