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A distribution modeling sums of squared normal variables.
DESCRIPTION
The Chi-squared distribution arises as the distribution of the sum of squares of independent standard normal variables. It depends on degrees of freedom and is a special case of the Gamma distribution. It underpins tests like goodness-of-fit, independence in contingency tables, and variance inference. In ANOVA and regression, related F-statistics are ratios of scaled chi-squared variables.
ALTERNATIVE DEFINITIONS
Wikipedia
In probability theory and statistics, the chi-squared distribution is the distribution of a sum of the squares of k independent standard normal random variables.
Formal definition via sum of squared standard normals.
Wolfram MathWorld
The chi-squared distribution is the distribution of a sum of squares of independent standard normal random variables.
Equivalent definition succinctly stated.
RELATED TERMS
Gamma Distribution Normal Distribution Degrees of Freedom Goodness of Fit Variance Estimation Hypothesis Testing Variance Analysis Quadratic Form Likelihood Ratio Test
COMPETING TERMS
KNOWLEDGE AREAS
TARGET AUDIENCE
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EXTERNAL SOURCES
- Wikipedia - Chi-squared distribution
- Statlect - Definition and statistical applications
- Penn State STAT 414 - Course notes on Chi-squared
- NIST - NIST overview of Chi-square
TOP QUESTIONS
Where is the Chi-squared Distribution used?
In goodness-of-fit tests, contingency table tests for independence, and variance confidence intervals under normality.
Category: Example
How is it related to the Gamma Distribution?
Chi-squared with ν degrees of freedom equals Gamma(shape=ν/2, scale=2).
Category: Explanation
What’s a common pitfall in chi-square tests?
Using expected counts that are too small; combine categories or use exact tests when expected frequencies are low.
Category: Mistakes
How do degrees of freedom affect shape?
Low degrees produce heavy right skew; as ν increases, the distribution becomes more symmetric.
Category: Explanation
What distribution competes conceptually with Chi-squared in ANOVA?
The F-Distribution, which is a ratio of scaled chi-squared variables and used for comparing variances.
Category: Alternatives
How is Chi-squared used in model selection?
Likelihood ratio tests rely on asymptotic chi-squared distributions of test statistics under regularity conditions.
Category: How-To
Why is normality important for Chi-squared inference?
Many derivations assume normal errors; violations can inflate Type I error, suggesting robust or exact alternatives.
Category: Risk
Can Chi-squared approximate other distributions?
With appropriate scaling, sums of squared normals approximate Chi-squared; central limit style results justify asymptotics.
Category: Explanation
How to compute p-values for Chi-squared tests?
Use software CDF functions or tables with degrees of freedom and observed statistic.
Category: How-To