Applications
The chi-squared distribution has numerous applications in inferential statistics, for instance in chi-squared tests and in estimating variances. It enters the problem of estimating the mean of a normally distributed population and the problem of estimating the slope of a regression line via its role in Student’s t-distribution. It enters all analysis of variance problems via its role in the F-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.
Following are some of the most common situations in which the chi-squared distribution arises from a Gaussian-distributed sample.
- if X1, ..., Xn are i.i.d. N(μ, σ2) random variables, then where .
- The box below shows probability distributions with name starting with chi for some statistics based on Xi ∼ Normal(μi, σ2i), i = 1, ⋯, k, independent random variables:
| Name | Statistic |
|---|---|
| chi-squared distribution | |
| noncentral chi-squared distribution | |
| chi distribution | |
| noncentral chi distribution |
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