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Approximate formula for median (from the Wilson–Hilferty transformation) compared with numerical quantile (top); and difference (blue) and relative difference (red) between numerical quantile and approximate formula (bottom). For the chi-squared distribution, only the positive integer numbers of degrees of freedom (circles) are meaningful.
Where and are the cdf and pdf of the corresponding random variables. Then Y = X 2 ∼ χ 1 2 . {\displaystyle Y=X^{2}\sim \chi _{1}^{2}.} Alternative proof directly using the change of variable formula
A chi-squared test (also chi-square or χ 2 test) is a statistical hypothesis test used in the analysis of contingency tables when the sample sizes are large. In simpler terms, this test is primarily used to examine whether two categorical variables ( two dimensions of the contingency table ) are independent in influencing the test statistic ...
It is the distribution of the positive square root of a sum of squared independent Gaussian random variables. Equivalently, it is the distribution of the Euclidean distance between a multivariate Gaussian random variable and the origin. The chi distribution describes the positive square roots of a variable obeying a chi-squared distribution.
The chi-squared statistic can then be used to calculate a p-value by comparing the value of the statistic to a chi-squared distribution. The number of degrees of freedom is equal to the number of cells , minus the reduction in degrees of freedom, . The chi-squared statistic can be also calculated as
From this representation, the noncentral chi-squared distribution is seen to be a Poisson-weighted mixture of central chi-squared distributions. Suppose that a random variable J has a Poisson distribution with mean λ / 2 {\displaystyle \lambda /2} , and the conditional distribution of Z given J = i is chi-squared with k + 2 i degrees of freedom.
revert: user:Niceoboe had replaced the pdf with the cdf: 12:52, 26 April 2013: 600 × 400 (23 KB) Niceoboe: Changed path ids (more than one named curve-1l) to pass conformity test at validator.w3.org. 17:21, 31 March 2010: 600 × 400 (23 KB) Geek3: chi square distribution Category:Chi-square distribution
In statistics, the reduced chi-square statistic is used extensively in goodness of fit testing. It is also known as mean squared weighted deviation ( MSWD ) in isotopic dating [ 1 ] and variance of unit weight in the context of weighted least squares .