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Chi-squared distribution, showing χ2 on the x -axis and p -value (right tail probability) on the y -axis. A chi-squared test (also chi-square or χ2 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 ...
Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value.
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 , and the conditional distribution of Z given J = i is chi-squared with k + 2 i degrees of freedom.
The minimum chi-square estimate of the population mean λ is the number that minimizes the chi-square statistic. where a is the estimated expected number in the "> 8" cell, and "20" appears because it is the sample size. The value of a is 20 times the probability that a Poisson-distributed random variable exceeds 8, and it is easily calculated ...
The usual rule for deciding whether the chi-squared approximation is good enough is that the chi-squared test is not suitable when the expected values in any of the cells of a contingency table are below 5, or below 10 when there is only one degree of freedom (this rule is now known to be overly conservative [5]).
Chi distribution. In probability theory and statistics, the chi distribution is a continuous probability distribution over the non-negative real line. 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 ...
Given the null hypothesis that the observed frequencies result from random sampling from a distribution with the given expected frequencies, the distribution of G is approximately a chi-squared distribution, with the same number of degrees of freedom as in the corresponding chi-squared test.