Use this tag for questions about (1) distributions of a sum of squares of independent standard normal random variables or (2) statistical hypothesis tests with such a sampling distribution if the null hypothesis is true.
Distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or χ$^2$ distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing or in construction of confidence intervals.
Test
A chi-squared test is any statistical hypothesis test for which the sampling distribution of the test statistic is a chi-squared distribution if the null hypothesis is true. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories.
In standard applications of the test, observations are classified into mutually exclusive classes, and there is a null hypothesis that gives the probability that any observation falls into the corresponding class. The purpose of the test is to evaluate how likely the observations would be assuming the null hypothesis is true.
Chi-squared tests are often constructed from a sum of squared errors or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally-distributed data, which is valid in many cases. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent.
