This article is about the mathematics of the chi-squared distribution. For its uses in statistics, see recurrence relation tutorial pdf-squared test. The chi-squared distribution is used primarily in hypothesis testing.

Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-squared distribution is not as often applied in the direct modeling of natural phenomena. It arises in the following hypothesis tests, among others. It is also a component of the definition of the t-distribution and the F-distribution used in t-tests, analysis of variance, and regression analysis. The primary reason that the chi-squared distribution is used extensively in hypothesis testing is its relationship to the normal distribution. Many hypothesis tests use a test statistic, such as the t-statistic in a t-test. A sample drawn at random from Z is a sample from the distribution shown in the graph of the standard normal distribution. Define a new random variable Q.

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To generate a random sample from Q, take a sample from Z and square the value. The subscript 1 indicates that this particular chi-squared distribution is constructed from only 1 standard normal distribution. A chi-squared distribution constructed by squaring a single standard normal distribution is said to have 1 degree of freedom. Thus, as the sample size for a hypothesis test increases, the distribution of the test statistic approaches a normal distribution, and the distribution of the square of the test statistic approaches a chi-squared distribution.

Lancaster shows the connections among the binomial, normal, and chi-squared distributions, as follows. De Moivre and Laplace established that a binomial distribution could be approximated by a normal distribution. 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 the normal or the chi-squared distribution. Further properties of the chi-squared distribution can be found in the box at the upper right corner of this article. For derivations of the pdf in the cases of one, two and k degrees of freedom, see Proofs related to chi-squared distribution. Tables of the chi-squared cumulative distribution function are widely available and the function is included in many spreadsheets and all statistical packages.