Chi-squared distribution

In probability theory and statistics, the chi-square distribution (also chi-squared or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square 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. When it is being distinguished from the more general noncentral chi-square distribution, this distribution is sometimes called the central chi-square distribution. In probability theory and statistics, the chi-square distribution (also chi-squared or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square 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. When it is being distinguished from the more general noncentral chi-square distribution, this distribution is sometimes called the central chi-square distribution. The chi-square distribution is used in the common chi-square tests for goodness of fit of an observed distribution to a theoretical one, the independence of two criteria of classification of qualitative data, and in confidence interval estimation for a population standard deviation of a normal distribution from a sample standard deviation. Many other statistical tests also use this distribution, such as Friedman's analysis of variance by ranks. If Z1, ..., Zk are independent, standard normal random variables, then the sum of their squares, is distributed according to the chi-square distribution with k degrees of freedom. This is usually denoted as The chi-square distribution has one parameter: k, a positive integer that specifies the number of degrees of freedom (the number of Zi’s). The chi-square distribution is used primarily in hypothesis testing, and to a lesser extent for confidence intervals for population variance when the underlying distribution is normal. Unlike more widely known distributions such as the normal distribution and the exponential distribution, the chi-square 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-square 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. For these hypothesis tests, as the sample size, n, increases, the sampling distribution of the test statistic approaches the normal distribution (central limit theorem). Because the test statistic (such as t) is asymptotically normally distributed, provided the sample size is sufficiently large, the distribution used for hypothesis testing may be approximated by a normal distribution. Testing hypotheses using a normal distribution is well understood and relatively easy. The simplest chi-square distribution is the square of a standard normal distribution. So wherever a normal distribution could be used for a hypothesis test, a chi-square distribution could be used. Suppose that Z is a random variable sampled from the standard normal distribution, where the mean equals to 0 and the variance equals to 1: Z ~ N(0,1). Now, consider the random variable Q = Z2. The distribution of the random variable Q is an example of a chi-square distribution:   Q   ∼   χ 1 2 . {displaystyle Q sim chi _{1}^{2}.} The subscript 1 indicates that this particular chi-square distribution is constructed from only 1 standard normal distribution. A chi-square 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-square distribution. Just as extreme values of the normal distribution have low probability (and give small p-values), extreme values of the chi-square distribution have low probability.