Studentized range distribution

In probability and statistics, studentized range distribution is the continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population. Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ2) and suppose that is the smallest of these sample means and is the largest of these sample means, and suppose s2 is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution. Differentiating the cumulative distribution function with respect to q gives the probability density function. Note that in the outer part of the integral, the equation was used to replace an exponential factor. The cumulative distribution function is given by If k is 2 or 3, the studentized range probability distribution function can be directly evaluated, where is the standard normal probability density function and is the standard normal cumulative distribution function. When the degrees of freedom approaches infinity the studentized range cumulative distribution can be calculated for any k using the standard normal distribution. Critical values of the studentized range distribution are used in Tukey's range test. The studentized range is used to calculate significance levels for results obtained by data mining, where one selectively seeks extreme differences in sample data, rather than only sampling randomly. The Studentized range distribution has applications to hypothesis testing and multiple comparisons procedures. For example, Tukey's range test and Duncan's new multiple range test (MRT), in which the sample x1, ..., xn is a sample of means and q is the basic test-statistic, can be used as post-hoc analysis to test between which two groups means there is a significant difference (pairwise comparisons) after rejecting the null hypothesis that all groups are from the same population (i.e. all means are equal) by the standard analysis of variance. When only the equality of the two groups means is in question (i.