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 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 random variable has a Studentized range distribution.

Definition

Probability density function

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.

Cumulative distribution function

The cumulative distribution function is given by

Special cases

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.

Uses

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.

Derivation of the studentized range distribution function

The studentized range distribution function arises from re-scaling the sample range R by the sample standard deviation s, since the studentized range is customarily tabulated in units of standard deviations, with the variable. The derivation begins with a perfectly general form of the distribution function of the sample range, which applies to any sample data distribution.
In order to obtain the distribution in terms of the "studentized" range q, we will change variable from R to s and q. Assuming the sample data is normally distributed, the standard deviation s will be distributed. By further integrating over s we can remove s as a parameter and obtain the re-scaled distribution in terms of q alone.

General form

For any probability density function f, the range probability density f is:
What this means is that we are adding up the probabilities that, given k draws from a distribution, two of them differ by r, and the remaining k − 2 draws all fall between the two extreme values.
If we change variables to u where is the low-end of the range, and define F as the cumulative distribution function of f, then the equation can be simplified:
We introduce a similar integral, and notice that differentiating under the integral-sign gives
which recovers the integral above, so that last relation confirms
because for any continuous cdf

Special form for normal data

The range distribution is most often used for confidence intervals around sample averages, which are asymptotically normally distributed by the central limit theorem.
In order to create the studentized range distribution for normal data, we first switch from the generic f and F to the distribution functions φ and Φ for the standard normal distribution, and change the variable r to s·q, where q is a fixed factor that re-scales r by scaling factor s:
Choose the scaling factor s to be the sample standard deviation, so that q becomes the number of standard deviations wide that the range is. For normal data s is chi distributed and the distribution function f of the chi distribution is given by:
Multiplying the distributions f and f and integrating to remove the dependence on the standard deviation s gives the studentized range distribution function for normal data:
where
The equation for the pdf shown in the sections above comes from using
to replace the exponential expression in the outer integral.