# Studentized range

In

statistics , the**studentized range**computed from a list "x"_{1}, ..., "x"_{"n"}of numbers is: $frac\{max\{,x\_1,dots,x\_n,\}\; -\; min\{,x\_1,dots,x\_n,\{s\},$

where

:$s^2\; =\; frac\{1\}\{n\; -\; 1\}sum\_\{i=1\}^n\; (x\_i\; -\; overline\{x\})^2,$

is the

sample variance and:$overline\{x\}\; =\; frac\{x\_1\; +\; cdots\; +\; x\_n\}\{n\}$

is the

sample mean .Generally, "studentized" means adjusted by dividing by an estimate of a population

standard deviation ; see alsostudentized residual . The concept is named afterWilliam Sealey Gosset , who wrote under the pseudonym "Student". The fact that the variance is a sample variance rather than the population variance, and thus something that differs from one random sample to the next, is essential to the definition, and complicates the problem of finding the probability distribution of any statistic that is studentized.If "X"

_{1}, ..., "X"_{"n"}areindependent identically distributed random variable s that are normally distributed, the probability distribution of their studentized range is what is usually called the**studentized range distribution**. This probability distribution is the same regardless of theexpected value andstandard deviation of the normal distribution from which the sample is drawn. This probability distribution has applications tohypothesis testing andmultiple comparisons .**References and further reading*** John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman, "Applied Linear Statistical Models", fourth edition, McGraw-Hill, 1996, page 726.

* John A. Rice, "Mathematical Statistics and Data Analysis", second edition, Duxbury Press, 1995, pages 451–452.

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