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 also studentized residual. The concept is named after William 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" are independent identically distributed random variables 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 the expected value and standard deviation of the normal distribution from which the sample is drawn. This probability distribution has applications to hypothesis testing and multiple 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.