Studentized range

Instatistics,thestudentized range,denotedq,is the difference between the largest and smallest data in asamplenormalizedby thesample standard deviation. It is named afterWilliam Sealy Gosset(who wrote under the pseudonym "Student"), and was introduced by him in 1927. The concept was later discussed by Newman (1939),Keuls (1952),andJohn Tukeyin some unpublished notes. Its statistical distribution is thestudentized range distribution,which is used formultiple comparisonprocedures, such as the single step procedureTukey's range test,theNewman–Keuls method,and the Duncan's step down procedure, and establishingconfidence intervalsthat are still valid afterdata snoopinghas occurred.

Description

The value of thestudentized range,most often represented by the variableq,can be defined based on a random samplex₁,...,x_nfrom theN(0, 1) distribution of numbers, and another random variablesthat is independent of all thex_i,andνs²has aχ²distribution withνdegrees of freedom. Then

$q_{n,\nu }={\frac {\max\{\,x_{1},\ \dots,\ x_{n}\,\}-\min\{\,x_{1},\ \dots,\ x_{n}\}}{s}}=\max _{i,j=1,\dots,n}\left\{{\frac {x_{i}-x_{j}}{s}}\right\}$

has the Studentized range distribution forngroups andνdegrees of freedom. In applications, thex_iare typically the means of samples each of sizem,s²is thepooled variance,and the degrees of freedom areν=n(m− 1).

The critical value ofqis based on three factors:

1. α(the probability of rejecting a truenull hypothesis)
2. n(the number of observations or groups)
3. ν(the degrees of freedom used to estimate thesample variance)

Distribution

Main article:Studentized range distribution

IfX₁,...,X_nareindependent identically distributedrandom variablesthat arenormally distributed,the probability distribution of their studentized range is what is usually called thestudentized range distribution.Note that the definition ofqdoes not depend on theexpected valueor thestandard deviationof the distribution from which the sample is drawn, and therefore its probability distribution is the same regardless of those parameters.

Studentization

Main article:Studentization

Generally, the termstudentizedmeans that the variable's scale was adjusted by dividing by anestimateof a populationstandard deviation(see alsostudentized residual). The fact that the standard deviation is asamplestandard deviation rather than thepopulationstandard deviation, and thus something that differs from one random sample to the next, is essential to the definition and the distribution of theStudentizeddata. The variability in the value of thesamplestandard deviation contributes additional uncertainty into the values calculated. This complicates the problem of finding the probability distribution of any statistic that isstudentized.

See also

• Studentized range distribution
• Tukey's range test

Further reading

• Pearson, E.S.; Hartley, H.O. (1970)Biometrika Tables for Statisticians, Volume 1, 3rd Edition,Cambridge University Press.ISBN0-521-05920-8
• John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman (1996)Applied Linear Statistical Models,fourth edition, McGraw-Hill, page 726.
• John A. Rice (1995)Mathematical Statistics and Data Analysis,second edition, Duxbury Press, pages 451–452.
• Douglas C. Montgomery (2013) "Design and Analysis of Experiments", eighth edition, Wiley, page 98.