View source for Studentized range

In [[Statistics|statistics]], the '''studentized range''', denoted ''q'', is the difference between the largest and smallest data in a [[Sample (statistics)|sample]] [[Normalization (statistics)|normalized]] by the [[Standard deviation|sample standard deviation]].
It is named after [[Biography:William Sealy Gosset|William Sealy Gosset]] (who wrote under the pseudonym "''Student''"), and was introduced by him in 1927.<ref>{{cite journal |author=Student |title= Errors of routine analysis|journal= Biometrika|volume=19 |issue= 1/2|pages=151–164 |year=1927 | doi = 10.2307/2332181 |jstor= 2332181}}</ref> 
The concept was later discussed by Newman (1939),<ref>{{cite journal |author= Newman D. |title= The Distribution of Range in Samples from a Normal Population Expressed in Terms of an Independent Estimate of Standard Deviation |journal= Biometrika |volume=31 |issue= 1–2 |pages=20–30 |year=1939 |doi=10.1093/biomet/31.1-2.20}}</ref> Keuls (1952),<ref>{{cite journal |author= Keuls M.|title= The Use of the "Studentized Range" in Connection with an Analysis of Variance |journal= Euphytica |volume=1 |issue= 2 |pages=112–122 |year=1952 |doi=10.1007/bf01908269|s2cid= 19365087 }}</ref> and [[Biography:John Tukey|John Tukey]] in some unpublished notes.
Its statistical distribution is the ''[[Studentized range distribution|studentized range distribution]]'', which is used for [[Multiple comparisons problem|multiple comparison]] procedures, such as the single step procedure [[Tukey's range test]], the [[Newman–Keuls method]], and the Duncan's step down procedure, and establishing [[Confidence interval|confidence interval]]s that are still valid after data snooping has occurred.<ref>{{cite journal |author=John A. Rafter |title=Multiple Comparison Methods for Means |journal=SIAM Review|volume=44 |issue=2 |pages=259–278 |year=2002 |doi=10.1137/s0036144501357233|citeseerx=10.1.1.132.2976 |bibcode=2002SIAMR..44..259R }}</ref>

== Description ==
The value of the '''studentized range''', most often represented by the variable ''q'', can be defined based on a random sample ''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> from the ''N''(0, 1) distribution of numbers, and another random variable ''s'' that is independent of all the ''x<sub>i</sub>'', and ''νs''<sup>2</sup> has a ''χ''<sup>2</sup> distribution with ''ν'' degrees of freedom. Then

: <math>
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\}</math>

has the Studentized range distribution for ''n'' groups and ''ν'' degrees of freedom. In applications, the ''x<sub>i</sub>'' are typically the means of samples each of size ''m'', ''s''<sup>2</sup> is the [[Pooled variance|pooled variance]], and the degrees of freedom are&nbsp;''ν''&nbsp;=&nbsp;''n''(''m''&nbsp;−&nbsp;1).

The critical value of ''q'' is based on three factors:
#''α'' (the probability of rejecting a true [[Null hypothesis|null hypothesis]])
#''n'' (the number of observations or groups) 
#''ν'' (the degrees of freedom used to estimate the [[Sample variance|sample variance]])

== Distribution ==
{{main|Studentized range distribution}}

If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are independent identically distributed [[Random variable|random variable]]s that are [[Normal distribution|normally distributed]], the probability distribution of their studentized range is what is usually called the ''studentized range distribution''. Note that the definition of ''q'' does not depend on the [[Expected value|expected value]] or the [[Standard deviation|standard deviation]] of the distribution from which the sample is drawn, and therefore its probability distribution is the same regardless of those parameters. 

== ''Studentization''==
{{main|Studentization}}
Generally, the term ''studentized'' means that the variable's scale was adjusted by dividing by an [[Estimation theory|estimate]] of a population [[Standard deviation|standard deviation]] (see also [[Studentized residual|studentized residual]]). The fact that the standard deviation is a ''sample'' standard deviation rather than the ''population'' standard deviation, and thus something that differs from one random sample to the next, is essential to the definition and the distribution of the ''Studentized'' data. The variability in the value of the ''sample'' standard deviation contributes additional uncertainty into the values calculated. This complicates the problem of finding the probability distribution of any statistic that is ''studentized''.

==See also==
*[[Studentized range distribution]]
*[[Tukey's range test]]

==References==
{{reflist}}

== Further reading ==
* Pearson, E.S.; Hartley, H.O. (1970) ''Biometrika Tables for Statisticians, Volume 1, 3rd Edition'', Cambridge University Press. {{ISBN|0-521-05920-8}}
* 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&ndash;452.
* Douglas C. Montgomery (2013) "Design and Analysis of Experiments", eighth edition, Wiley, page 98.

{{DEFAULTSORT:Studentized Range}}
[[Category:Summary statistics]]
[[Category:Multiple comparisons]]
[[Category:Statistical ratios]]

{{Sourceattribution|Studentized range}}