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| In [[statistics]], the '''studentized range''' is the difference between the largest and smallest data in a [[sample (statistics)|sample]] measured in units of [[standard deviation|sample standard deviation]]s.
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| The '''studentized range''', ''q'', which was presented by Newman (1939) and Keuls (1952) and [[John Tukey]] in some unpublished notes, is the base statistic for the '''[[studentized range distribution]]''', which is used for multiple comparison procedures, such as the single step procedure [[Tukey's range test]] and the Duncan's step down procedure and establishing [[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 |volume=44 |issue=2 |pages=259–278 |year=2002 }}</ref>
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| == Description ==
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| The value of the '''studentized range''' is most often represented by the variable ''q''.
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| The '''studentized range''' computed from a list ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> of numbers is given by the formulas
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| : <math>
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| 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>
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| where
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| :<math> s^2 = \frac{1}{n - 1}\sum_{i=1}^n (x_i - \overline{x})^2, </math>
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| is the [[sample variance]], the square of the sample [[standard deviation]] ''s'', and
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| :<math> \overline{x} = \frac{x_1 + \cdots + x_n}{n} </math> | |
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| is the [[sample mean]].
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| The critical value of ''q'' based on three factors:
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| #α (the probability of rejecting a true [[null hypothesis]])
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| #''n'' (the number of observations or groups)
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| #''v'' (degrees of freedom in the second sample)
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| If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[independent identically distributed]] [[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]]'''. This probability distribution is the same regardless of the [[expected value]] and [[standard deviation]] of the normal distribution from which the sample is drawn: tables are available.<ref>Pearson & Hartley (1970, Section 14, Table 29)</ref> This probability distribution has applications to [[hypothesis testing]] and [[multiple comparisons]]. For example, [[Tukey's range test]] and [[Duncan's new multiple range test]] '''(MRT)''', which uses '''q statistics''', can be used as [[post-hoc analysis]] to test between which two groups there is a significant difference after rejecting [[null hypothesis]] by [[analysis of variance]].<ref>Pearson & Hartley (1970, Section 14.2)</ref>
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| When only two groups need to be compared, the '''[[studentized range distribution]]''' is similar to the [[Student's t distribution]], differing only in that it takes into account the number of means under consideration. The more means under consideration, the larger the critical value is. This makes sense since the more means there are, the greater the likelihood that at least some differences between pairs of means will be large due to chance alone.
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| == ''Studentized'' data ==
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| Generally, the term ''[[Studentization|studentized]]'' means that the variable's scale was adjusted by dividing by an [[estimation theory|estimate]] of a population [[standard deviation]] (see also [[studentized residual]]).
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| The concept is named after [[William Sealey Gosset]], who wrote under the pseudonym "Student". 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.
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| The variability in the value of the ''sample'' [[standard deviation]] introduces additional uncertainty into the values calculated. This complicates the problem of finding the probability distribution of any statistic that is ''studentized''.
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| ==See also==
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| *[[Studentized range distribution]]
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| *[[Tukey's range test]]
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| ==Notes==
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| {{reflist}}
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| {{More footnotes|date=November 2010}}
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| ==References==
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| * Pearson, E.S.; Hartley, H.O. (1970) ''Biometrika Tables for Statisticians, Volume 1, 3rd Edition'', Cambridge University Press. ISBN 0-521-05920-8
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| == Further reading ==
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| * John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman (1996) ''Applied Linear Statistical Models'', fourth edition, McGraw-Hill, page 726.
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| * John A. Rice (1995) ''Mathematical Statistics and Data Analysis'', second edition, Duxbury Press, pages 451–452.
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| {{DEFAULTSORT:Studentized Range}}
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| [[Category:Statistical terminology]]
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| [[Category:Summary statistics]]
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| [[Category:Multiple comparisons]]
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| [[Category:Statistical ratios]]
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