Studentized range

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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Range (statistics) — In descriptive statistics, the range is the length of the smallest interval which contains all the data. It is calculated by subtracting the smallest observations from the greatest and provides an indication of statistical dispersion. It is… …   Wikipedia

  • Studentized residual — In statistics, a studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student , is a residual adjusted by dividing it by an estimate of its standard deviation. Studentization of residuals is an important… …   Wikipedia

  • Duncan's new multiple range test — In statistics, Duncan s new multiple range test (MRT) is a multiple comparison procedure developed by David B. Duncan in 1955. Duncan s MRT belongs to the general class of multiple comparison procedures that use the studentized range statistic qr …   Wikipedia

  • Tukey's test — Tukey s test, named after John Tukey, is a statistical test generally used in conjunction with an ANOVA to find which means are significantly different from one another. It compares all possible pairs of means, and is based on a studentized range …   Wikipedia

  • Tukey-Kramer method — The Tukey method (also known as Tukey s Honest Significance Test), named for John Tukey, is a single step multiple comparison procedure which applies simultaneously to the set of all pairwise comparisons:mu i mu jThe confidence coefficient for… …   Wikipedia

  • Newman–Keuls method — In statistics, the Newman–Keuls method (named after D. Newman (1939),[1] and M. Keuls (1952)[2]) is a post hoc test used for comparisons after the performed F test (analysis of variance) is found to be significant. The Newman–Keuls method is very …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Level of measurement — The levels of measurement , or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled On the theory of… …   Wikipedia

  • Deviation (statistics) — In mathematics and statistics, deviation is a measure of difference for interval and ratio variables between the observed value and the mean. The sign of deviation (positive or negative), reports the direction of that difference (it is larger… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”