Studentized range

In statistics, the studentized range is the difference between the largest and smallest data in a sample measured in units of sample standard deviations.

The studentized range, q, is named for William Sealy Gosset (who wrote under the pseudonym "Student"), and was initially evoked by him (1927).^[1] The concept was later presented by a number of actual students, Newman (1939) ^[2] and Keuls (1952) ^[3] and John Tukey in some unpublished notes. q is the basic statistic for the studentized range distribution, which is used for 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 intervals that are still valid after data snooping has occurred.^[4]

Description

The value of the studentized range is most often represented by the variable q.

The studentized range computed from a list x₁, ..., x_n of numbers is given by the formulas

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\}

where

s^2 = \frac{1}{n - 1}\sum_{i=1}^n (x_i - \overline{x})^2,

is the sample variance, an unbiased estimator of the population variance and the square of the sample standard deviation s, and

\overline{x} = \frac{x_1 + \cdots + x_n}{n}

is the sample mean. The critical value of q is based on three factors:

α (the probability of rejecting a true null hypothesis)
n (the number of observations or groups)
v (the degrees of freedom used to estimate the sample variance)

Distribution (normal data) and applications

If X₁, ..., 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. Note that the definition of "q" does not depend on the expected value or the standard deviation of the distribution from which the sample is drawn, and therefore its probability distribution is the same regardless of those parameters. tables of the distribution quantiles are available here.

The Studentized range distribution has applications to hypothesis testing and multiple comparisons procedures. For example, Tukey's range test and Duncan's new multiple range test (MRT), in which the sample x₁, ..., x_n is a sample of means and q is the basic test-statistic, can be used as post-hoc analysis to test between which two groups means there is a significant difference (pairwise comparisons) after rejecting the null hypothesis that all groups are from the same population (i.e. all means are equal) by the standard analysis of variance.^[5]

When only the equality of the two groups means is in question (i.e whether μ₁ = μ₂), the studentized range distribution is similar to the Student's t distribution, differing only in that the first takes into account the number of means under consideration, and the critical value is adjusted accordingly. The more means under consideration, the larger the critical value is. This makes sense since the more means there are, the greater the probability that at least some differences between pairs of means will be significantly large due to chance alone.

Studentized data

Generally, the term studentized means that the variable's scale was adjusted by dividing by an estimate of a population standard deviation (see also 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.

Notes

↑ Student (1927). "Errors of routine analysis". Biometrika. 19 (1/2): 151–164. doi:10.2307/2332181.
↑ Newman D. (1939). "The Distribution of Range in Samples from a Normal Population Expressed in Terms of an Independent Estimate of Standard Deviation". Biometrika. 31: 20–30. doi:10.1093/biomet/31.1-2.20.
↑ Keuls M. (1952). "The Use of the "Studentized Range" in Connection with an Analysis of Variance". Euphytica. 1: 112–122. doi:10.1007/bf01908269.
↑ John A. Rafter (2002). "Multiple Comparison Methods for Means". SIAM. 44 (2): 259–278. doi:10.1137/s0036144501357233.
↑ Pearson & Hartley (1970, Section 14.2)

References

Pearson, E.S.; Hartley, H.O. (1970) Biometrika Tables for Statisticians, Volume 1, 3rd Edition, Cambridge University Press. ISBN 0-521-05920-8