# Tukey

0th

Percentile

##### The Studentized Range Distribution

Functions of the distribution of the studentized range, $R/s$, where $R$ is the range of a standard normal sample and $df*s^2$ is independently distributed as chi-squared with $df$ degrees of freedom, see pchisq.

Keywords
distribution
##### Usage
ptukey(q, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
qtukey(p, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
##### Arguments
q
vector of quantiles.
p
vector of probabilities.
nmeans
sample size for range (same for each group).
df
degrees of freedom for $s$ (see below).
nranges
number of groups whose maximum range is considered.
log.p
logical; if TRUE, probabilities p are given as log(p).
lower.tail
logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
##### Details

If $ng =$nranges is greater than one, $R$ is the maximum of $ng$ groups of nmeans observations each.

##### Value

ptukey gives the distribution function and qtukey its inverse, the quantile function.The length of the result is the maximum of the lengths of the numerical arguments. The other numerical arguments are recycled to that length. Only the first elements of the logical arguments are used.

##### Note

A Legendre 16-point formula is used for the integral of ptukey. The computations are relatively expensive, especially for qtukey which uses a simple secant method for finding the inverse of ptukey. qtukey will be accurate to the 4th decimal place.

##### Source

qtukey is in part adapted from Odeh and Evans (1974).

##### References

Copenhaver, Margaret Diponzio and Holland, Burt S. (1988) Multiple comparisons of simple effects in the two-way analysis of variance with fixed effects. Journal of Statistical Computation and Simulation, 30, 1--15.

Odeh, R. E. and Evans, J. O. (1974) Algorithm AS 70: Percentage Points of the Normal Distribution. Applied Statistics 23, 96--97.

Distributions for standard distributions, including pnorm and qnorm for the corresponding functions for the normal distribution.

• Tukey
• ptukey
• qtukey
##### Examples
library(stats) if(interactive()) curve(ptukey(x, nm = 6, df = 5), from = -1, to = 8, n = 101) (ptt <- ptukey(0:10, 2, df = 5)) (qtt <- qtukey(.95, 2, df = 2:11)) ## The precision may be not much more than about 8 digits: summary(abs(.95 - ptukey(qtt, 2, df = 2:11))) 
Documentation reproduced from package stats, version 3.2.2, License: Part of R 3.2.2

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