Tukey {stats}R Documentation

The Studentized Range Distribution


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


ptukey(q, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)
qtukey(p, nmeans, df, nranges = 1, lower.tail = TRUE, log.p = FALSE)


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 <= x], otherwise, P[X > x].


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


ptukey gives the distribution function and qtukey its inverse, the quantile function.


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.


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.

See Also

pnorm and qnorm for the corresponding functions for the normal distribution.


  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)))

[Package stats version 2.2.1 Index]