Density, distribution function, quantile function and random
generation for the t distribution with
df degrees of freedom
(and optional non-centrality parameter
1 2 3 4
vector of quantiles.
vector of probabilities.
number of observations. If
degrees of freedom (> 0, maybe non-integer).
non-centrality parameter delta;
currently except for
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].
The t distribution with
df = n degrees of
freedom has density
f(x) = Γ((n+1)/2) / (√(n π) Γ(n/2)) (1 + x^2/n)^-((n+1)/2)
for all real x. It has mean 0 (for n > 1) and variance n/(n-2) (for n > 2).
The general non-central t
with parameters (df, Del)
= (df, ncp)
is defined as the distribution of
T(df, Del) := (U + Del) / √(V/df)
where U and V are independent random
variables, U ~ N(0,1) and
V ~ χ^2(df) (see Chisquare).
The most used applications are power calculations for t-tests:
Let T= (mX - m0) / (S/sqrt(n)) where mX is the
mean and S the sample standard
sd) of X_1, X_2, …, X_n which are
i.i.d. N(μ, σ^2)
Then T is distributed as non-central t with
df= n - 1
degrees of freedom and non-centrality parameter
ncp = (μ - m0) * sqrt(n)/σ.
dt gives the density,
pt gives the distribution function,
qt gives the quantile function, and
rt generates random deviates.
Invalid arguments will result in return value
NaN, with a warning.
The length of the result is determined by
rt, and is the maximum of the lengths of the
numerical arguments for the other functions.
The numerical arguments other than
n are recycled to the
length of the result. Only the first elements of the logical
arguments are used.
ncp = 0 uses the algorithm for the non-central
distribution, which is not the same algorithm used if
omitted. This is to give consistent behaviour in extreme cases with
ncp very near zero.
The code for non-zero
ncp is principally intended to be used
for moderate values of
ncp: it will not be highly accurate,
especially in the tails, for large values.
dt is computed via an accurate formula
provided by Catherine Loader (see the reference in
For the non-central case of
dt, C code contributed by
Claus Ekstrøm based on the relationship (for
x != 0) to the cumulative distribution.
For the central case of
pt, a normal approximation in the
tails, otherwise via
For the non-central case of
pt based on a C translation of
Lenth, R. V. (1989). Algorithm AS 243 — Cumulative distribution function of the non-central t distribution, Applied Statistics 38, 185–189.
This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant.
qt, a C translation of
Hill, G. W. (1970) Algorithm 396: Student's t-quantiles. Communications of the ACM, 13(10), 619–620.
altered to take account of
Hill, G. W. (1981) Remark on Algorithm 396, ACM Transactions on Mathematical Software, 7, 250–1.
The non-central case is done by inversion.
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (Except non-central versions.)
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapters 28 and 31. Wiley, New York.
Distributions for other standard distributions, including
df for the F distribution.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
require(graphics) 1 - pt(1:5, df = 1) qt(.975, df = c(1:10,20,50,100,1000)) tt <- seq(0, 10, len = 21) ncp <- seq(0, 6, len = 31) ptn <- outer(tt, ncp, function(t, d) pt(t, df = 3, ncp = d)) t.tit <- "Non-central t - Probabilities" image(tt, ncp, ptn, zlim = c(0,1), main = t.tit) persp(tt, ncp, ptn, zlim = 0:1, r = 2, phi = 20, theta = 200, main = t.tit, xlab = "t", ylab = "non-centrality parameter", zlab = "Pr(T <= t)") plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32), main = "Non-central t - Density", yaxs = "i")