The Student t Distribution

Density, distribution function, quantile function and random generation for the t distribution with df degrees of freedom (and optional non-centrality parameter ncp).

dt(x, df, ncp, log = FALSE)
pt(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
qt(p, df, ncp, lower.tail = TRUE, log.p = FALSE)
rt(n, df, ncp)
x, q

vector of quantiles.


vector of probabilities.


number of observations. If length(n) > 1, the length is taken to be the number required.


degrees of freedom (\(> 0\), maybe non-integer). df = Inf is allowed.


non-centrality parameter \(\delta\); currently except for rt(), only for abs(ncp) <= 37.62. If omitted, use the central t distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p).


logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).


The \(t\) distribution with df \(= \nu\) degrees of freedom has density $$ f(x) = \frac{\Gamma ((\nu+1)/2)}{\sqrt{\pi \nu} \Gamma (\nu/2)} (1 + x^2/\nu)^{-(\nu+1)/2}% $$ for all real \(x\). It has mean \(0\) (for \(\nu > 1\)) and variance \(\frac{\nu}{\nu-2}\) (for \(\nu > 2\)).

The general non-central \(t\) with parameters \((\nu, \delta)\) = (df, ncp) is defined as the distribution of \(T_{\nu}(\delta) := (U + \delta)/\sqrt{V/\nu}\) where \(U\) and \(V\) are independent random variables, \(U \sim {\cal N}(0,1)\) and \(V \sim \chi^2_\nu\) (see Chisquare).

The most used applications are power calculations for \(t\)-tests: Let \(T = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}\) where \(\bar{X}\) is the mean and \(S\) the sample standard deviation (sd) of \(X_1, X_2, \dots, X_n\) which are i.i.d. \({\cal N}(\mu, \sigma^2)\) Then \(T\) is distributed as non-central \(t\) with df\({} = n-1\) degrees of freedom and non-centrality parameter ncp\({} = (\mu - \mu_0) \sqrt{n}/\sigma\).


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 n for 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.


Supplying ncp = 0 uses the algorithm for the non-central distribution, which is not the same algorithm used if ncp is omitted. This is to give consistent behaviour in extreme cases with values of 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.


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.

See Also

Distributions for other standard distributions, including df for the F distribution.

  • TDist
  • dt
  • pt
  • qt
  • rt
library(stats) # NOT RUN { 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") # }
Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2

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