# TDist {stats}

### Description

Density, distribution function, quantile function and random generation for the t distribution with `df`

degrees of freedom (and optional non-centrality parameter `ncp`

).

### Usage

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)

### Arguments

- x, q
- vector of quantiles.
- p
- vector of probabilities.
- n
- number of observations. If
`length(n) > 1`

, the length is taken to be the number required. - df
- degrees of freedom (> 0, maybe non-integer).
`df = Inf`

is allowed. - ncp
- 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).
- lower.tail
- logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

### Details

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 deviation (`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 **n**on-**c**entrality **p**arameter `ncp`

= (μ - m0) * sqrt(n)/σ.

### Values

`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 parameters for the other functions. The numerical parameters other than `n`

are recycled to the length of the result. Only the first elements of the logical parameters are used.

### References

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.

### Note

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.

### See Also

Distributions for other standard distributions, including `df`

for the F distribution.

### Examples

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 R 3.0.2. License: GPL-2.