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<table width="100%" summary="page for TDist {stats}"><tr><td>TDist {stats}</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>The Student t Distribution</h2>

<h3>Description</h3>

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

<h3>Usage</h3>

<pre>
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)
</pre>

<h3>Arguments</h3>

<table summary="R argblock">
<tr valign="top"><td><code>x, q</code></td>
<td>
vector of quantiles.
</td></tr>
<tr valign="top"><td><code>p</code></td>
<td>
vector of probabilities.
</td></tr>
<tr valign="top"><td><code>n</code></td>
<td>
number of observations. If <code>length(n) &gt; 1</code>, the length
is taken to be the number required.
</td></tr>
<tr valign="top"><td><code>df</code></td>
<td>
degrees of freedom (&gt; 0, maybe non-integer). <code>df
 = Inf</code> is allowed.
</td></tr>
<tr valign="top"><td><code>ncp</code></td>
<td>
non-centrality parameter delta;
currently except for <code>rt()</code>, only for <code>abs(ncp) &lt;= 37.62</code>.
If omitted, use the central t distribution.
</td></tr>
<tr valign="top"><td><code>log, log.p</code></td>
<td>
logical; if TRUE, probabilities p are given as log(p).
</td></tr>
<tr valign="top"><td><code>lower.tail</code></td>
<td>
logical; if TRUE (default), probabilities are
P[X &le; x], otherwise, P[X &gt; x].
</td></tr>
</table>

<h3>Details</h3>

The t distribution with <code>df</code> = n degrees of
freedom has density

f(x) = &Gamma;((n+1)/2) / (&radic;(n &pi;) &Gamma;(n/2)) (1 + x^2/n)^-((n+1)/2)

for all real x.
It has mean 0 (for n &gt; 1) and
variance n/(n-2) (for n &gt; 2).

The general non-central t
with parameters (df, Del) <code>= (df, ncp)</code>
is defined as the distribution of
T(df, Del) := (U + Del) / &radic;(V/df) 
where U and V are independent random
variables, U ~ N(0,1) and
V ~ &chi;^2(df) (see <a href="Chisquare.html">Chisquare</a>).

The most used applications are power calculations for t-tests: 
Let T= (mX - m0) / (S/sqrt(n))
where
mX is the <code><a href="../../base/html/mean.html">mean</a></code> and S the sample standard
deviation (<code><a href="sd.html">sd</a></code>) of X_1, X_2, &hellip;, X_n which are
i.i.d. N(&mu;, &sigma;^2)
Then T is distributed as non-central t with
<code>df</code>= n - 1
degrees of freedom and non-centrality parameter
<code>ncp</code> = (&mu; - m0) * sqrt(n)/&sigma;.

<h3>Value</h3>

<code>dt</code> gives the density,
<code>pt</code> gives the distribution function,
<code>qt</code> gives the quantile function, and
<code>rt</code> generates random deviates.

Invalid arguments will result in return value <code>NaN</code>, with a warning.

The length of the result is determined by <code>n</code> for
<code>rt</code>, and is the maximum of the lengths of the
numerical arguments for the other functions. 

The numerical arguments other than <code>n</code> are recycled to the
length of the result. Only the first elements of the logical
arguments are used.

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

The code for non-zero <code>ncp</code> is principally intended to be used
for moderate values of <code>ncp</code>: it will not be highly accurate,
especially in the tails, for large values.

<h3>Source</h3>

The central <code>dt</code> is computed via an accurate formula
provided by Catherine Loader (see the reference in <code><a href="Binomial.html">dbinom</a></code>).

For the non-central case of <code>dt</code>, C code contributed by
Claus Ekstrøm based on the relationship (for
x != 0) to the cumulative distribution.

For the central case of <code>pt</code>, a normal approximation in the
tails, otherwise via <code><a href="Beta.html">pbeta</a></code>.

For the non-central case of <code>pt</code> based on a C translation of

Lenth, R. V. (1989). Algorithm AS 243 &mdash;
Cumulative distribution function of the non-central t distribution,
Applied Statistics 38, 185&ndash;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.

For central <code>qt</code>, a C translation of

Hill, G. W. (1970) Algorithm 396: Student's t-quantiles.
Communications of the ACM, 13(10), 619&ndash;620.

altered to take account of

Hill, G. W. (1981) Remark on Algorithm 396, ACM Transactions on
Mathematical Software, 7, 250&ndash;1.

The non-central case is done by inversion.

<h3>References</h3>

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
The New S Language.
Wadsworth &amp; 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.

<a href="Distributions.html">Distributions</a> for other standard distributions, including
<code><a href="Fdist.html">df</a></code> for the F distribution.

<h3>Examples</h3>

<pre>
require(graphics)

1 - pt(1:5, df = 1)
qt(.975, df = c(1:10,20,50,100,1000))

tt &lt;- seq(0, 10, len = 21)
ncp &lt;- seq(0, 6, len = 31)
ptn &lt;- outer(tt, ncp, function(t, d) pt(t, df = 3, ncp = d))
t.tit &lt;- "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 &lt;= t)")

plot(function(x) dt(x, df = 3, ncp = 2), -3, 11, ylim = c(0, 0.32),
 main = "Non-central t - Density", yaxs = "i")
</pre>

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