\HeaderA{quantile}{Sample Quantiles}{quantile}
\methaliasA{quantile.default}{quantile}{quantile.default}
\keyword{univar}{quantile}
\begin{Description}\relax
The generic function \code{quantile} produces sample quantiles
corresponding to the given probabilities.
The smallest observation corresponds to a probability of 0 and the
largest to a probability of 1.
\end{Description}
\begin{Usage}
\begin{verbatim}
quantile(x, ...)

## Default S3 method:
quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE,
         names = TRUE, type = 7, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] numeric vectors whose sample quantiles are wanted.  Missing
values are ignored.
\item[\code{probs}] numeric vector of probabilities with values in \eqn{[0,1]}{}.
\item[\code{na.rm}] logical; if true, any \code{\LinkA{NA}{NA}} and \code{NaN}'s
are removed from \code{x} before the quantiles are computed.
\item[\code{names}] logical; if true, the result has a \code{\LinkA{names}{names}}
attribute.  Set to \code{FALSE} for speedup with many \code{probs}.
\item[\code{type}] an integer between 1 and 9 selecting one of the
nine quantile algorithms detailed below to be used.
\item[\code{...}] further arguments passed to or from other methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
A vector of length \code{length(probs)} is returned;
if \code{names = TRUE}, it has a \code{\LinkA{names}{names}} attribute.

\code{\LinkA{NA}{NA}} and \code{\LinkA{NaN}{NaN}} values in \code{probs} are
propagated to the result.
\end{Details}
\begin{Section}{Types}
\code{quantile} returns estimates of underlying distribution quantiles
based on one or two order statistics from the supplied elements in
\code{x} at probabilities in \code{probs}.  One of the nine quantile
algorithms discussed in Hyndman and Fan (1996), selected by
\code{type}, is employed.

Sample quantiles of type \eqn{i}{} are defined by
\deqn{Q_{i}(p) = (1 - \gamma)x_{j} + \gamma x_{j+1}}{Q[i](p) = (1 - gamma) x[j] + gamma x[j+1],}
where \eqn{1 \le i \le 9}{1 <= i <= 9},
\eqn{\frac{j - m}{n} \le p < \frac{j - m + 1}{n}}{(j-m)/n <= p < (j-m+1)/ n},
\eqn{x_{j}}{x[j]} is the \eqn{j}{}th order statistic, \eqn{n}{} is the
sample size, and \eqn{m}{} is a constant determined by the sample
quantile type. Here \eqn{\gamma}{} depends on the fractional part
of \eqn{g = np+m-j}{}.

For the continuous sample quantile types (4 through 9), the sample
quantiles can be obtained by linear interpolation between the \eqn{k}{}th
order statistic and \eqn{p(k)}{}:
\deqn{p(k) = \frac{k - \alpha} {n - \alpha - \beta + 1}}{p(k) = (k - alpha) / (n - alpha - beta + 1),} where
\eqn{\alpha}{} and \eqn{\beta}{} are constants determined by
the type.  Further, \eqn{m = \alpha + p \left( 1 - \alpha - \beta
\right)}{m = alpha + p(1 - alpha - beta)}, and \eqn{\gamma = g}{}.

\strong{Discontinuous sample quantile types 1, 2, and 3}

\describe{
\item[Type 1] Inverse of empirical distribution function.
\item[Type 2] Similar to type 1 but with averaging at discontinuities.
\item[Type 3] SAS definition: nearest even order statistic.
}


\strong{Continuous sample quantile types 4 through 9}

\describe{
\item[Type 4] \eqn{p(k) = \frac{k}{n}}{p(k) = k / n}.
That is, linear interpolation of the empirical cdf.


\item[Type 5] \eqn{p(k) = \frac{k - 0.5}{n}}{p(k) = (k - 0.5) / n}.
That is a piecewise linear function where the knots are the values
midway through the steps of the empirical cdf. This is popular
amongst hydrologists.


\item[Type 6] \eqn{p(k) = \frac{k}{n + 1}}{p(k) = k / (n + 1)}.
Thus \eqn{p(k) = \mbox{E}[F(x_{k})]}{p(k) = E[F(x[k])]}.
This is used by Minitab and by SPSS.


\item[Type 7] \eqn{p(k) = \frac{k - 1}{n - 1}}{p(k) = (k - 1) / (n - 1)}.
In this case, \eqn{p(k) = \mbox{mode}[F(x_{k})]}{p(k) = mode[F(x[k])]}.
This is used by S.


\item[Type 8] \eqn{p(k) = \frac{k - \frac{1}{3}}{n + \frac{1}{3}}}{p(k) = (k - 1/3) / (n + 1/3)}.
Then \eqn{p(k) \approx \mbox{median}[F(x_{k})]}{p(k) =~ median[F(x[k])]}.
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of \code{x}.


\item[Type 9] \eqn{p(k) = \frac{k - \frac{3}{8}}{n + \frac{1}{4}}}{p(k) = (k - 3/8) / (n + 1/4)}.
The resulting quantile estimates are approximately unbiased for
the expected order statistics if \code{x} is normally distributed.

}
Hyndman and Fan (1996) recommend type 8.
The default method is type 7, as used by S and by \R{} < 2.0.0.
\end{Section}
\begin{Author}\relax
of the version used in \R{} >= 2.0.0, Ivan Frohne and Rob J Hyndman.
\end{Author}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical
packages, \emph{American Statistician}, \bold{50}, 361--365.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{ecdf}{ecdf}} for empirical distributions of which
\code{quantile} is the \dQuote{inverse};
\code{\LinkA{boxplot.stats}{boxplot.stats}} and \code{\LinkA{fivenum}{fivenum}} for computing
\dQuote{versions} of quartiles, etc.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
quantile(x <- rnorm(1001))# Extremes & Quartiles by default
quantile(x,  probs=c(.1,.5,1,2,5,10,50, NA)/100)

### Compare different types
p <- c(0.1,0.5,1,2,5,10,50)/100
res <- matrix(as.numeric(NA), 9, 7)
for(type in 1:9) res[type, ] <- y <- quantile(x,  p, type=type)
dimnames(res) <- list(1:9, names(y))
round(res, 3)
\end{ExampleCode}
\end{Examples}

