quantile
Sample Quantiles
The generic function 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.
- Keywords
- univar
Usage
quantile(x, ...)
"quantile"(x, probs = seq(0, 1, 0.25), na.rm = FALSE, names = TRUE, type = 7, ...)
Arguments
- x
- numeric vector whose sample quantiles are wanted, or an
object of a class for which a method has been defined (see also
details).
NA
andNaN
values are not allowed in numeric vectors unlessna.rm
isTRUE
. - probs
- numeric vector of probabilities with values in $[0,1]$. (Values up to 2e-14 outside that range are accepted and moved to the nearby endpoint.)
- na.rm
- logical; if true, any
NA
andNaN
's are removed fromx
before the quantiles are computed. - names
- logical; if true, the result has a
names
attribute. Set toFALSE
for speedup with manyprobs
. - type
- an integer between 1 and 9 selecting one of the nine quantile algorithms detailed below to be used.
- ...
- further arguments passed to or from other methods.
Details
A vector of length length(probs)
is returned;
if names = TRUE
, it has a names
attribute.
NA
and NaN
values in probs
are
propagated to the result.
The default method works with classed objects sufficiently like
numeric vectors that sort
and (not needed by types 1 and 3)
addition of elements and multiplication by a number work correctly.
Note that as this is in a namespace, the copy of sort
in
base will be used, not some S4 generic of that name. Also note
that that is no check on the correctly, and so
e.g. quantile
can be applied to complex vectors which (apart
from ties) will be ordered on their real parts.
There is a method for the date-time classes (see
"POSIXt"
). Types 1 and 3 can be used for class
"Date"
and for ordered factors.
Types
quantile
returns estimates of underlying distribution quantiles
based on one or two order statistics from the supplied elements in
x
at probabilities in probs
. One of the nine quantile
algorithms discussed in Hyndman and Fan (1996), selected by
type
, is employed. All sample quantiles are defined as weighted averages of
consecutive order statistics. Sample quantiles of type $i$
are defined by:
$$Q_{i}(p) = (1 - \gamma)x_{j} + \gamma x_{j+1}$$
where $1 \le i \le 9$,
$(j-m)/n \le p < (j-m+1)/n$,
$x[j]$ is the $j$th order statistic, $n$ is the
sample size, the value of $\gamma$ is a function of
$j = floor(np + m)$ and $g = np + m - j$,
and $m$ is a constant determined by the sample quantile type. Discontinuous sample quantile types 1, 2, and 3 For types 1, 2 and 3, $Q[i](p)$ is a discontinuous
function of $p$, with $m = 0$ when $i = 1$ and $i =
2$, and $m = -1/2$ when $i = 3$.
- Type 1
- Inverse of empirical distribution function. $\gamma = 0$ if $g = 0$, and 1 otherwise.
- Type 2
- Similar to type 1 but with averaging at discontinuities. $\gamma = 0.5$ if $g = 0$, and 1 otherwise.
- Type 3
- SAS definition: nearest even order statistic. $\gamma = 0$ if $g = 0$ and $j$ is even, and 1 otherwise.
- Type 4
- $m = 0$. $p[k] = k / n$. That is, linear interpolation of the empirical cdf.
- Type 5
- $m = 1/2$. $p[k] = (k - 0.5) / n$. 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.
- Type 6
- $m = p$. $p[k] = k / (n + 1)$. Thus $p[k] = E[F(x[k])]$. This is used by Minitab and by SPSS.
- Type 7
- $m = 1-p$. $p[k] = (k - 1) / (n - 1)$. In this case, $p[k] = mode[F(x[k])]$. This is used by S.
- Type 8
- $m = (p+1)/3$.
$p[k] = (k - 1/3) / (n + 1/3)$.
Then $p[k] =~ median[F(x[k])]$.
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of
x
. - Type 9
- $m = p/4 + 3/8$.
$p[k] = (k - 3/8) / (n + 1/4)$.
The resulting quantile estimates are approximately unbiased for
the expected order statistics if
x
is normally distributed.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician 50, 361--365.
See Also
ecdf
for empirical distributions of which
quantile
is an inverse;
boxplot.stats
and fivenum
for computing
other versions of quartiles, etc.
Examples
library(stats)
quantile(x <- rnorm(1001)) # Extremes & Quartiles by default
quantile(x, probs = c(0.1, 0.5, 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)