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.
quantile(x, …)# S3 method for default
quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE,
names = TRUE, type = 7, …)
numeric vector whose sample quantiles are wanted, or an
object of a class for which a method has been defined (see also
‘details’). NA and NaN values are not
allowed in numeric vectors unless na.rm is TRUE.
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.)
logical; if true, any NA and NaN's
are removed from x before the quantiles are computed.
logical; if true, the result has a names
attribute. Set to FALSE for speedup with many probs.
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.
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 \(1 \le i \le 9\), \(\frac{j - m}{n} \le p < \frac{j - m + 1}{n}\), \(x_{j}\) is the \(j\)th order statistic, \(n\) is the sample size, the value of \(\gamma\) is a function of \(j = \lfloor np + m\rfloor\) 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 \(i = 3\).
Inverse of empirical distribution function. \(\gamma = 0\) if \(g = 0\), and 1 otherwise.
Similar to type 1 but with averaging at discontinuities. \(\gamma = 0.5\) if \(g = 0\), and 1 otherwise.
SAS definition: nearest even order statistic. \(\gamma = 0\) if \(g = 0\) and \(j\) is even, and 1 otherwise.
Continuous sample quantile types 4 through 9
For types 4 through 9, \(Q_i(p)\) is a continuous function of \(p\), with \(\gamma = g\) and \(m\) given below. The sample quantiles can be obtained equivalently by linear interpolation between the points \((p_k,x_k)\) where \(x_k\) is the \(k\)th order statistic. Specific expressions for \(p_k\) are given below.
\(m = 0\). \(p_k = \frac{k}{n}\). That is, linear interpolation of the empirical cdf.
\(m = 1/2\). \(p_k = \frac{k - 0.5}{n}\). 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.
\(m = p\). \(p_k = \frac{k}{n + 1}\). Thus \(p_k = \mbox{E}[F(x_{k})]\). This is used by Minitab and by SPSS.
\(m = 1-p\). \(p_k = \frac{k - 1}{n - 1}\). In this case, \(p_k = \mbox{mode}[F(x_{k})]\). This is used by S.
\(m = (p+1)/3\).
\(p_k = \frac{k - 1/3}{n + 1/3}\).
Then \(p_k \approx \mbox{median}[F(x_{k})]\).
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x.
\(m = p/4 + 3/8\).
\(p_k = \frac{k - 3/8}{n + 1/4}\).
The resulting quantile estimates are approximately unbiased for
the expected order statistics if x is normally distributed.
Further details are provided in Hyndman and Fan (1996) who recommended type 8. The default method is type 7, as used by S and by R < 2.0.0.
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.
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.
ecdf for empirical distributions of which
quantile is an inverse;
boxplot.stats and fivenum for computing
other versions of quartiles, etc.
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
quantAll <- function(x, prob, ...)
t(vapply(1:9, function(typ) quantile(x, prob=prob, type = typ, ...), quantile(x, prob, type=1)))
p <- c(0.1, 0.5, 1, 2, 5, 10, 50)/100
signif(quantAll(x, p), 4)
## for complex numbers:
z <- complex(re=x, im = -10*x)
signif(quantAll(z, p), 4)
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