Sample Quantiles
Description
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.
Usage
1 2 3 4 5 
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’). 
probs 
numeric vector of probabilities with values in [0,1]. (Values up to 2e14 outside that range are accepted and moved to the nearby endpoint.) 
na.rm 
logical; if true, any 
names 
logical; if true, the result has a 
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.\sspacequantile
can be applied to complex vectors which (apart
from ties) will be ordered on their real parts.
There is a method for the datetime 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  γ) x[j] + γ x[j+1],
where 1 ≤ i ≤ 9, (jm)/n ≤ p < (jm+1)/n, x[j] is the jth order statistic, n is the sample size, the value of γ 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. γ = 0 if g = 0, and 1 otherwise.
 Type 2
Similar to type 1 but with averaging at discontinuities. γ = 0.5 if g = 0, and 1 otherwise.
 Type 3
SAS definition: nearest even order statistic. γ = 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 kth order statistic. Specific expressions for p[k] are given below.
 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 = 1p
. 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 medianunbiased 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.
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.
Author(s)
of the version used in R >= 2.0.0, Ivan Frohne and Rob J Hyndman.
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
1 2 3 4 5 6 7 8 9  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)
