quantile {stats}  R Documentation 
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
quantile(x, ...)
## Default S3 method:
quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE,
names = TRUE, type = 7, digits = 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’). 
probs 
numeric vector of probabilities with values in

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. 
digits 
used only when 
... 
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 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  \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
.
 Type 1
Inverse of empirical distribution function.
\gamma = 0
ifg = 0
, and 1 otherwise. Type 2
Similar to type 1 but with averaging at discontinuities.
\gamma = 0.5
ifg = 0
, and 1 otherwise (SAS default, see Wicklin (2017)). Type 3
Nearest even order statistic (SAS default till ca. 2010).
\gamma = 0
ifg = 0
andj
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.
 Type 4
m = 0
.p_k = \frac{k}{n}
. That is, linear interpolation of the empirical cdf. Type 5
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. Type 6
m = p
.p_k = \frac{k}{n + 1}
. Thusp_k = \mbox{E}[F(x_{k})]
. This is used by Minitab and by SPSS. Type 7
m = 1p
.p_k = \frac{k  1}{n  1}
. In this case,p_k = \mbox{mode}[F(x_{k})]
. This is used by S. Type 8
m = (p+1)/3
.p_k = \frac{k  1/3}{n + 1/3}
. Thenp_k \approx \mbox{median}[F(x_{k})]
. The resulting quantile estimates are approximately medianunbiased regardless of the distribution ofx
. Type 9
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 ifx
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. Makkonen argues for type 6, also as already proposed by Weibull in 1939. The Wikipedia page contains further information about availability of these 9 types in software.
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. doi:10.2307/2684934.
Wicklin, R. (2017) Sample quantiles: A comparison of 9 definitions; SAS Blog. https://blogs.sas.com/content/iml/2017/05/24/definitionssamplequantiles.html
Wikipedia: https://en.wikipedia.org/wiki/Quantile#Estimating_quantiles_from_a_sample
See Also
ecdf
for empirical distributions of which
quantile
is an inverse;
boxplot.stats
and fivenum
for computing
other versions of quartiles, etc.
Examples
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, probs = 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)
## 0% and 100% are equal to min(), max() for all types:
stopifnot(t(quantAll(x, prob=0:1)) == range(x))
## for complex numbers:
z < complex(real = x, imaginary = 10*x)
signif(quantAll(z, p), 4)