| quantile | R Documentation |
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, ...)
## Default S3 method:
quantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE,
names = TRUE, type = 7, digits = 7, ...)
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 2e-14 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. |
digits |
used only when |
... |
further arguments passed to or from other methods. |
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.
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, (j-m)/n ≤ p < (j-m+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.
Inverse of empirical distribution function. γ = 0 if g = 0, and 1 otherwise.
Similar to type 1 but with averaging at discontinuities. γ = 0.5 if g = 0, and 1 otherwise (SAS default, see Wicklin(2017)).
Nearest even order statistic (SAS default till ca. 2010). 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.
m = 0. p[k] = k / n. That is, linear interpolation of the empirical cdf.
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.
m = p. p[k] = k / (n + 1). Thus p[k] = E[F(x[k])]. This is used by Minitab and by SPSS.
m = 1-p. p[k] = (k - 1) / (n - 1). In this case, p[k] = mode[F(x[k])]. This is used by S.
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.
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. 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.
of the version used in R >= 2.0.0, Ivan Frohne and Rob J Hyndman.
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. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_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/definitions-sample-quantiles.html
Wikipedia: https://en.wikipedia.org/wiki/Quantile#Estimating_quantiles_from_a_sample
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, 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)
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