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Sample Quantiles
Read the amounts before the curve
The minimum, maximum, median and quartiles use the entered dollar amounts in ascending order. For the median, an even number of records uses the arithmetic midpoint of the two central amounts. Quartiles use type 7 linear interpolation: for probability p, the zero-based position is (n − 1)p; the result interpolates between the two neighboring amounts. Calculations retain unrounded values; monetary display rounds to cents. The chart places these dollar results on its logarithmic axis afterward.
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Complete quantile help page, including all nine definitions, defaults, limitations, examples, references and version footer.
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R: Sample Quantiles
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,
fuzz = if(type == 7L) 0 else 4 * .Machine$double.eps,
...)
Arguments
x
numeric vector whose sample quantiles are wanted, or an
object of a class for which a method has been defined (see also
‘details’). NA and NaN values are not
allowed in numeric vectors unless na.rm is TRUE.
probs
numeric vector of probabilities with values in
[0,1]. (Values up to ‘2e-14’ outside that
range are accepted and moved to the nearby endpoint.)
na.rm
logical; if true, any NA and NaN's
are removed from x before the quantiles are computed.
names
logical; if true, the result has a names
attribute. Set to FALSE for speedup with many probs.
type
an integer between 1 and 9 selecting one of the
nine quantile algorithms detailed below to be used.
digits
used only when names is true: the precision to use
when formatting the percentages. In R versions up to 4.0.x, this had
been set to max(2, getOption("digits")), internally.
fuzz
small non-negative number to protect against rounding errors
when j <- floor(np + m), (np “a version of”
n * probs, see the formula below), is computed.
...
further arguments passed to or from other methods.
Details
A vector of length length(probs) is returned;
if names = TRUE, it has a names attribute.
NA and NaN values in probs are
propagated to the result.
The default method works with classed objects sufficiently like
numeric vectors that sort and (not needed by types 1 and 3)
addition of elements and multiplication by a number work correctly.
Note that as this is in a namespace, the copy of sort in
base will be used, not some S4 generic of that name. Also note
that that is no check on the ‘correctly’, and so
e.g. quantile can be applied to complex vectors which (apart
from ties) will be ordered on their real parts.
There is a method for the date-time classes (see
"POSIXt"). Types 1 and 3 can be used for class
"Date" and for ordered factors.
fuzz := 4 * .Machine$double.eps has been hard coded and used for
type = 4,5,6, 8,9 since types were introduced,
and is used, since R 4.5.0, also for the other types (but with default
0 for the default type = 7 for back compatibility reasons).
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 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 (SAS default, see
Wicklin (2017)).
Type 3
Nearest even order statistic (SAS default till ca. 2010).
\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.
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}.
Thus p_k = \mbox{E}[F(x_{k})].
This is used by Minitab and by SPSS.
Type 7
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.
Type 8
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.
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 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 and Pajari (2014) argue 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;
tweaks, notably use of fuzz, by the R Core Team.
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.
Eric Langford (2006) Quartiles in Elementary Statistics, Journal
of Statistics Education 14 3; doi:10.1080/10691898.2006.11910589
Makkonen, L. and Pajari, M. (2014) Defining Sample Quantiles by the True
Rank Probability, Journal of Probability and Statistics; Hindawi Publ.Corp.
doi:10.1155/2014/326579
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
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)
[Package stats version 4.5.0 Index]