## 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, …)`# S3 method for defaultquantile(x, probs = seq(0, 1, 0.25), na.rm = FALSE, names = TRUE, type = 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’). `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.

…

further arguments passed to or from other methods.

## 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.

- Type 3
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.

- 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.

## 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.

## 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. 10.2307/2684934.

## See Also

`ecdf`

for empirical distributions of which `quantile`

is an inverse; `boxplot.stats`

and `fivenum`

for computing other versions of quartiles, etc.

## Examples

`# NOT RUN {quantile(x <- rnorm(1001)) # Extremes & Quartiles by defaultquantile(x, probs = c(0.1, 0.5, 1, 2, 5, 10, 50, NA)/100)### Compare different typesquantAll <- 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)/100signif(quantAll(x, p), 4)## for complex numbers:z <- complex(re=x, im = -10*x)signif(quantAll(z, p), 4)# }`

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