# quantile {stats}

### 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 class 'default': quantile((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.

### 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.

### 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, (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 , 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 = 1-p. 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 median-unbiased 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.

### 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

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)

Documentation reproduced from R 3.0.2. License: GPL-2.