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, ...)
```

### Arguments

 `x` numeric vectors whose sample quantiles are wanted. Missing values are ignored. `probs` numeric vector of probabilities with values in [0,1]. `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.

### 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.

Sample quantiles of type i are defined by

Q[i](p) = (1 - gamma) x[j] + gamma 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, and m is a constant determined by the sample quantile type. Here gamma depends on the fractional part of g = np+m-j.

For the continuous sample quantile types (4 through 9), the sample quantiles can be obtained by linear interpolation between the kth order statistic and p(k):

p(k) = (k - alpha) / (n - alpha - beta + 1),

where α and β are constants determined by the type. Further, m = alpha + p(1 - alpha - beta), and gamma = g.

Discontinuous sample quantile types 1, 2, and 3

Type 1
Inverse of empirical distribution function.
Type 2
Similar to type 1 but with averaging at discontinuities.
Type 3
SAS definition: nearest even order statistic.

Continuous sample quantile types 4 through 9

Type 4
p(k) = k / n. That is, linear interpolation of the empirical cdf.
Type 5
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
p(k) = k / (n + 1). Thus p(k) = E[F(x[k])]. This is used by Minitab and by SPSS.
Type 7
p(k) = (k - 1) / (n - 1). In this case, p(k) = mode[F(x[k])]. This is used by S.
Type 8
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
p(k) = (k - 3/8) / (n + 1/4). The resulting quantile estimates are approximately unbiased if `x` is normally distributed.

Hyndman and Fan (1996) recommend type 8. The default method is type 7, as used by S and by R < 2.0.0.

### 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.

`ecdf` for empirical distributions of which `quantile` is the “inverse”; `boxplot.stats` and `fivenum` for computing “versions” of quartiles, etc.

### Examples

```quantile(x <- rnorm(1001))# Extremes & Quartiles by default
quantile(x,  probs=c(.1,.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)
```

[Package stats version 2.2.1 Index]