quantile | R Documentation |

### Description

The generic function `quantile`

produces sample quantilescorresponding to the given probabilities.The smallest observation corresponds to a probability of 0 and thelargest 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, ...)

### Arguments

`x` | numeric vector whose sample quantiles are wanted, or anobject of a class for which a method has been defined (see also‘details’). |

`probs` | numeric vector of probabilities with values in |

`na.rm` | logical; if true, any |

`names` | logical; if true, the result has a |

`type` | an integer between 1 and 9 selecting one of thenine quantile algorithms detailed below to be used. |

`digits` | used only when |

`...` | 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`

arepropagated to the result.

The default method works with classed objects sufficiently likenumeric 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`

inbase will be used, not some S4 generic of that name. Also notethat that is no check on the ‘correctly’, and soe.g. `quantile`

can be applied to complex vectors which (apartfrom 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 quantilesbased on one or two order statistics from the supplied elements in`x`

at probabilities in `probs`

. One of the nine quantilealgorithms discussed in Hyndman and Fan (1996), selected by`type`

, is employed.

All sample quantiles are defined as weighted averages ofconsecutive 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 *j*th order statistic, *n* is thesample 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 discontinuousfunction 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.

*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 (SAS default, seeWicklin(2017)).- Type 3
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 functionof *p*, with *gamma = g* and *m* given below. Thesample quantiles can be obtained equivalently by linear interpolationbetween 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] = 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 valuesmidway through the steps of the empirical cdf. This is popularamongst 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-unbiasedregardless 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 forthe 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 ofthese 9 types in software.

### 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 statisticalpackages, *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

### See Also

`ecdf`

for empirical distributions of which`quantile`

is an inverse;`boxplot.stats`

and `fivenum`

for computingother versions of quartiles, etc.

### Examples

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, probs = prob, type = typ, ...), quantile(x, prob, type=1, ...)))p <- c(0.1, 0.5, 1, 2, 5, 10, 50)/100signif(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)