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

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)