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, probs=seq(0, 1, 0.25), na.rm=FALSE, names = TRUE)

Details

A vector of length length(probs) is returned; if names = TRUE, it has a names attribute.

quantile(x,p) as a function of p linearly interpolates the points ( (i-1)/(n-1), ox[i] ), where ox <- order(x) (the “order statistics”) and n <- length(x).

This gives quantile(x, p) == (1-f)*ox[i] + f*ox[i+1], where r <- 1 + (n-1)*p, i <- floor(r), f <- r - i and ox[n+1] := ox[n].

Examples

quantile(x <- rnorm(1001))# Extremes & Quartiles by default
quantile(x,  probs=c(.1,.5,1,2,5,10,50)/100)

n <- length(x) ## the following is exact, because 1/(1001-1) is exact:
all(abs(sort(x) == quantile(x, probs = ((1:n)-1)/(n-1), names=F)))# TRUE

n <- 777
ox <- sort(x <- round(rnorm(n),1))# round() produces ties
ox <- c(ox, ox[n]) #- such that ox[n+1] := ox[n]
p <- c(0,1,runif(100))
i <- floor(r <- 1 + (n-1)*p)
f <- r - i
all(abs(quantile(x,p) - ((1-f)*ox[i] + f*ox[i+1])) < 20*.Machine$double.eps)