Empirical Cumulative Distribution Function

Compute an empirical cumulative distribution function, with several methods for plotting, printing and computing with such an “ecdf” object.

hplot, dplot

# S3 method for ecdf plot(x, …, ylab="Fn(x)", verticals = FALSE, col.01line = "gray70", pch = 19)

# S3 method for ecdf print(x, digits= getOption("digits") - 2, …)

# S3 method for ecdf summary(object, …) # S3 method for ecdf quantile(x, …)

x, object

numeric vector of the observations for ecdf; for the methods, an object inheriting from class "ecdf".

arguments to be passed to subsequent methods, e.g., plot.stepfun for the plot method.


label for the y-axis.


see plot.stepfun.


numeric or character specifying the color of the horizontal lines at y = 0 and 1, see colors.


plotting character.


number of significant digits to use, see print.


The e.c.d.f. (empirical cumulative distribution function) \(F_n\) is a step function with jumps \(i/n\) at observation values, where \(i\) is the number of tied observations at that value. Missing values are ignored.

For observations x\(= (\)\(x_1,x_2\), … \(x_n)\), \(F_n\) is the fraction of observations less or equal to \(t\), i.e., $$F_n(t) = \#\{x_i\le t\}\ / n = \frac1 n\sum_{i=1}^n \mathbf{1}_{[x_i \le t]}.$$

The function plot.ecdf which implements the plot method for ecdf objects, is implemented via a call to plot.stepfun; see its documentation.


For ecdf, a function of class "ecdf", inheriting from the "stepfun" class, and hence inheriting a knots() method.

For the summary method, a summary of the knots of object with a "header" attribute.

The quantile(obj, ...) method computes the same quantiles as quantile(x, ...) would where x is the original sample.


The objects of class "ecdf" are not intended to be used for permanent storage and may change structure between versions of R (and did at R 3.0.0). They can usually be re-created by

    eval(attr(old_obj, "call"), environment(old_obj))

since the data used is stored as part of the object's environment.

See Also

stepfun, the more general class of step functions, approxfun and splinefun.

  • ecdf
  • plot.ecdf
  • print.ecdf
  • summary.ecdf
  • quantile.ecdf
library(stats) # NOT RUN { ##-- Simple didactical ecdf example : x <- rnorm(12) Fn <- ecdf(x) Fn # a *function* Fn(x) # returns the percentiles for x tt <- seq(-2, 2, by = 0.1) 12 * Fn(tt) # Fn is a 'simple' function {with values k/12} summary(Fn) ##--> see below for graphics knots(Fn) # the unique data values {12 of them if there were no ties} y <- round(rnorm(12), 1); y[3] <- y[1] Fn12 <- ecdf(y) Fn12 knots(Fn12) # unique values (always less than 12!) summary(Fn12) summary.stepfun(Fn12) ## Advanced: What's inside the function closure? ls(environment(Fn12)) ##[1] "f" "method" "n" "x" "y" "yleft" "yright" utils::ls.str(environment(Fn12)) stopifnot(all.equal(quantile(Fn12), quantile(y))) ###----------------- Plotting -------------------------- require(graphics) op <- par(mfrow = c(3, 1), mgp = c(1.5, 0.8, 0), mar = .1+c(3,3,2,1)) F10 <- ecdf(rnorm(10)) summary(F10) plot(F10) plot(F10, verticals = TRUE, do.points = FALSE) plot(Fn12 , lwd = 2) ; mtext("lwd = 2", adj = 1) xx <- unique(sort(c(seq(-3, 2, length = 201), knots(Fn12)))) lines(xx, Fn12(xx), col = "blue") abline(v = knots(Fn12), lty = 2, col = "gray70") plot(xx, Fn12(xx), type = "o", cex = .1) #- plot.default {ugly} plot(Fn12, col.hor = "red", add = TRUE) #- plot method abline(v = knots(Fn12), lty = 2, col = "gray70") ## luxury plot plot(Fn12, verticals = TRUE, col.points = "blue", col.hor = "red", col.vert = "bisque") ##-- this works too (automatic call to ecdf(.)): plot.ecdf(rnorm(24)) title("via simple plot.ecdf(x)", adj = 1) par(op) # }
Documentation reproduced from package stats, version 3.6.1, License: Part of R 3.6.1

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