# ecdf: Empirical Cumulative Distribution Function

### Description

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

### Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```ecdf(x) ## S3 method for class 'ecdf' plot(x, ..., ylab="Fn(x)", verticals = FALSE, col.01line = "gray70", pch = 19) ## S3 method for class 'ecdf' print(x, digits= getOption("digits") - 2, ...) ## S3 method for class 'ecdf' summary(object, ...) ## S3 method for class 'ecdf' quantile(x, ...) ```

### Arguments

 `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. `ylab` label for the y-axis. `verticals` see `plot.stepfun`. `col.01line` numeric or character specifying the color of the horizontal lines at y = 0 and 1, see `colors`. `pch` plotting character. `digits` number of significant digits to use, see `print`.

### Details

The e.c.d.f. (empirical cumulative distribution function) Fn 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`= (x1,x2, ... xn), Fn is the fraction of observations less or equal to t, i.e.,

Fn(t) = #{xi <= t}/n = 1/n sum(i=1,n) Indicator(xi <= t).

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

### Value

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.

### Note

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

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

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

### Author(s)

Martin Maechler; fixes and new features by other R-core members.

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

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52``` ```##-- 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) ```

Questions? Problems? Suggestions? or email at ian@mutexlabs.com.

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