Logistic {stats}R Documentation

The Logistic Distribution


Density, distribution function, quantile function and random generation for the logistic distribution with parameters location and scale.


dlogis(x, location = 0, scale = 1, log = FALSE)
plogis(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qlogis(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rlogis(n, location = 0, scale = 1)


x, q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number required.
location, scale location and scale parameters.
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


If location or scale are omitted, they assume the default values of 0 and 1 respectively.

The Logistic distribution with location = m and scale = s has distribution function

F(x) = 1 / (1 + exp(-(x-m)/s))

and density

f(x) = 1/s exp((x-m)/s) (1 + exp((x-m)/s))^-2.

It is a long-tailed distribution with mean m and variance pi^2 /3 s^2.


dlogis gives the density, plogis gives the distribution function, qlogis gives the quantile function, and rlogis generates random deviates.


qlogis(p) is the same as the well known ‘logit’ function, logit(p) = log(p/(1-p)), and plogis(x) has consequently been called the “inverse logit”.

The distribution function is a rescaled hyperbolic tangent, plogis(x) == (1+ tanh(x/2))/2, and it is called sigmoid function in contexts such as neural networks.


[dpr]logis are calculated directly from the definitions.

rlogis uses inversion.


Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, chapter 23. Wiley, New York.


var(rlogis(4000, 0, s = 5))# approximately (+/- 3)
pi^2/3 * 5^2

[Package stats version 2.5.0 Index]