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.


vector of probabilities.


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).


logical; if TRUE (default), probabilities are \(P[X \le x]\), 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 \(= \mu\) and scale \(= \sigma\) has distribution function $$ F(x) = \frac{1}{1 + e^{-(x-\mu)/\sigma}}% $$ and density $$ f(x)= \frac{1}{\sigma}\frac{e^{(x-\mu)/\sigma}}{(1 + e^{(x-\mu)/\sigma})^2}% $$

It is a long-tailed distribution with mean \(\mu\) and variance \(\pi^2/3 \sigma^2\).


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

The length of the result is determined by n for rlogis, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.


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 a sigmoid function in contexts such as neural networks.


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.

See Also

Distributions for other standard distributions.

  • Logistic
  • dlogis
  • plogis
  • qlogis
  • rlogis
library(stats) # NOT RUN { var(rlogis(4000, 0, scale = 5)) # approximately (+/- 3) pi^2/3 * 5^2 # }
Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2

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