Skip to Content

Logistic {stats}

The Logistic Distribution
R 3.0.2


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 ≤ x], otherwise, P[X > x].


If location or scale are omitted, they assume the default values of   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 π^2 /3 s^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 parameters for the other functions. The numerical parameters other than n are recycled to the length of the result. Only the first elements of the logical parameters are used.


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.


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.

See Also

Distributions for other standard distributions.


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

Documentation reproduced from R 3.0.2. License: GPL-2.