Logistic Distribution Calculator

Compute the probability density, cumulative distribution, upper tail probability, mean, and variance of a logistic distribution with location mu and scale s, with charts.

Input

Compute probabilities for a logistic distribution with location mu and scale s. Enter x, mu, and s.

Value x

The value of x to evaluate

Location mu

Center of the distribution (the mean)

Scale s

A positive value that controls the spread

Result

Lower tail probability F(x) = P(X ≤ 1)

0.73105858

Upper tail 1 − F(x)

0.26894142

Density f(x)

0.19661193

Mean

Variance

3.28986813

Probability density function f(x)

Cumulative distribution function F(x)

How it works

The logistic distribution is defined by a location parameter mu and a scale parameter s. It resembles the normal distribution with a symmetric bell shape but has heavier tails.
The density is f(x) = e^(−z) / ( s (1 + e^(−z))² ) and the cumulative distribution is F(x) = 1 / (1 + e^(−z)), where z = (x − mu) / s.
The mean equals mu and the variance equals pi² s² / 3. The scale s must be a positive value.
The upper tail probability is 1 − F(x). Because the CDF is available in closed form, no numerical approximation is required.

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