Logistic Distribution Calculator

The logistic distribution is a continuous probability distribution whose cumulative distribution function is the logistic function. It resembles the normal distribution in shape but has heavier tails. It is parameterized by a location parameter (μ) and a scale parameter (s > 0).

Both distributions are symmetric and bell-shaped, but the logistic distribution has heavier tails than the normal distribution. This means extreme values are slightly more probable under the logistic distribution. Their CDFs also have different mathematical forms — the logistic CDF is a simple closed-form sigmoid function, whereas the normal CDF requires numerical integration.

The location parameter μ determines the center (mean and median) of the logistic distribution. Shifting μ moves the entire distribution left or right along the x-axis without changing its shape.

The scale parameter s controls the spread or dispersion of the distribution. A larger value of s produces a flatter, wider distribution, while a smaller s produces a taller, narrower peak. The standard deviation of the logistic distribution equals s·π/√3, and the variance equals s²·π²/3.

The PDF of the logistic distribution is f(x) = e^(-(x−μ)/s) / [s · (1 + e^(-(x−μ)/s))²]. It gives the relative likelihood of the random variable taking a specific value x.

The CDF is F(x) = 1 / (1 + e^(-(x−μ)/s)), commonly known as the sigmoid or logistic function. It returns the probability that a random variable X is less than or equal to x.

The probability that X falls between a and b is P(a < X < b) = F(b) − F(a), where F is the CDF. Enter values for both a and b in this calculator to compute this probability automatically.

The logistic distribution is widely used in logistic regression for binary classification, in epidemiology to model growth curves, in survival analysis, and in economics. Its CDF — the sigmoid function — is foundational in machine learning and neural networks.