Logistic Distribution Calculator

Logistic Distribution Calculator. Enter a percentile (x), location parameter (μ), and scale parameter (s) to evaluate the Logistic Distribution. This calculator returns the Probability Density Function (PDF), Lower Cumulative Distribution Function (CDF), Upper CDF, and optional interval probability P(a < X < b) for any valid inputs. Also try the calculate Chi-Square Distribution.

Results

Probability Density Function (PDF)

Lower CDF — P(X ≤ x)

Upper CDF — P(X > x)

Interval Probability — P(a < X < b)

Mean (μ)

Variance (π²s²/3)

What is the logistic distribution?

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). See also our use the Gumbel Distribution Calculator.

What is the difference between the logistic distribution and the normal distribution?

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.

What does the location parameter (μ) represent?

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.

What does the scale parameter (s) represent?

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. You might also find our Standard Normal Distribution Calculator useful.

How is the Probability Density Function (PDF) calculated?

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.

What is the cumulative distribution function (CDF) of the logistic distribution?

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.

How do I calculate the interval probability P(a < X < b)?

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.

What are common applications of the logistic distribution?

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.