Weibull Distribution Calculator
Enter the shape and scale parameters of a Weibull distribution plus an x value to get the probability density, cumulative probability, and key summary statistics.
What this calculator does
The Weibull distribution is widely used in reliability engineering, weather modeling (wind speeds), and survival analysis to describe how long something lasts or how likely an event is at a given value of x. This calculator finds the probability density f(x), the cumulative probability F(x), and the distribution’s mean, standard deviation, and median for the shape (k) and scale (λ) parameters you enter.
How the calculation works
The probability density function is f(x) = (k/λ)(x/λ)^(k-1) e^(-(x/λ)^k), and the cumulative distribution function is F(x) = 1 – e^(-(x/λ)^k). The mean and variance use the gamma function: mean = λ Γ(1 + 1/k), and variance = λ²[Γ(1 + 2/k) – Γ(1 + 1/k)²]. A shape parameter k below 1 models a decreasing failure rate, k = 1 reduces to the exponential distribution, and k above 1 models an increasing failure rate.
This tool is for general statistical and educational use. For safety-critical reliability analysis, verify results with dedicated statistical software.