Weibull Distribution Calculator
Enter the scale parameter (lambda), shape parameter (k), and a value of interest (x) to compute the Weibull probability density, cumulative probability, survival function, and hazard rate all at once. Switch to Quantile mode to find the x that corresponds to any given probability. The distribution mean, median, mode, variance, and standard deviation update live regardless of mode.
Formula
Worked example
For lambda = 1 and k = 2 (Rayleigh), evaluated at x = 1: z = 1/1 = 1, z^2 = 1, PDF = (2/1)(1)^1 * e^(-1) = 2e^-1 ≈ 0.7358, CDF = 1 - e^(-1) ≈ 0.6321, Survival = e^(-1) ≈ 0.3679, Hazard = (2/1)(1)^1 = 2. Mean = 1 * Gamma(1.5) = 1 * 0.8862 ≈ 0.8862.
What is the Weibull distribution?
The Weibull distribution is a two-parameter continuous probability distribution used extensively in reliability engineering, survival analysis, and failure-time modelling. It was developed by Waloddi Weibull in 1951 to describe the breaking strength of materials, and it has since become the standard tool for modelling time-to-failure data. Its key advantage is flexibility: by adjusting the shape parameter k, a single distribution family can describe components that fail early, randomly, or through gradual wear-out. The scale parameter lambda (sometimes written eta) sets the characteristic life, which is the time by which approximately 63.2% of a population is expected to have failed regardless of the shape.
PDF, CDF, survival function, and hazard rate explained
The probability density function (PDF) f(x) gives the relative likelihood of a continuous random variable taking a particular value. The cumulative distribution function (CDF) F(x) = P(X <= x) tells you what fraction of a population fails by time x. The survival function S(x) = 1 - F(x) = P(X > x) is its complement: the probability of surviving past x, which is the reliability in engineering contexts. The hazard rate h(x) = f(x) / S(x) measures the instantaneous risk of failure at time x given survival up to that point. For the Weibull distribution, h(x) = (k / lambda)(x / lambda)^(k-1), which is a pure power function of time. When k = 1 this equals 1/lambda, a constant, recovering the exponential distribution. When k > 1 the hazard increases with time, modelling wear-out. When k < 1 it decreases, modelling infant-mortality patterns where the weakest items fail first and survivors improve.
How to use this calculator
Set the scale parameter lambda and shape parameter k for your Weibull model, then choose a calculation mode. In PDF/CDF/Survival/Hazard mode, enter the time or measurement value x to get all four quantities at once. In Quantile mode, enter a cumulative probability p (such as 0.90 for the 90th percentile) to find the x at which that fraction of the population has failed. In Range mode, enter a lower and upper bound to compute the probability that the random variable falls within that interval. The mean, median, mode, variance, and standard deviation are always displayed regardless of mode. The shape-parameter reference table gives guidance on choosing k for your application, and the interactive chart shows the PDF and survival curves so you can see the full distribution shape.
Moments: mean, median, mode, and variance
The mean (expected value) of the Weibull distribution is lambda * Gamma(1 + 1/k), where Gamma is the gamma function. The median is lambda * ln(2)^(1/k). The mode - the most likely value - is lambda * ((k-1)/k)^(1/k) when k > 1 and equals zero when k <= 1. The variance is lambda^2 * [Gamma(1 + 2/k) - Gamma(1 + 1/k)^2]. As k increases toward 3.4, the distribution becomes more symmetric and these quantities converge, approximating the normal distribution. For k = 1, all formulas reduce to those of the exponential distribution with rate 1/lambda, and the mean equals lambda.
Shape parameter k and distribution behaviour
| k value | Special case / name | Hazard rate | Typical application |
|---|---|---|---|
| k < 1 | Stretched exponential | Decreasing | Infant-mortality, early-life failures |
| k = 1 | Exponential distribution | Constant (memoryless) | Random failures, electronic components |
| k = 2 | Rayleigh distribution | Linearly increasing | Wind speed, mechanical wear |
| k = 2.5 | (intermediate) | Increasing | Fatigue failures |
| k = 3.4 | Normal approximation | Increasing | Wear-out, aging, material fatigue |
| k > 4 | Steep wear-out | Rapidly increasing | Ball bearings, brittle fracture |
The shape parameter determines the failure-rate pattern and gives the Weibull distribution its flexibility.
Frequently asked questions
What do the scale and shape parameters mean?
The scale parameter lambda (also called eta or the characteristic life) stretches or compresses the distribution along the x-axis. It equals the value at which the CDF is exactly 1 - 1/e ≈ 63.2% regardless of k, making it a natural "typical failure time." The shape parameter k controls the curvature of the distribution and the behaviour of the hazard rate. A value below 1 gives a decreasing hazard (infant mortality), k = 1 gives a constant hazard (exponential), and k > 1 gives an increasing hazard (wear-out).
How is the Weibull distribution used in reliability engineering?
Engineers fit the Weibull distribution to observed failure times to estimate metrics such as B10 life (the time by which 10% of units fail), the mean time to failure (MTTF), and warranty coverage. The survival function S(x) directly gives the fraction of units expected to survive to time x, and the hazard rate shows how quickly risk accumulates over the product life. By plotting failure data on Weibull probability paper (or fitting parameters computationally), engineers can determine whether a design fails by random causes or by wear-out, and plan maintenance schedules accordingly.
What is the quantile function and how do I use it?
The quantile function Q(p) is the inverse of the CDF: given a probability p, it returns the value x such that P(X <= x) = p. In reliability, Q(0.1) is the B10 life (10% failure point). In quality control, Q(0.999) is the x value below which 99.9% of measurements fall. Set the calculator to Quantile mode, enter p as a decimal between 0 and 1, and the calculator returns Q(p) = lambda * (-ln(1-p))^(1/k).
What is the hazard rate and why does it matter?
The hazard rate h(x) - also called the failure rate or force of mortality - is the conditional probability density of failure at exactly time x, given that the item has survived to time x. In the Weibull model, h(x) = (k/lambda)(x/lambda)^(k-1). For engineering, a rising hazard rate means that older components are at increasing risk, so preventive maintenance should replace parts before they reach a high-hazard age. A decreasing hazard rate (k < 1) means survival improves with time, often seen in electronic components where early "burn-in" failures thin out the weak units.
What are the special cases of the Weibull distribution?
When k = 1, the Weibull distribution becomes the exponential distribution with rate 1/lambda - a memoryless distribution where past survival gives no information about future risk. When k = 2, it becomes the Rayleigh distribution, widely used to model wind speeds and signal amplitudes in communications. When k is approximately 3.4, the distribution closely resembles a normal (Gaussian) distribution in shape. These special cases make the Weibull family exceptionally versatile for modelling diverse physical phenomena.
How do I choose k and lambda for my data?
In practice, k and lambda are estimated from observed failure data, usually by maximum likelihood estimation (MLE) or by linear regression on a Weibull probability plot. If failures follow a decreasing trend, k < 1 is indicated. If failures are roughly constant (exponential-looking), k = 1 is a good starting point. If failures cluster around a central wear-out time, k > 2 is typical. Statistical software packages provide MLE fitting with confidence intervals. For a first approximation, the sample mean and variance can be used to derive moment-matched estimates.