What this calculator does
The Weibull distribution is the standard model for time to failure, and its shape parameter k encodes which kind of failure you are dealing with. Below 1 the failure rate decreases over time, at exactly 1 it is constant, and above 1 it increases.
That single parameter maps onto the classic bathtub curve of reliability engineering: k below 1 is infant mortality, where early faults are weeded out; k equal to 1 is random failure, the exponential case; k above 1 is wear-out, where ageing dominates. Fitting a Weibull to failure data and reading off k tells you which regime a component is in.
The formula
The cumulative distribution is 1 minus e to the minus (x/λ) to the power k, and the density is its derivative. The mean is λ times the gamma function of (1 + 1/k), which is why the mean is not simply the scale parameter. Setting k to 1 reduces the whole thing to the exponential distribution.
| Term | Meaning |
|---|---|
| k (shape) | Determines the failure rate behaviour: decreasing below 1, constant at 1, increasing above 1. |
| λ (scale) | The characteristic life. About 63.2% of items fail by this point regardless of k. |
| Bathtub curve | The three-phase reliability pattern the shape parameter distinguishes between. |
| Characteristic life | x = λ, where the cumulative probability is always 1 − 1/e. |
The inputs explained
| Field | What to enter |
|---|---|
| x value | The value at which to evaluate, typically a time or a number of cycles. |
| Shape (k) | Shape parameter. Below 1 for early-life failures, 1 for random, above 1 for wear-out. |
| Scale (λ) | Scale parameter, the characteristic life. |
When to use it
Reliability engineering
Modelling component lifetimes and deciding between preventive replacement and run-to-failure.
Wind resource assessment
Wind speed distributions are conventionally modelled as Weibull, usually with a shape around 2.
Survival analysis
Time-to-event data in medicine and engineering, where the hazard rate may rise or fall over time.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the shape parameter change things?
Evaluated at the scale parameter for several shapes.
| Shape (k) | CDF, P(X ≤ x) | Mean | Standard deviation |
|---|---|---|---|
| k = 0.7 | 0.632121 | 2.532 | 3.702 |
| k = 1 | 0.632121 | 2.000 | 2.000 |
| k = 1.5 | 0.632121 | 1.805 | 1.226 |
| k = 3 | 0.632121 | 1.786 | 0.6491 |
Questions
What does the shape parameter tell me?
Which failure regime applies. Below 1 the failure rate falls with age, indicating manufacturing defects showing up early. At exactly 1 it is constant, meaning failures are random and age-independent. Above 1 it rises, indicating wear-out. Fitting k to real data identifies which is happening.
Why is 63.2% significant?
Because at x equal to the scale parameter, the cumulative probability is 1 − 1/e, about 0.6321, for every shape value. That invariance is what makes λ meaningful as a characteristic life and why it is the standard summary quoted for a Weibull fit.
How does Weibull relate to the exponential distribution?
The exponential is the Weibull with k = 1. In that case the failure rate is constant and the distribution is memoryless, so a component that has survived a year is exactly as likely to fail in the next hour as a new one.
Why is the mean not the scale parameter?
Because the distribution is asymmetric. The mean involves the gamma function of (1 + 1/k) and equals λ only when that gamma term happens to be 1, which occurs at k = 1. At k = 3 with λ = 2 the mean is 1.786, below the scale parameter.
For a related two-dimensional case, see the Rayleigh distribution calculator. For normal probabilities, see the normal distribution calculator.