Weibull Distribution Calculator
Calculate Weibull PDF, CDF, reliability, and hazard rate from shape and scale parameters.
Find mean, median, and mode for reliability and failure analysis.
Weibull Distribution
The Weibull distribution is a versatile probability distribution widely used in reliability engineering, survival analysis, and failure time modelling. By adjusting the shape parameter k, it can model increasing, constant, or decreasing failure rates.
Parameters
- k (shape / β): Controls the failure rate behavior
- λ (scale / η): Characteristic life, the time by which 63.2% of units will have failed
Key Formulas
| Function | Formula |
|---|---|
| f(x) = (k/λ)(x/λ)^(k−1) × exp(−(x/λ)^k) | |
| CDF | F(x) = 1 − exp(−(x/λ)^k) |
| Reliability | R(x) = exp(−(x/λ)^k) |
| Hazard rate | h(x) = (k/λ)(x/λ)^(k−1) |
| Mean | μ = λ × Γ(1 + 1/k) |
| Median | m = λ × (ln 2)^(1/k) |
| Mode | (k−1)^(1/k) × λ/k^(1/k) for k > 1, else 0 |
Shape Parameter Interpretation
| k | Failure Rate | Common Scenario |
|---|---|---|
| k < 1 | Decreasing | Infant mortality / early failures |
| k = 1 | Constant | Random / exponential distribution |
| k = 2 | Increasing (linear) | Wear-out failures |
| k = 3–4 | Bell-shaped | Near-normal, aging products |
The 63.2% Rule
At x = λ (the scale parameter), the CDF always equals 1 − e⁻¹ ≈ 63.2%, whatever k happens to be. Substitute x = λ into the formula and the ratio (x/λ) becomes 1, so 1 raised to any power is still 1 and the shape parameter drops out entirely. That is what makes λ a “characteristic life” rather than an average: it is the one point on the curve that means the same thing across every shape.
Note that λ is not the mean. The mean is λ × Γ(1 + 1/k), so the gap between them depends on the shape. At k = 1 the two are equal; the mean then dips to a minimum of about 0.886 λ near k = 2.2 and creeps back toward λ as k grows large. For the common k = 2 case with λ = 100, the mean life is 88.6 while 63.2% have already failed by 100.
Reading the shape parameter
The whole reason engineers reach for Weibull rather than the exponential distribution is that k lets one formula describe three completely different failure stories, which together trace the bathtub curve every reliability textbook opens with.
Below 1, things fail early and the survivors improve with age. Manufacturing defects behave this way, and it is the entire justification for burn-in testing: run the population hard for a short time, let the weak units die on your bench instead of the customer’s.
At exactly 1 the hazard rate is flat, so a part that has run for a year is neither more nor less likely to fail tomorrow than a new one. Scheduled replacement is money wasted here, because the replacement is no better than what you pulled out.
Above 1 the hazard rate climbs with age. This is wear-out, and it is the only regime where replacing a part before it fails actually buys you anything.
Where it gets used
Wind resource assessment fits a Weibull to measured wind speeds, usually landing near k = 2. Bearing manufacturers quote B10 life, the point at which 10% of a batch has failed, which is the inverse CDF at p = 0.10. Medical survival analysis, battery cycle life and extreme weather return periods all lean on the same two parameters.
Fitting it to your own data
This calculator takes k and λ as given. Getting them from failure data is a separate job: plot the failure times on Weibull probability paper, or run a maximum-likelihood fit. The classic quick method is a regression of ln(−ln(1−F)) against ln(x): the slope is k directly, and λ falls out of the intercept as exp(−intercept / k). A visibly bent line on that plot usually means two failure mechanisms are competing, and a single Weibull will not describe either of them well.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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