Weibull Distribution Calculator

The shape parameter is the hazard story

Everything that makes the Weibull useful lives in k, and it says one thing: how the risk of failing next changes as a unit ages.

k < 1 — falling hazard. The longer something survives, the safer it becomes. This is infant mortality: manufacturing defects show up early, and units that get past burn-in were the good ones all along. Electronics and software failures often look like this.

k = 1 — constant hazard. Age is irrelevant. This is exactly the exponential distribution — not approximately, identically, and the suite asserts it. Failures arrive from external shocks rather than accumulating damage.

k > 1 — rising hazard. Wear-out. Bearings, batteries, structural fatigue. The steeper k is, the more sharply failures cluster around the characteristic life, and the more a preventive replacement schedule earns its keep.

The tool plots h(t) directly so this is visible rather than inferred. At k = 3.5 the instantaneous risk at 1.5λ is many times the risk at 0.5λ; at k = 0.6 it is a fraction of it. Same family, opposite engineering conclusions.

Two special cases worth recognising: k = 2 is the Rayleigh distribution, whose hazard rises linearly, and k ≈ 3.6 makes the Weibull look almost symmetric — close enough to a normal that people sometimes mistake it for one.

Fitting a Weibull to real failure data and reading off k is therefore a diagnostic, not just a curve fit. k near 3 says wear-out and points at maintenance intervals; k near 0.5 says your burn-in process is leaking defects into the field. Those call for different fixes.

The bathtub curve is three Weibulls

Reliability engineering’s standard picture of a product’s lifetime has three phases, and each is a Weibull with a different k.

Infant mortality at the start: hazard falling, k < 1. Defective units failing fast.

Useful life in the middle: hazard flat, k = 1. Random external causes only. This is the phase — and only this phase — that an exponential model describes correctly.

Wear-out at the end: hazard rising, k > 1. Accumulated damage.

This explains a practice that otherwise looks wasteful. Burn-in deliberately runs units before shipping, so that infant-mortality failures happen in the factory rather than at a customer. It only makes sense because the hazard is falling there — a unit that survives burn-in is genuinely more reliable than an untested one, which is not true of anything with a constant or rising hazard.

Preventive replacement is the mirror image. Replacing a part on a schedule only helps when the hazard is rising: swapping a component with a constant hazard for a new one buys nothing, because the new one is exactly as likely to fail. Under a falling hazard it is actively harmful — you have traded a survivor for an untested unit.

So “should we replace this on a schedule?” is answered by k, and by nothing else in the model. That is why fitting it matters more than the probability the calculator returns.

Why λ is called the characteristic life

The scale parameter has a property that is genuinely useful and frequently misread.

P(T > λ) = e⁻¹ = 0.3679, at every value of k. Whatever the shape, 63.2% of units have failed by time λ. The suite asserts this across four shapes; it is exact, not an approximation.

That is why λ is the characteristic life rather than the mean. It is a fixed percentile of the distribution, and it does not move when k changes.

The mean is not λ. It is λ·Γ(1 + 1/k), which depends on the shape. At k = 1 that is exactly λ; at k = 2 it is about 0.886λ; at k = 3.5 about 0.900λ. Reporting λ as “average life” is wrong for every k except 1.

The median is not λ either, being λ(ln2)¹⃗ᵏ — at k = 2, about 0.833λ.

The practical reading: λ answers “by when will roughly two thirds have failed?”, which is often the question warranty planning actually needs. If you want a mean or a B10 life (the time by which 10% fail), compute those separately — the tool prints the mean and median alongside so the gap is visible.

Reading the shape off your own data

The classical method predates computers and still explains what a Weibull fit is doing better than any output table.

Take the survival function twice through logs: ln(−ln(P(T > t))) = k·ln t − k·ln λ. That is a straight line in ln t with slope k.

So plotting failure data on Weibull probability paper — ln(−ln(1−F)) against ln t — gives a line whose slope reads off the shape directly and whose intercept gives the scale. No fitting software required, and it is why those charts existed.

Curvature is the useful signal. A genuinely Weibull dataset plots straight. A bend means the shape changes over the range, which usually means two failure modes are mixed — infant mortality early and wear-out later, each with its own k. That is a finding rather than a nuisance, and it argues for separating the populations rather than fitting one curve through both.

A caution that applies to nearly all real reliability data: it is censored. Units still running when the test ended have not failed yet, and discarding them biases the estimate badly toward short lives. Proper Weibull fitting uses maximum likelihood with censoring handled explicitly, which is beyond what a browser calculator should pretend to do — this page computes probabilities from parameters you supply, and fitting parameters from censored data is a separate exercise.

When the simpler model is good enough

The Weibull nests the exponential, so the honest question is whether the extra parameter earns itself.

Use the exponential when the hazard really is flat. Random external shocks, radioactive decay, arrivals. One parameter, closed-form everything, and no risk of over-fitting a shape that is not there.

Use the Weibull when age plausibly matters. Anything mechanical, anything that fatigues, anything with a burn-in period. Assuming a constant hazard there does not produce a slightly wrong answer — it produces one wrong in a specific direction, understating late-life failures, which is exactly the region maintenance and warranty decisions live in.

The test is cheap: fit a Weibull and look at k. If the confidence interval for k comfortably contains 1, the exponential is adequate and simpler. If it does not, the extra parameter is telling you something real about the physics.

Because k = 1 is a genuine special case rather than a limit, this is a clean nested comparison — which is the same structure as the Poisson sitting inside the negative binomial for overdispersed counts. In both, the simpler model is a boundary case and the extra parameter is a testable claim.

Where it is used, beyond reliability

The Weibull is the standard tool for time-to-failure, and it turns up in a few places that have nothing to do with failure at all.

Reliability and warranty. The core application. B10 life — the time by which 10% have failed — is a Weibull quantile, and warranty reserves are set from the same curve.

Wind energy. Wind speeds at a site are conventionally modelled as Weibull, typically with k near 2. Turbine energy yield is computed by integrating the power curve against that distribution, so the shape parameter directly affects a site’s valuation.

Materials science. Brittle fracture strength follows a Weibull because failure is governed by the weakest flaw present — and the distribution arises naturally as an extreme-value law for minima, which is the deeper reason it fits so many failure processes.

Survival analysis. One of the standard parametric survival models in medicine, where k distinguishes a hazard that climbs after diagnosis from one that does not.

The unifying idea is weakest link. When a system fails as soon as its most vulnerable component does, the minimum of many random strengths tends toward a Weibull — the same kind of limiting argument that makes sums tend toward a normal, applied to minima instead.

What it cannot do

Three limits, and the first two are the ones that bite in practice.

It cannot fit a full bathtub curve with one k. The hazard is monotone by construction: falling, flat or rising, never falling then rising. A product exhibiting both infant mortality and wear-out needs a mixture of Weibulls, and forcing one curve through both phases gives a shape parameter that describes neither.

It assumes a fixed population. If units come from different production batches with different quality, you have a mixture, and the fitted k will be pulled toward whichever mode dominates the observation window.

Parameters from small or censored samples are uncertain. k in particular is hard to pin down: with twenty failures the interval around it is wide enough to span “falling” and “rising”, which are opposite engineering conclusions. A point estimate of k without an interval is close to meaningless for decision-making.

And the general caveat this page shares with every distribution here: these probabilities are exact given the parameters you supply. Whether those parameters describe your process is a question about your data, and it is the harder half.

Sources and methodology

Reliability references for the shape parameter and the hazard framing.

• NIST/SEMATECH e-Handbook — the Weibull distribution in reliability
• NIST/SEMATECH e-Handbook — the bathtub curve and the three hazard regimes
• NIST/SEMATECH e-Handbook — Weibull probability plotting and reading the shape off a line
• Exponential Distribution Calculator — exactly this distribution at k = 1
• Poisson Distribution Calculator — the same nested-model structure, for counts
• How we calculate — methodology

Method. Survival is computed as e⁻⁽ᵗˈᵤ⁾ᵏ directly rather than as 1 − CDF, so a long survival time does not lose precision to cancellation. The mean uses the log-gamma function rather than a table of Γ values, which keeps it accurate for large or fractional shapes. The hazard function is plotted from its closed form beside the density, because the shape parameter is only interpretable through it. The suite asserts the two identities this page rests on: that k = 1 reproduces the exponential CDF to within 1e-14, and that P(T > λ) = e⁻¹ = 0.367879 at every shape tested — which is what makes λ the characteristic life rather than the mean. That engine is verified on every change against 219 assertions. The count and the per-case breakdown are published on the formula verification page.

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