The Weibull distribution is reliability engineering’s workhorse: it models time-to-failure for components that may wear in, fail randomly, or wear out, depending on one shape parameter. This Weibull Distribution Calculator takes the scale λ and shape k plus a time x, and returns the cumulative failure probability, the reliability (survival probability), the density, and the mean and median life — the numbers lifetime decisions are actually made from.
Worked Example: λ = 100 Hours, k = 1.5, at x = 100
At exactly the characteristic life x = λ = 100, the calculator reports CDF 0.6321 and reliability 0.3679 — no coincidence: F(λ) = 1 − e^(−1) ≈ 63.2% for every shape k, which is why λ is called the characteristic life, the age by which about two-thirds of units have failed. The density there is f(100) = 0.005518 per hour. The mean life comes out at 90.27 hours — λΓ(1 + 1/k) = 100 × Γ(1.6667) ≈ 100 × 0.9027 — and the median life at 78.32 hours, λ(ln 2)^(1/k). With k = 1.5 the hazard grows with age (the wear-out regime), so the median falls visibly short of the mean: half the units are gone by hour 78, while a long-lived minority pulls the average to 90. A reliability engineer reading these three numbers together already knows whether to schedule preventive replacement before the median or run to failure.
The Formulas and the Shape Parameter’s Three Regimes
CDF: F(x) = 1 − exp(−(x/λ)^k). Reliability: R(x) = exp(−(x/λ)^k) — the two sum to 1 at every x, as they must. Density: f(x) = (k/λ)(x/λ)^(k−1) · exp(−(x/λ)^k). Mean: λΓ(1 + 1/k); median: λ(ln 2)^(1/k); the hazard rate h(x) = (k/λ)(x/λ)^(k−1) carries the physical story. When k < 1 the hazard decreases with age — infant mortality: weak units fail early, survivors are the strong ones (electronics burn-in). When k = 1 the hazard is constant and the Weibull collapses to the exponential distribution — random failures, no aging. When k > 1 the hazard increases — wear-out: bearings, insulation, fatigue cracks. Estimating k from failure data (the slope on a Weibull probability plot) is thus a diagnosis, not just a fit: it tells maintenance whether to burn in, ignore age, or schedule replacements — which is why this two-parameter family covers so much of lifetime analysis.
Where the Weibull Works
Component and warranty life: manufacturers fit Weibull models to failure times to set warranty periods at a chosen reliability — say the age where R = 0.95 — rather than at the mean. Wind energy: wind-speed distributions are routinely Weibull with k near 2 (the Rayleigh special case), feeding turbine output forecasts. Materials strength: the weakest-link theory behind Weibull’s original 1939 paper makes fracture strength of brittle materials Weibull-distributed, with size effects built in. Survival analysis in medicine and hydrology (extreme flows, drought durations) round out the territory. In all of them the calculator’s outputs map directly to decisions: F(x) prices warranty risk, R(x) schedules maintenance, the median plans spares.
Common Mistakes
Quoting λ as “the average life” — it is the 63.2-percentile life; the mean is λΓ(1 + 1/k), which sits above λ when k is below 1, equals λ exactly at k = 1 (because Γ(2) = 1), and sits below λ for every k above 1 — read the mean line rather than assuming. Mixing parameterizations: some software uses (α, β) with scale and shape swapped relative to this page’s (λ, k) — always check which parameter the CDF formula shows. Fitting a single Weibull to data with two failure modes (early defects plus wear-out), which no single k can bend to fit — the probability plot kinks, and a mixture model is the honest fix. And extrapolating reliability far beyond the observed failure ages, where the tail is assumption, not evidence.
Speaking B-Life: The Percentile Language of Reliability
Reliability datasheets quote B-lives: the B10 life is the age by which 10% of units have failed — the solution of F(x) = 0.10, which the CDF formula inverts to x = λ(−ln 0.9)^(1/k). For this page’s example (λ = 100, k = 1.5) the B10 life is about 22.31 hours — a sobering contrast with the mean life of 90.27: one unit in ten is dead before a quarter of the “average” lifetime has passed. Bearing catalogs rate L10 life on exactly this basis. Whenever a vendor quotes a single lifetime number, the first question is which percentile it is; the calculator’s CDF lets you translate any quoted age into its failure percentage and compare like with like.
Related Tools
Continue with the Lognormal Distribution Calculator, the Beta Distribution Calculator, the Benford’s Law Calculator, or browse the full Statistics Calculators hub.
Frequently Asked Questions
What do λ and k mean in a Weibull distribution?
λ (lambda) is the scale or characteristic life: by age λ, 63.2% of units have failed, for any shape. k is the shape: below 1 means early-life failures dominate, 1 means random (exponential), above 1 means wear-out.
What is the reliability at the characteristic life?
R(λ) = e^(−1) ≈ 36.79%, always. Correspondingly F(λ) ≈ 63.21% — the defining property of λ, independent of k.
What is the mean of a Weibull distribution?
λΓ(1 + 1/k), where Γ is the gamma function. For λ = 100 and k = 1.5 that is 100 × Γ(1.6667) ≈ 90.27 — less than λ here, because Γ(1 + 1/k) is below 1 for every k above 1 (the gamma function dips under 1 for arguments between 1 and 2).
How does the Weibull relate to the exponential distribution?
It contains it: at k = 1 the Weibull CDF becomes 1 − e^(−x/λ), the exponential with mean λ. The exponential is the constant-hazard special case.
What is the median life formula?
λ(ln 2)^(1/k). For λ = 100, k = 1.5: 100 × 0.6931^(2/3) ≈ 78.32 — the age by which half the units have failed.
Why do engineers prefer percentiles over the mean here?
Decisions attach to reliability targets: the age at which 95% still survive, not the average. The CDF on this page gives any percentile by solving F(x) = p — or read R(x) directly for survival guarantees.