Weibull Distribution Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The Weibull distribution models a component's reliability over time: R(t) = exp(−(t/λ)^k), where λ is the scale parameter (the characteristic lifetime) and k the shape parameter. At t=λ with k=2, reliability is about 36.8%, whatever the value of λ.
Explanation
The Weibull distribution is the reference tool in reliability analysis for modeling the time to failure of a component, machine, or system. Its strength lies in its shape parameter k, which can represent three very different failure regimes with a single family of formulas: k<1 describes a decreasing failure rate ("infant mortality", typical of manufacturing defects that show up early), k=1 describes a constant failure rate over time — in this specific case, the Weibull distribution reduces exactly to our exponential distribution calculator, as this page's second test case confirms — and k>1 describes an increasing failure rate, typical of the progressive wear of mechanical components. Reliability R(t) gives the probability that the component still works without failure after a duration t; the probability of failure F(t) = 1 − R(t) gives the complementary probability that it has failed before that time. This distribution is widely used in engineering to plan preventive maintenance, estimate a product's guaranteed lifetime, or size spare-part inventories. For a strictly positive quantity with a heavier tail, see instead our log-normal distribution calculator.
Example: t=λ=1000, k=2
Inputs
Time: 1000. Scale parameter: 1000. Shape parameter: 2.
Calculation
R(t) = exp(−(1000÷1000)²) = exp(−1) ≈ 0.3679. F(t) = 1 − 0.3679 = 0.6321.
Result
At this time, reliability is about 36.79% and the probability of failure about 63.21%.
Frequently asked questions
What does the shape parameter k mean?
It determines how the failure rate changes over time: k<1 indicates a decreasing rate (early defects becoming rarer), k=1 a constant rate (random failures unrelated to the component's age), and k>1 an increasing rate (progressive wear, material fatigue). It is this parameter that gives the Weibull distribution its great flexibility for modeling very different failure behaviors.
What does the scale parameter λ represent?
The scale parameter represents the component's characteristic lifetime: it is the time at which about 63.2% of units have already failed (and therefore 36.8% remain reliable), a property that holds whatever the shape parameter k, as this calculator's first test case shows.
Why does the Weibull distribution reduce to the exponential distribution when k=1?
Because with k=1, the exponent (t/λ)^k simplifies to t/λ, which gives R(t) = exp(−t/λ) — exactly the form of our exponential distribution calculator, with a rate λ_exp equal to the inverse of the scale parameter (1/λ). It is the signature of a purely random failure process, with no aging or run-in effect.