Weibull Reliability and Life Distribution Calculator

Screen component life behavior with a two- or three-parameter Weibull distribution and calculate reliability, hazard, B-life, and life moments.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Distribution inputs

Life-distribution result

Reliability at evaluation life, R(t)— %

Cumulative failures, F(t)— %

Target B-life— cycles

Hazard rate— 1/cycles

Density, f(t)— 1/cycles

Mean life— cycles

Median life— cycles

Mode— cycles

Standard deviation— cycles

Variance— cycles²

How to use

Select one consistent life unit, then enter positive shape β and characteristic life η. Enter location γ = 0 for the standard two-parameter model or γ ≥ 0 for a threshold-life model.
Enter evaluation life t ≥ γ and a target cumulative failure probability p strictly between 0 and 100%. For example, p = 10% returns B10 life.
Select Calculate. Read reliability as the surviving fraction, cumulative failures as the failed fraction, and hazard as the instantaneous conditional failure rate.
Review B-life, moments, and the probability plot in the selected life unit. A missing mode for β ≤ 1 is mathematically intentional.
Select Reset to restore the worked example. Correct any warning before interpreting cleared outputs.
Do not use this parameter-input screen to estimate parameters, analyze censoring, combine mixed populations, or extrapolate beyond evidence without a reviewed life-data analysis.

Worked example

For β = 2, η = 1000 cycles, γ = 0, and t = 500 cycles, the model gives F(t) = 22.12%, R(t) = 77.88%, hazard h(t) = 0.001 per cycle, and B10 ≈ 324.59 cycles. The increasing hazard indicates wear-out-like behavior; it does not by itself prove a physical failure mechanism.

Equations used

Let adjusted life be u = t − γ, with t ≥ γ, shape β > 0, scale η > 0, and target cumulative probability p in (0,1).

Reliability: R(t) = exp[−(u/η)^β]
CDF: F(t) = 1 − R(t)
Density: f(t) = (β/η)(u/η)^β−1R(t)
Hazard: h(t) = f(t)/R(t) = (β/η)(u/η)^β−1
B-life: B_p = γ + η[−ln(1−p)]^1/β
Mean: E[T] = γ + ηΓ(1+1/β)
Variance: Var[T] = η²{Γ(1+2/β) − Γ(1+1/β)²}
Standard deviation: σ_T = sqrt(Var[T])
Median: γ + η(ln 2)^1/β
Mode: γ + η[(β−1)/β]^1/β for β > 1; no finite interior mode is reported for β ≤ 1.

Γ is the Gamma function. η, γ, t, B-life, mean, median, mode, and standard deviation share the selected life unit. Density and hazard use its reciprocal; variance uses its square. Positive life advances toward failure. At t = γ, β < 1, density and hazard tend to infinity; for β = 1, both equal 1/η; for β > 1, both equal zero. Away from the threshold, β < 1 means decreasing hazard, β = 1 constant hazard, and β > 1 increasing hazard.

Theory and method

The Weibull distribution is a continuous parametric model. The location parameter shifts the support to t ≥ γ; setting γ = 0 yields the two-parameter form. This calculator evaluates supplied parameters only. It assumes independent items from one stationary population, consistent units, and a credible basis for β, η, and γ. Small samples, censoring, competing risks, repairs, mixtures, and changing duty require dedicated statistical analysis. The probability plot is generated directly from the same CDF and survival equations.

References: NIST/SEMATECH Weibull distribution; IEC 61649:2008, Weibull analysis. Functional background: Weibull distribution calculator. Continue in TriboSolver: for system-level interpretation or refined simulation, open TriboSolver.