Weibull Life Reliability Calculator

Purpose and model boundary

This model evaluates a two-parameter Weibull distribution with zero location from an entered shape β, characteristic life η, time, and target failed fraction. It does not estimate or fit Weibull parameters from observations.

Inputs and units

The user selects a consistent time unit and enters β, positive η, nonnegative evaluation time t, and a target cumulative failed fraction q strictly between zero and one. The selected unit labels all time-like outputs but does not alter the dimensionless distribution relationships.

Governing relationships

H(t) = (t/η)^β

R(t) = exp[-H(t)]

F(t) = 1 - R(t)

For t > 0:

f(t) = (β/η)(t/η)^(β-1) exp[-(t/η)^β]

h(t) = (β/η)(t/η)^(β-1)

mean life = η Gamma(1 + 1/β)

median life = η [ln(2)]^(1/β)

target percentile life = η [-ln(1-q)]^(1/β)

At t = 0, density and hazard use their finite exponential case when β = 1; the singular β < 1 case is hidden and checked rather than approximated. The chart plots the equation-derived reliability curve.

Calculation sequence

1. Validate time unit, β, η, evaluation time, and target failed fraction.
2. Calculate cumulative hazard, reliability, and failure probability.
3. Calculate density and hazard except at the decreasing-hazard origin singularity.
4. Calculate mean, median, target percentile, and qualitative hazard trend.
5. Evaluate status using the exact order below.

Outputs and interpretation

Reliability is survival beyond the evaluation time and failure probability is the cumulative failed fraction by that time. η is the 63.2% failed life, not generally the mean. The trend is decreasing for β < 1, constant for β = 1, and increasing for β > 1; it describes the entered model rather than proving a failure mechanism.

Validation and status logic

This is the exact Validation!B20 precedence. In the checked origin case, reliability and cumulative probability remain defined while density and hazard are intentionally not exposed.

Assumptions and limitations

• The entered parameters already represent the population, mission, and failure mode of interest.
• The location parameter is fixed at zero.
• The model does not fit censored data, estimate confidence intervals, compare distributions, or test goodness of fit.
• Extrapolation beyond the supporting observations can be misleading even when the equation is valid.
• A single Weibull distribution may not describe mixed failure mechanisms or populations.

Restrictions and non-computing states

β and η must be positive and within the published limits, time must be nonnegative, and the target failed fraction must be strictly between zero and one. The singular decreasing-hazard origin is a supported checked state, not a numeric approximation.

Errors and warnings

NOT VALID blocks interpretation. CHECK identifies the mathematical density/hazard singularity at t=0, β<1; the defined cumulative outputs remain available. A connection or calculation-service failure is not a Weibull result.

The distribution equations, origin behavior, visibility rules, named outputs, and status precedence were verified directly against the delivered workbook, its published input rules, tests, and reviewer packet. The workbook identifies these technical sources:

• NIST/SEMATECH e-Handbook — Weibull distribution — Two-parameter Weibull reliability, density, hazard, mean, median, and characteristic-life definitions.
• NIST copyrights and disclaimers — Source-rights and attribution context.