Weibull Reliability Calculator

Choose the workflow that matches the question

1. Calculator: plan a zero-failure test by solving for Lives / sample or Sample size, or evaluate an existing plan.
2. Sample Size Matrix: compare required exposure across sample sizes using the current R, C, and β targets.
3. Weibull Fit: enter complete failure-life observations to estimate Weibull shape, β, and characteristic life, η, then examine life predictions.

Use one consistent exposure basis within each workflow. Planner design-life units and Weibull Fit data units are independent and are never converted automatically.

Input definitions

Target reliability, R

The target surviving fraction at the specified design life or exposure.

Confidence, C

The statistical confidence requested for the zero-failure demonstration conclusion.

Weibull shape, β

The parameter describing how failure hazard changes with life.

Sample size, n

The number of independent specimens completing the required exposure.

Lives / sample, L

The duration completed by each specimen, expressed in durability lives.

Characteristic life, η

The fitted life at which cumulative failure is approximately 63.2% and reliability is approximately 36.8%.

Zero-failure reliability demonstration

The planner determines the exposure required to demonstrate a reliability target at a chosen confidence, assuming every specimen completes the specified exposure with ZERO FAILURES.

Solve for Lives / sample

Provide R, C, β, and Sample size. The result is the minimum required Lives / sample, L.

Solve for Sample size

Provide R, C, β, and available Lives / sample. The calculator returns the smallest conservative integer Sample size that satisfies the required Weibull exposure.

Acceptance criterion: every planned specimen must complete its exposure with ZERO FAILURES. If a failure occurs, the criterion has not been met; use an analysis appropriate to failed and censored data or revise the test plan.

Design-life conversion

A durability life is an engineering-defined reference exposure. For example, one life might equal 1,500,000 cycles, 5,000 hours, or 150,000 miles.

Physical exposure / sample = D × L

Total physical exposure = D × L × n

The planner reports the Minimum calculated physical exposure / sample from the exact mathematical L. This optional conversion does not alter the underlying Weibull calculation.

Fit units describe the failure-life observations entered in Weibull Fit. They remain independent from the planner’s design-life conversion.

Sample Size Matrix

The Matrix holds R, C, and β fixed while showing how the required lives per sample change with sample size. More specimens generally reduce the exposure required from each specimen.

Lives / sample is the exact mathematical result. Rounded test lives is conservatively rounded upward to the next 0.1 life, and the adjacent confidence is recalculated at that rounded duration.

When Design-life conversion is active, the Matrix’s secondary physical-exposure line uses the rounded test duration. It can therefore be slightly higher than the planner’s exact minimum calculated exposure.

Two-parameter Weibull probability-plot regression

Enter at least three positive failure lives from independent specimens representing a reasonably common population and dominant failure mechanism. Units may be cycles, hours, miles, or another consistent positive exposure unit.

Observations are sorted, assigned Bernard median ranks, transformed to Weibull probability coordinates, and fit by ordinary least squares.

Fᵢ = (i − 0.3) / (n + 0.4)

Xᵢ = ln(tᵢ)    Yᵢ = ln[−ln(1 − Fᵢ)]

Y = βX + b    η = exp(−b / β)

The slope estimates Weibull shape, β. The intercept determines Characteristic life, η. At t = η, F(t) is approximately 63.2% and R(t) is approximately 36.8%; η is neither the median nor generally the mean.

Weibull shape, β

• β < 1: decreasing hazard, often associated with early-life or infant-mortality behavior.
• β = 1: the exponential special case with mathematically constant hazard.
• β > 1: increasing hazard, commonly associated with wear-out behavior.

For the on-screen interpretation only, values approximately within 0.9–1.1 are described as approximately constant hazard. This is a display convention, not a formal statistical classification boundary.

Use representative historical failure data or a technically justified conservative assumption. Never choose β merely to lower the required test burden.

B-life

Bx is the modeled life by which x% of the population is expected to have failed. The calculator reports B1, B5, B10, and B50 immediately after a valid fit.

• B1: 1% failed and 99% predicted to survive.
• B5: 5% failed and 95% predicted to survive.
• B10: life by which 10% failed, equivalent to 90% reliability.
• B50: median life, with 50% failed and 50% surviving.

Bₓ = η[−ln(1 − x/100)]^(1/β)

These are point predictions from the fitted model, not confidence bounds.

Reliability at mission life

Reliability at mission life is the modeled probability that a unit survives beyond a specified life or exposure.

R(t) = exp[−(t / η)^β]

F(t) = 1 − R(t)

Enter a mission life in the same units as the fitted failure-life observations. At t = 0, R(t) = 1 and F(t) = 0.

Life at target reliability

This prediction solves for the exposure at which the fitted model expects the requested surviving fraction to remain.

t = η[−ln(R)]^(1/β)

• R = 0.99 → B1
• R = 0.95 → B5
• R = 0.90 → B10
• R = 0.50 → B50

An arbitrary reliability may be shown as an equivalent fractional B-life, such as B7.5.

Mean life (MTTF)

MTTF is the mathematical mean of the fitted two-parameter Weibull life distribution.

MTTF = η Γ(1 + 1/β)

Its units follow the failure-life dataset. MTTF is not generally equal to η; they are equal only in the β = 1 exponential special case.

Complete failures versus censored or suspended data

The Fit workflow accepts complete failure-life observations only: every entered value represents a specimen that actually failed at that life.

A specimen that survives to the end of a test is not a failure observation. It is right-censored or suspended data and requires methods that account for its known survival exposure. Those methods are outside the current calculator’s scope.

Do not enter suspension times as though they were failures; doing so can distort β, η, and every derived prediction.

Probability-plot linearity, R²

R² summarizes how closely the transformed observations follow the fitted straight line. It is a useful probability-plot linearity diagnostic, not formal proof that the Weibull distribution is correct.

A high R² does not correct a small or unrepresentative sample, mixed failure mechanisms, dependent observations, or incorrect treatment of suspended units.

Assumptions and limitations

• Two-parameter Weibull model with no threshold or location parameter.
• Complete failures only for fitting; no censored or suspended observations.
• Independent specimens from a reasonably common population and failure mechanism.
• Consistent exposure units within each analysis.
• Ordinary least-squares probability-plot fit, not maximum-likelihood estimation.
• β and η are point estimates; parameter and B-life confidence intervals are not included.
• Zero-failure planning assumes identical severity and exposure and requires every specimen to complete the planned duration.

Equation reference

Zero-failure demonstration

E = −ln(1 − C) / −ln(R)

L = (E / n)^(1/β)

n = smallest positive integer satisfying nL^β ≥ E

C achieved = 1 − exp[−nL^β(−ln R)]

R demonstrated = exp[ln(1 − C) / (nL^β)]

Fitted distribution

R(t) = exp[−(t / η)^β]    F(t) = 1 − R(t)

Bₓ = η[−ln(1 − x/100)]^(1/β)

MTTF = η Γ(1 + 1/β)

Glossary

Reliability

The modeled or demonstrated probability of surviving a specified life.

Confidence

The statistical strength attached to a reliability-demonstration conclusion.

Weibull shape, β

The parameter governing how hazard changes with life.

Characteristic life, η

The life at approximately 63.2% cumulative failures.

Design life

An engineering-defined reference exposure used to express durability lives.

B-life

The modeled life by which a stated percentage of the population has failed.

MTTF

The mathematical mean life of the fitted distribution.

Censoring / suspension

A known survival exposure without an observed failure at that time.

Probability plot

A transformed plot on which a two-parameter Weibull model appears as a straight line.