Weibull Reliability Calculator Guide

Quick Start

Choose the workflow that matches the question

Calculator: plan a zero-failure test by solving for Lives / sample or Sample size, or evaluate an existing plan.
Sample Size Matrix: compare required exposure across sample sizes using the current R, C, and β targets.
Weibull Fit: choose Probability Plot for complete failures or Maximum Likelihood for failures with right-censored suspensions, then estimate β and η.

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

Solver continuity

Why solver-mode round trips may not return the original value

When you switch solver modes, the calculator transfers the outgoing calculated complementary value at full internal precision and solves the newly selected problem. It does not restore a historical input.

Sample size is a discrete conservative integer, and a manually entered plan may contain feasibility slack. Near an exact capacity boundary, binary floating-point representation can also place delivered exposure microscopically below the requirement. The calculator does not add hidden epsilon corrections or cache an earlier answer to force reversibility.

Boundary example: R=0.95, C=0.90, β=2, n=6 gives L≈2.7352808350744913. Switching to solve n can conservatively return 7 because the represented delivered exposure is microscopically below the required exposure.

Feasibility-slack example: with the same targets and entered L=2, solve n returns 12. Switching back solves the minimum L for n=12, approximately 1.9341356269—not the historical entered 2.

Definitions

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%.

Calculator

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.

Optional conversion

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.

Comparison

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.

Failure-life observations

Two-parameter Weibull Fit

Enter positive life observations from independent specimens representing a reasonably common population and dominant failure mechanism. Units may be cycles, hours, miles, or another consistent positive exposure unit. Mark each observation as Failure or Suspended.

Choose Probability Plot — Bernard + OLS for complete failures only, or Maximum Likelihood — MLE when the dataset includes right-censored suspensions.

Fit Methods

Probability Plot and Maximum Likelihood

Probability Plot — Bernard + OLS

Probability Plot requires at least three finite positive failures and does not accept Suspended rows. Observations are sorted, assigned Bernard median ranks, transformed to Weibull probability coordinates, and fit by ordinary least squares.

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.

Maximum Likelihood — MLE

MLE fits the two-parameter Weibull likelihood using exact failure times and right-censored suspension exposures. This application requires at least two finite positive failures. That two-failure minimum is an application policy, not a claim that two failures ordinarily provide a defensible engineering population estimate.

ℓ(β, η) = Σ_failures[ln β − β ln η + (β − 1)ln tᵢ] − Σ_all(tᵢ / η)^β

η(β) = [Σ_all tᵢ^β / r]^(1/β)

The implementation solves the profile-likelihood score for β and derives η from the scale equation, where r is the number of failures. MLE does not display probability-plot R².

Interpretation

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.

Distribution predictions

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.

Forward prediction

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.

Inverse prediction

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.

Distribution mean

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.

Statistical boundary

Failure versus Suspended

A Failure row records an observed failure at that life. A Suspended row records a specimen known to have survived through that exposure without an observed failure; it is right-censored.

Probability Plot accepts complete failures only. Maximum Likelihood accounts for both Failures and right-censored Suspensions. The calculator does not support left censoring, interval censoring, or a three-parameter/location Weibull model.

Suspensions should reflect a planned test ending or another stopping reason unrelated to the specimen’s impending failure; selective removal of deteriorating specimens can bias the analysis.

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

site_admin only

Saved Analyses and CSV import

Authenticated site administrators can save, reopen, update, duplicate, and delete complete calculator workspaces. Saved records are private to the authenticated admin owner. Anonymous and private-user sessions have no Save or Import capability.

Auto detect recognizes plain single-column files as well as comma, semicolon, and tab delimiters. Choose Single column / no delimiter explicitly for unquoted decimal-comma values, where a comma could otherwise mean a column separator. First row defaults to Auto detect: numeric first rows stay in the data, recognized headings appear as excluded rows in the preview, and ambiguous rows require an explicit Header or Data choice. You can override the detected header with either choice.

Clear file removes the selected file and pending preview so you can select it again. It does not remove observations already imported into the Fit workspace.

CSV status rules

With no status column, every imported row is Failure. With a status column, accepted Failure aliases are failure, failed, f; accepted Suspended aliases are suspended, suspension, censored, right-censored, s, c. Matching is case-insensitive after trimming. Blank, missing, unknown, 0, and 1 statuses are rejected.

Preview confirms delimiter, header use, decimal/grouping convention, selected columns, unit, row interpretation, and Failure/Suspended totals. Every invalid row must be corrected before Apply. Replace discards current Fit rows; Append retains valid nonblank rows and requires exactly matching units. No unit conversion is performed, and import never changes Fit Method.

Imported rows retain source and numeric-convention provenance. The upload is parsed on the authorized server path; the raw file is not stored as a saved dataset.

Diagnostic

Probability-plot linearity, R²

R² is displayed only for Probability Plot. It 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.

Engineering use

Assumptions and limitations

Two-parameter Weibull model with no threshold or location parameter.
Probability Plot uses complete failures only; MLE supports right-censored Suspensions.
Independent specimens from a reasonably common population and failure mechanism.
Consistent exposure units within each analysis.
Fit Method is either Bernard/OLS Probability Plot or two-parameter right-censored MLE.
β and η are point estimates; parameter and B-life confidence intervals are not included.
No left censoring, interval censoring, or three-parameter Weibull fitting.
Zero-failure planning assumes identical severity and exposure and requires every specimen to complete the planned duration.

Reference

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)

Right-censored maximum likelihood

ℓ(β, η) = Σ_failures[ln β − β ln η + (β − 1)ln tᵢ] − Σ_all(tᵢ / η)^β

η(β) = [Σ_all tᵢ^β / r]^(1/β)

Reference

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.