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Life data analysis: Weibull, lognormal and exponential with censoring

Paste from Excel the hours (or cycles, km) to failure of each unit and those still running, and see the failure pattern, B10 life with confidence interval, mission reliability, warranty cost and how many units to test to prove the goal.

Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.

CheckRevised 11/03/2026
Life data
See what was read

Status: F or 1 = failure; C or 0 = right-censored (still running). Group (supplier, lot) and count columns are optional.

Analysis
Warranty (optional)
Demonstration test

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—Failure pattern and life
—

Probability plot with confidence interval

Each point is a failure at its modified Kaplan-Meier position; censored units enter the position calculation. Line = fitted distribution; band = CI of the percentiles. Points on the line: the distribution describes the data well.

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Distribution identification

The same data on four scales. Pick the one with the straightest points: lowest adjusted Anderson-Darling and highest correlation.

Estimates, characteristics and percentiles

Parameters with standard error and confidence interval, mean and quartiles of the fitted distribution and the percentile table (Bx life).

Parameter estimates

Characteristics of distribution

Table of percentiles

Table of survival probabilities

Survival function R(t)

Steps = Kaplan-Meier (data, no distribution assumed; + = censored); curve = fitted distribution with CI. Agreement confirms the fit.

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Hazard function h(t)

Instantaneous failure risk of units still running. Rising = wear-out (preventive replacement pays); constant = random; falling = infant mortality (improve assembly and start-up).

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Probability density function f(t)

Where failures concentrate in time. Ticks on the axis = observed failures.

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Nonparametric analysis (Kaplan-Meier)

Kaplan-Meier table and empirical hazard

Group comparison

Each group fitted separately, tests for equal parameters (likelihood ratio) and equal survival curves (log-rank and Wilcoxon).

Probability plot by group

Parallel lines = same shape (same failure mode); the one shifted right lasts longer. Crossing lines = different failure modes.

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Survival by group

Kaplan-Meier for each group (steps) and fitted distribution (dashed). The B10 life difference between groups shows directly on the chart.

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Expected warranty failures

Expected cumulative failures in the units sold over time, with CI; the vertical line is the end of the warranty.

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Demonstration test plan

How many units and how much test time to prove the goal at the chosen confidence (Weibull with known β).

Probability of passing the test

x axis = how much the true life (η) exceeds the goal. A design just at the goal passes with probability 1 − C; allowing failures (m > 0) protects a design slightly better than the goal.

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Test time per unit × number of units

Each curve is an equally valid plan: pick the combination that fits your test rig and schedule.

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Next step

Take this page’s results to another calculator with the fields already filled in.

Optimal preventive interval with β and η →Fleet MTBF, availability and R(t) →Cost of poor quality with the warranty provision →

The calculation report gathers this page's inputs, results, equations, assumptions and references, with fields for project, engineer and revision. The full report also includes the charts and interpretations, in color. In the print dialog, choose Save as PDF.

Method, assumptions and references

Equations used

Weibull: F(t) = 1 − exp[−(t/η)^β]; lognormal: F(t) = Φ[(ln t − μ)/σ]; exponential: F(t) = 1 − exp(−t/θ); normal: F(t) = Φ[(t − μ)/σ]
Likelihood: ln L = Σ failures ln f(tᵢ) + Σ censored ln R(tⱼ) + Σ intervals ln[F(bₖ) − F(aₖ)]; parameter CIs from the observed information matrix (normal on ln η, ln σ, ln β)
Percentile: t_p = exp(μ + σ·w_p), w_p = ln[−ln(1 − p)] (Weibull) or Φ⁻¹(p) (lognormal); Var(ln t_p) = Var μ + 2w_p Cov(μ, σ) + w_p² Var σ
Plotting positions: modified Kaplan-Meier, F = 1 − [S(tᵢ) + S(tᵢ₋₁)]/2; LSXY: regression of ln t on w(F); adjusted Anderson-Darling = n·∫(Fₙ − F)²/[F(1 − F)] dF
Kaplan-Meier: S(t) = Π(1 − dⱼ/nⱼ), Greenwood SE; demonstration test: p = Beta⁻¹(C; m + 1, n − m), T = η₀·[−ln(1 − p)]^(1/β)

Assumptions and limits

  • Independent units, same failure mode and same operating conditions; censoring does not depend on imminent failure (removing units “about to fail” biases the result).
  • For repairable systems (the same equipment failing repeatedly), times between failures only apply here if repair leaves it as good as new; with degradation, use process models (Crow-AMSAA).
  • Intervals use the normal approximation of maximum likelihood; with fewer than 10 failures they are optimistic. For LSXY, intervals use the information matrix evaluated at the LSXY estimates.
  • Demonstration test: known Weibull β, units tested for the same time, without replacement; 3-parameter distributions and accelerated life are not covered here.

Standards and technical documents

  • Meeker, W. Q.; Escobar, L. A. Statistical Methods for Reliability Data. Wiley, 1998 (cap. 3, 7, 8 e 10)
  • Abernethy, R. B. The New Weibull Handbook, 5th ed., 2006
  • Nelson, W. Applied Life Data Analysis. Wiley, 1982
  • Klein, J. P.; Moeschberger, M. L. Survival Analysis, 2nd ed. Springer, 2003 (Kaplan-Meier, log-rank)
  • IEC 61649:2008 Weibull analysis
  • IEC 61124:2023 Reliability testing — Compliance tests for constant failure rate and constant failure intensity
  • IEC 61810-2 / VDI 4002 (B10 e ensaios de vida)

U.S. regulations

  • ISO 14224:2016 Collection and exchange of reliability and maintenance data for equipment
  • Magnuson-Moss Warranty Act (15 U.S.C. 2301 et seq.)

Frequently asked questions

What is Weibull analysis and what is it for?

It fits a life distribution to a component’s times to failure. The shape β tells whether failures are early (β < 1), random (β ≈ 1) or wear-out (β > 1), and the scale η is the age by which 63.2% have failed. From that you estimate B10 life, mission reliability, preventive replacement interval and warranty cost.

What do I do with units that have not failed yet?

Include them all, marked with status C (right-censored). Dropping them or treating them as failures underestimates life: in the bearing example, 12 of Supplier A’s 26 units were still running. Maximum likelihood uses the fact that they lasted at least that long.

Maximum likelihood or least squares (LSXY)?

Use maximum likelihood by default, especially with heavy censoring: it handles censoring exactly and gives confidence intervals. LSXY, the line on the probability plot, often does better with few failures (fewer than 10 to 15) and little censoring.

How do I choose between Weibull, lognormal and exponential?

Compare the adjusted Anderson-Darling and correlation in the distribution identification: the lowest AD and highest correlation win. If two tie, choose by physics: Weibull for fatigue, wear and bearings; lognormal for cracks, corrosion and repairs; exponential only when the failure rate is constant, as in electronics after burn-in.

What is B10 life?

It is the time by which 10% of units fail. Bearings are rated by L10 life (= B10), and warranties are usually set below B1 or B5. Use the lower bound of the confidence interval for conservative decisions, since the estimate comes from a sample.

How do I compare two suppliers or lots?

Paste the group column. The page fits each group, tests whether the parameters are equal (likelihood ratio) and whether the survival curves differ (log-rank and Wilcoxon). In the example, Supplier A has a much higher B10 than B, with log-rank p ≈ 0.03: the difference is real.

How many parts must I test to prove a reliability?

Use the demonstration test plan: enter the goal (for example, 90% at 4,000 h), the confidence, the known β and how many failures you accept. The page returns the test time per unit for a given number of parts, or the number of parts for a given time, and shows the chance that a good design passes.

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