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eduKateSG · Why Mathematics? · Reliability Engineering: Weibull Models, Hazard Rates and Censored Lifetimes
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 9–16
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 17–24
Check assumptions
See where design and calibration matter.Route 4 · Chapters 25–32
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 33–43
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when some items fail and others are still working when a test ends? Reliability engineering uses survival functions, hazard rates, Weibull models and censoring to learn from every unit. The result is a time-based account of performance with explicit assumptions and uncertainty.
Did You Know? In a two-parameter Weibull model, the scale eta is the time by which about 63.2 percent have failed, not the mean or maximum life. With beta=2 and eta=1000 hours, modelled median life is about 832.6 hours and B10 life about 324.6 hours.
Full chapter index
Chapters 1–8 · Understand the question
Chapters 9–16 · Reproduce the example
Chapters 17–24 · Check assumptions
Chapters 25–32 · Explore applications
Chapters 33–43 · Study and verify
Reliability concerns whether an item performs its defined function for a stated time under stated conditions. Every word matters. A component in storage and one under cyclic load have different clocks and stresses. “Reliable” without a function, population, environment and time is incomplete. Reliability mathematics turns failure times and surviving test units into survival probabilities, hazard rates and life quantiles. It does not remove engineering judgement; it makes the assumptions and uncertainty visible enough to test.
Choose a time origin such as installation, first use or completion of burn-in. Define the event: first functional failure, a threshold crossing or another objective condition. Units can be hours, cycles, kilometres or demands. Changing the definition changes the data. Repairable systems need event-process methods, while this article focuses on non-repairable units or time to first failure. Record competing events and withdrawals because they affect what was observed and what remains unknown.
The survival function S(t) is the probability that lifetime T exceeds time t. It begins near one and decreases as time grows. It is not the proportion surviving forever; it refers to a specified t and population model. Empirically, Kaplan–Meier methods can estimate survival with censoring. Parametric models such as Weibull impose a shape that supports smoothing and extrapolation. Plot survival with confidence intervals and the number at risk so readers see where information becomes sparse.
The cumulative distribution F(t)=P(T≤t)=1−S(t) gives the probability of failure by time t under the model. It accumulates from zero toward one. In the worked Weibull example, S(800)=0.5273, so F(800)=0.4727. This does not mean a 47.27 percent chance of failing during the next hour; it covers the whole interval from the time origin through 800 hours. Clear time language prevents a common interpretation error.
The hazard h(t) is an instantaneous failure rate among units that have survived to t. It is a rate per unit time, not a probability, so values can exceed one depending on units and interval. Roughly, h(t) times a very short interval approximates the conditional chance of failure in that interval. Hazard describes how risk changes with age, while survival describes accumulated endurance. Two populations can have similar survival at one time and different hazard patterns.
For the two-parameter Weibull model, S(t)=exp[−(t/eta)^beta] for t≥0. Eta is the scale or characteristic life, the time at which survival equals exp(−1), about 0.3679. Beta is the shape. The hazard is (beta/eta)(t/eta)^(beta−1). The model is flexible and widely used, but flexibility does not prove fit. Check failure modes, plots and alternative distributions before using it for important extrapolation.
| Quantity | Expression | Value |
|---|---|---|
| Survival at 800 h | exp(−0.8²) | 0.5273 |
| Failure probability by 800 h | 1−S(800) | 0.4727 |
| Hazard at 800 h | (2/1000)(0.8) | 0.0016 per h |
| Median life | 1000 sqrt(ln 2) | 832.6 h |
| B10 life | 1000 sqrt(−ln 0.9) | 324.6 h |
Let beta=2 and eta=1000 hours. At t=800, (t/eta)^beta=(0.8)^2=0.64. Survival is exp(−0.64)=0.5273. Cumulative failure probability is 1−0.5273=0.4727. These values are conditional on the fitted model, population and operating conditions. Parameter estimates also have uncertainty. Reporting 0.5273 without a confidence interval can imply more precision than a small life test supports.
With beta=2 and eta=1000, h(800)=(2/1000)(0.8)^(1)=0.0016 per hour. This is the instantaneous rate among survivors near 800 hours. Multiplying by 10 hours gives an approximate 0.016 conditional probability only when the interval is short enough that hazard does not change much. For an exact interval probability, use survival ratios: P(800<T≤810 | T>800)=1−S(810)/S(800).
Median life solves S(t)=0.5. For the Weibull model, t50=eta(ln2)^(1/beta). With beta=2 and eta=1000, t50≈832.6 hours. Half the modelled population fails by this time and half survives longer. Median is not mean; for beta=2, the mean is eta times Gamma(1.5), about 886.2 hours. Life summaries answer different questions and can diverge greatly for skewed distributions.
B10 is the time by which 10 percent are expected to fail, so F(t)=0.10 and S(t)=0.90. For beta=2, eta=1000, B10=1000 sqrt(−ln0.9)=324.6 hours. Lower-tail quantiles can matter for warranties or safety, but estimating them from few failures may require substantial extrapolation. Report an interval and the model. B10 is not a guarantee that every unit lasts at least that long.
When beta<1, Weibull hazard decreases with time; beta=1 gives a constant hazard and the exponential distribution; beta>1 gives an increasing hazard. These shapes can be consistent with early defects, random failures or wear-out, but beta alone does not identify a physical cause. Mixed failure mechanisms can mimic a changing hazard. Use teardown, physics and failure-mode records alongside the fitted shape. Statistical labels should guide investigation rather than replace it.
Eta is not the maximum life and not the mean in general. At t=eta, cumulative failure probability is 1−exp(−1), about 63.2 percent, regardless of beta. Calling eta “characteristic life” is clearer than calling it average life. Changing time units rescales eta and hazard but leaves beta unchanged. Always display units; an eta of 1000 means something very different in cycles, hours and kilometres.
Right censoring occurs when a unit has not failed by its last observed time or leaves observation for an unrelated reason. A unit still running at 1200 hours contributes evidence that lifetime exceeds 1200; it is not a failure at 1200 and should not be discarded. Store observed time and an event indicator. The censoring mechanism should be plausibly independent of unobserved failure time conditional on modelled information. Removing weak-looking units for inspection can violate that assumption.
Left censoring means failure occurred before a detection time but the exact time is unknown. Interval censoring places failure between inspections. Left truncation occurs when units enter observation only after surviving to an entry time. These mechanisms require appropriate likelihood contributions. Treating inspection time as exact failure time creates bias. Document inspection schedules, entry criteria and withdrawals before choosing software. A neat spreadsheet of times may hide different observation mechanisms.
| Unit | Observed time (h) | Event indicator |
|---|---|---|
| A | 410 | failure |
| B | 690 | failure |
| C | 910 | failure |
| D | 1200 | right-censored |
| E | 1200 | right-censored |
For independent right-censored observations under a parametric model, a failed unit contributes density f(t), while a right-censored unit contributes survival S(t). An event indicator delta gives contribution f(t)^delta S(t)^(1−delta). Multiplying contributions or summing log contributions uses every unit correctly. This is why censored survivors still affect eta and beta. Maximum likelihood finds parameter values that make the observed combination of failures and survival most plausible under the model.
Kaplan–Meier estimation builds a stepwise survival curve without choosing a parametric lifetime distribution. At each failure time, survival is multiplied by the fraction surviving that event among those at risk. Right-censored units reduce the later risk set but do not create steps. It is valuable for descriptive comparison and model checking. When few units remain, intervals widen and the tail becomes uncertain. Display numbers at risk rather than drawing a confident-looking line into sparse territory.
A Weibull probability or hazard plot transforms survival so a two-parameter Weibull model appears approximately linear under suitable plotting positions. The slope relates to beta and the intercept to eta. NIST notes that Weibull hazard and Weibull plots can handle censored data, while a simple probability plot may not. Curvature, multiple slopes or grouped points can indicate poor fit or mixed failure modes. A visually straight line is evidence, not proof, especially with small samples.
The exponential model has constant hazard. The lognormal model often describes multiplicative degradation and can have a non-monotone hazard. Gamma and other models offer different shapes. Compare physical plausibility, graphical diagnostics, information criteria and predictions at decision times. A model that fits central failures similarly can produce very different lower-tail quantiles. Sensitivity analysis is essential when warranty or safety decisions depend on extrapolation beyond the observed range.
An item may fail through several modes, such as bearing wear, insulation breakdown or software lockup. Treating all failures as one homogeneous Weibull population can blur mechanisms. Record failure mode using a consistent adjudication process. Cause-specific or cumulative-incidence methods may be appropriate. Removing failures from other causes as if they were ordinary independent censoring can answer a different hypothetical question. State the estimand: all-cause reliability, mode-specific behaviour or a competing-risk probability.
Production lots, suppliers, environments or designs may form different lifetime populations. A pooled Weibull fit can show curvature or an apparently changing hazard that is really a mixture. Stratify known sources, use covariates or hierarchical models and examine interaction with stress. Decide whether the delivered population is genuinely mixed. A single attractive beta can conceal a small high-risk subgroup, so raw failure-mode and source tables matter.
Accelerated tests increase temperature, voltage, load or another stress to produce failures sooner. A stress-life model then relates accelerated conditions to use conditions. Acceleration must preserve relevant failure mechanisms; excessive stress can create failures that never occur in service. Randomise and block test positions, monitor actual stress and include use-level confirmation where feasible. Extrapolation across stress is an additional model layer beyond the Weibull lifetime model.
When a system can be repaired and returned to service, successive event times may be analysed as a recurrence process rather than independent lifetimes. Choices include minimal repair, renewal and trend models. Treating every inter-failure time as a fresh independent item can be wrong because age and repair quality persist. Define system boundaries and what “as good as new” means. This article’s non-repairable Weibull formulas should not be transferred automatically to repairable fleets.
An increasing hazard can support age-based maintenance only when replacement cost, downtime, consequences and failure mechanisms are considered. Preventive replacement too early wastes useful life; too late raises failure risk. Decision models combine survival with costs and constraints. A fitted beta>1 does not prescribe one interval by itself. Validate the policy with operational data and account for imperfect inspection, repair and changing environments.
Students can model the lifetime of identical low-cost components in a safe classroom test, recording failures and units still running at the end. The focus should be on devices, not people. Human persistence, achievement or wellbeing must not be reduced to component “failure” times. The experiment can teach censoring, survival and hazard while prompting an ethical discussion about where engineering metaphors do and do not belong.
Time to service incident, storage-device failure or first software crash can be studied with survival methods when the event and exposure are defined. Software updates, workload and hardware generations act as covariates or regime changes. Telemetry censoring may be informative if users disable reporting after problems. Fleet data are often left truncated because only devices that connect enter observation. Reliability engineering begins with these data mechanisms, not with fitting a Weibull button.
NIST gives the Weibull reliability function, failure rate, mean, median and cumulative hazard, with scale and shape parameters. It also explains that censored data arise when not all units fail during a reliability test. Read the NIST Weibull formulas and censoring guidance. These primary references support the calculations; design-specific claims still require test records and domain evidence.
Parameter estimates, survival curves, quantiles and hazard values all have uncertainty. Likelihood intervals, bootstrap methods or Bayesian credible intervals can be used under stated assumptions. With few failures, beta and lower-tail life may be weakly identified even when many units were tested. Confidence bounds can be asymmetric and very wide. Report the number of failures and censoring pattern beside every estimate. A four-decimal survival probability from six failures is presentation precision, not knowledge precision.
Recalculate S(800)=0.5273, F(800)=0.4727, h(800)=0.0016 per hour, median 832.6 hours and B10 324.6 hours. Verify time units. Build a five-unit dataset with three failures and two right-censored observations at 1200 hours. Fit Weibull and lognormal models, compare plots and predictions, then perturb one censoring time. Check formulas with a second tool and simulate data from known beta and eta to test recovery.
A Weibull model cannot prove a physical failure mechanism, guarantee life, handle every censoring process automatically or justify unlimited extrapolation. Small samples, mixtures, competing risks and changing stresses can distort parameters. Right-censoring is informative if withdrawal relates to impending failure. A fitted hazard is conditional on surviving and on the modelled population. Preserve raw histories and failure investigations so statistical summaries remain connected to engineering reality.
Report population, function, environment, time origin, unit, event definition, sample size, failure count, censoring and truncation, failure modes, model, parameterisation, beta and eta with intervals, survival and quantiles at decision times, hazard interpretation, diagnostics, alternative-model sensitivity and extrapolation distance. Show a survival plot with confidence bands and numbers at risk. State what the evidence does not establish. Reliability language should make uncertainty clearer, not hide it.
Reliability mathematics gives surviving units a voice instead of discarding them and gives failure times a structure beyond an average. Students see exponentials, logarithms, rates and probability working together. Engineers gain a framework for tests, maintenance and design improvement. The hopeful outcome is not a promise that nothing will fail; it is a better, testable understanding of when, how and under which conditions performance changes.
Use one row per unit with entry time, observed time, event indicator, mode, stress and covariates. Validate that event indicators match failure records and that time is positive. Build Kaplan–Meier steps with a documented add-in or formulas and show numbers at risk. For Weibull calculations, place beta, eta and time in labelled cells and test S, F, h, median and B10 against the worked values. Do not replace censored times with blanks or failures. Keep parameterisation visible because software packages use different symbols.
Store each unit’s observation interval and event code in tidy data. Fit Kaplan–Meier and Weibull models with libraries that support censoring, then print the exact parameterisation. Unit-test formulas at beta=2, eta=1000. Plot survival, cumulative hazard and Weibull diagnostics; compare lognormal predictions at decision times. Preserve convergence messages, seeds and versions. Simulate from known parameters to validate code, and run sensitivity analyses for ambiguous failure modes, withdrawal assumptions and alternative time origins.
Reliability engineering supports aerospace, electronics, energy, medical devices, transport, manufacturing, telecommunications, infrastructure and software operations. Roles combine probability, materials, physics, testing, maintenance, safety, standards and communication. Mathematics helps quantify time and uncertainty, but it does not guarantee automatic career access or safe products. Students can build foundations through algebra, logarithms, probability, coding and laboratory documentation, then explore engineering, statistics and data roles with their options open.
Run a safe life test on identical inexpensive components under one controlled condition. Randomise positions, define failure before starting and record time consistently. End the test at a fixed time so some units are right-censored. Build a Kaplan–Meier curve and fit a Weibull model, then calculate survival at a time inside the observed range. Inspect failed components for modes. Repeat with a second stress only if safe, and explain why a stress-life model is required before predicting normal use.
Hazard is not cumulative probability. Eta is not average life or maximum life. A right-censored unit did not fail at its censoring time. Beta does not prove a failure mechanism. A straight Weibull plot does not establish truth. More surviving units do not replace the information in failures. B10 is a modelled quantile, not a guarantee. Pooling different failure modes can create a misleading shape. Repairable-system events are not automatically independent non-repairable lifetimes.
The statistical process control article monitors a process through ordered data; reliability follows time to an event and retains censoring. The robust parameter design article studies sensitivity to conditions that may also affect life. The D-optimal design article helps choose informative constrained tests. Reliability connects those designs to lifetime evidence.
Students should first compute the Weibull example by hand, then draw survival and hazard curves for beta below, equal to and above one. Parents can ask what the clock, event and censoring indicator mean. Use devices or simulated data, never people as “failed components.” Encourage learners to explain why a surviving unit at test end still contributes information. The most useful habit is to state the population and condition every time a reliability percentage is quoted.
1. Recalculate the five worked quantities. 2. Convert hazard per hour to a short-interval approximation and label it. 3. Build a likelihood contribution for one failure and one censored unit. 4. Draw hazards for beta 0.7, 1 and 2. 5. Explain eta at 63.2 percent cumulative failure. 6. Compare Weibull and lognormal B10 predictions. 7. Identify informative censoring. 8. Write a result with uncertainty and no guarantee language.
### What does a censored unit tell us? Its lifetime exceeded the observed censoring time.
### Is hazard a probability? No. It is an instantaneous conditional rate per time unit.
### What does beta greater than one mean? The fitted Weibull hazard increases with time; physical interpretation needs evidence.
### Can Weibull model every lifetime? No. Fit and mechanism must be checked against alternatives.
Report function and population; environment and stress; time origin and unit; event definition; entry and withdrawal rules; sample size; failure and censored counts; failure-mode classification; inspection schedule; censoring and truncation assumptions; model and parameterisation; beta, eta and intervals; survival, hazard and life quantiles at decision times; numbers at risk; diagnostics; alternative models; acceleration model if used; software; extrapolation distance; validation and scope.
Trace every event and censoring code to a source record. Verify time origins, units and entry times. Recalculate the worked formulas and fit the five-unit example with two independent tools. Compare Kaplan–Meier and fitted survival inside the observed range. Inspect Weibull plot curvature, residuals, failure modes and source strata. Test lognormal and plausible alternative parameterisations. Review withdrawals for links to condition. Challenge lower-tail predictions by changing one early failure. If the decision changes materially, report that sensitivity and collect more informative data.
Continue through the Mathematics Learning Hub. NIST’s reliability data plotting guide and Weibull plot page show model-checking routes, while its Weibull maximum-likelihood example demonstrates censored estimation. Next reproduce one small analysis from an event-coded dataset and compare it with a nonparametric survival curve.
Extended practice laboratory
Practice case 1: Compute survival
Setup: Use Weibull beta 2, eta 1000 hours and t 800 hours.
Task: Calculate S=exp(−0.64)=0.5273.
Checking target: Interpret this as modelled survival beyond 800 hours for the stated population.
Practice case 2: Compute failure probability
Setup: Use the same parameters.
Task: Calculate F=1−S=0.4727.
Checking target: Do not confuse cumulative failure probability with instantaneous hazard.
Practice case 3: Compute hazard
Setup: Evaluate h=(2/1000)(0.8)=0.0016 per hour.
Task: State the time unit and conditional meaning.
Checking target: A hazard is a rate among survivors, not a probability by itself.
Practice case 4: Find median life
Setup: Solve S(t)=0.5.
Task: Calculate 1000 sqrt(ln 2)=832.6 hours.
Checking target: Distinguish median from mean and characteristic life.
Practice case 5: Find B10 life
Setup: Solve F(t)=0.10.
Task: Calculate 1000 sqrt(−ln 0.9)=324.6 hours.
Checking target: State the fitted distribution and parameter uncertainty.
Practice case 6: Read beta
Setup: Compare beta 0.7, 1 and 2.
Task: Describe decreasing, constant and increasing Weibull hazards.
Checking target: Do not assign a physical failure mechanism from beta alone.
Practice case 7: Represent right censoring
Setup: Record one unit still operating at 1200 hours.
Task: Store time 1200 with a censoring indicator, not as a failure.
Checking target: Explain the survival information the unit contributes.
Practice case 8: Build a likelihood
Setup: Use three failures and two right-censored observations.
Task: Multiply density terms for failures and survival terms for censored items.
Checking target: Keep the indicator and time paired for every unit.
Practice case 9: Check model fit
Setup: Create a Weibull plot and residual or graphical checks.
Task: Compare a lognormal alternative and inspect failure modes.
Checking target: A straight-looking plot is evidence, not proof.
Practice case 10: Report the analysis
Setup: State population, time origin, failures, censoring and model.
Task: Report beta, eta, uncertainty, survival estimates and validation.
Checking target: Avoid promises beyond the observed conditions and failure mechanisms.