What the bathtub curve is
The bathtub curve shows how a machine's failure rate changes over its life: high at the start, low and steady in the middle, and rising at the end. The name comes from the shape of the chart, which has three distinct phases.
Each phase has its own failure causes and therefore calls for a different maintenance action. The Weibull distribution describes all three phases with a single shape parameter, beta (β).
Visual map of the bathtub curve
The map summarizes the 3 phases, the value of β in each one, the matching maintenance action and the worked example from this article.
The 3 phases of the curve
| Phase | Failure rate | Weibull β | Typical causes | Action |
|---|---|---|---|---|
| 1. Infant mortality | High and decreasing | β < 1 | Assembly error, manufacturing defect, poor installation, initial adjustment | Incoming inspection, commissioning, run-in test, training |
| 2. Useful life | Low and constant | β ≈ 1 | Random failures: overload, contamination, operating error | Lubrication, cleaning, inspection, autonomous maintenance |
| 3. Wear-out | Increasing | β > 1 | Fatigue, abrasion, corrosion, ageing | Predictive maintenance, replacement by age or condition, overhaul |
Only in phase 2 is the failure rate constant. There the exponential model applies, and MTBF is the inverse of the failure rate (λ = 1 ÷ MTBF). Outside that phase, treating MTBF as if the rate were constant gives wrong predictions.
Weibull: failure rate and reliability
With two parameters, the Weibull describes any of the three phases:
R(t) = e−(t/η)β h(t) = (β ÷ η) × (t ÷ η)β−1
- R(t): reliability, the probability of running without failure up to time t.
- h(t): instantaneous failure rate (failures per hour among the units still running).
- β (shape): below 1, decreasing rate; equal to 1, constant; above 1, increasing.
- η (characteristic life): the time by which 63.2% of units have failed. At that point R = e−1 = 36.8%, whatever the β.
With β = 1 the Weibull becomes the exponential and η is the MTBF. Two more useful measures: the B10 life, the time to 10% failures, calculated as η × (−ln 0.90)1/β; and the mean time to failure, η × Γ(1 + 1/β), where Γ is the gamma function.
Worked example: pump bearing
Fictional example. Failure data for a pump bearing, fitted to a Weibull, gave β = 2.0 and η = 1,200 hours. Since β > 1, the bearing is in wear-out.
| Time t (h) | R(t) = e−(t/1200)² | h(t) (failures per 1,000 h) |
|---|---|---|
| 100 | 99.31% | 0.14 |
| 300 | 93.94% | 0.42 |
| 500 | 84.06% | 0.69 |
| 800 | 64.12% | 1.11 |
| 1,200 | 36.79% | 1.67 |
Key calculations:
- R(500): e−(500÷1200)² = e−0.1736 = 84.06%.
- B10 life: 1,200 × (−ln 0.90)1/2 = 1,200 × 0.3246 ≈ 390 hours. By then 10% of the bearings will have failed.
- Mean time to failure: 1,200 × Γ(1.5) = 1,200 × 0.8862 ≈ 1,063 hours.
- Conditional probability: the bearing has already run 500 h without failure. Chance of reaching 600 h: R(600) ÷ R(500) = 77.88% ÷ 84.06% = 92.6%.
If someone treated this bearing as exponential with an MTBF of 1,063 h, they would predict R(500) = e−500÷1063 = 62.5%. The real value is 84.06%. The failure rate grows with use, so the constant-rate model is wrong in both directions.
What to do in each phase
- Phase 1: confirm the installation (alignment, torque, clearances) and run the unit in before releasing it to production. If early failures repeat, investigate the supplier and the assembly with the 5 Whys.
- Phase 2: the goal is not to shorten useful life. Correct lubrication, cleaning and operator inspection prevent random failures. Preventive replacement here is waste, because the new part restarts phase 1.
- Phase 3: track vibration, temperature and noise, and schedule replacement before the point where the rate takes off. TPM treats this monitoring as routine.
To decide which assets to analyze first, build a Pareto chart of downtime hours. The FMEA helps list the failure modes of each phase, and the measured availability feeds into OEE.
How to estimate β and η
- Record the operating time to failure of each unit (hours, cycles or km), including units that have not failed yet as censored data.
- Sort the times and compute the cumulative failure probability F for each (for example with Benard's median ranks).
- Plot ln(t) against ln(−ln(1 − F)). On Weibull paper, the points form a straight line.
- The slope of the line is β. The value of η is the time where the line crosses F = 63.2%.
Spreadsheets and statistics software do the fit. Rule of thumb: with few recorded failures (fewer than about 6 to 10), the estimated β is very uncertain and decisions should be made with caution.
Common mistakes and limitations
- Using MTBF as a replacement interval. With β > 1, replacing at the MTBF means accepting a large share of failures first (54% of the bearings in the example). Use the B10 life or an acceptable failure rate.
- Mixing failure modes. Fatigue and contamination failures in the same data set give a meaningless β. Separate them by mode.
- Ignoring units that did not fail. They carry information and must enter as censored data.
- Assuming every item follows the bathtub. Classic reliability-centered maintenance studies (Nowlan and Heap) showed that most complex components have no defined wear-out phase, and that infant mortality is common.
- Replacing parts unnecessarily. The intervention itself is a source of phase 1 failures.
Frequently asked questions
What is the bathtub curve in reliability?
It is the chart of failure rate over life: high at the start (infant mortality), low and constant in useful life, and rising in wear-out.
What does the Weibull parameter β mean?
It is the shape parameter. β below 1 means a decreasing failure rate, equal to 1 means constant, and above 1 means increasing.
What is the characteristic life η?
It is the time by which 63.2% of units have failed. At that point reliability is 36.8%, for any β.
When can I use MTBF?
When the failure rate is roughly constant (β close to 1), that is, in the useful-life phase. Outside it, MTBF hides the trend.
How do I reduce infant mortality?
With incoming inspection, standardized commissioning, a run-in period and training for the people who assemble and operate the equipment.
How many failures do I need to fit a Weibull?
There is no exact number, but with fewer than 6 to 10 failures the estimate is very uncertain. The more data, the more reliable the β.
Sources
- WEIBULL, W. A statistical distribution function of wide applicability. Journal of Applied Mechanics, v. 18, 1951.
- NOWLAN, F. S.; HEAP, H. F. Reliability-Centered Maintenance. United Airlines / U.S. Department of Defense, 1978.
- ABERNETHY, R. B. The New Weibull Handbook. 5th ed. 2004.
- O'CONNOR, P.; KLEYNER, A. Practical Reliability Engineering. 5th ed. Wiley, 2012.
Keep studying: trusted sources and links
- NIST e-Handbook: Assessing Product Reliability
- ABRAMAN: Associação Brasileira de Manutenção e Gestão de Ativos (in Portuguese)
- Wikipedia: Reliability engineering
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About the author
Lean Manufacturing & Behavioral Management Specialist
Over 20 years in the automotive and metalworking industries (GM and Dana), Lean Manufacturing practitioner since 2006. SENAI instructor and mentor in Brazil’s Brasil Mais Produtivo program, delivering consulting, training and audits for 50+ companies, combining quality, productivity and people development.
