The Weibull distribution fits failure data with two parameters. β (shape / slope) tells you the failure pattern: β<1 infant mortality (hazard falling), β=1 random/constant hazard, β>1 wear-out (hazard rising). η (scale / characteristic life) is the age by which 63.2% have failed.
β is the decision-maker: only when β>1 does age-based preventive replacement help. At β=1 it’s useless (failures are random — use CBM); at β<1 it makes things worse (you’re replacing good parts with infant-mortality-prone new ones).
From the fit you read B10 life (10% failed), MTTF = η·Γ(1+1/β), and the whole reliability curve R(t). The three regions of β are exactly the three zones of the bathtub curve.
Why one curve isn’t enough
“The pump lasts 2,000 hours” is almost meaningless on its own. Do they all fail near 2,000 hours (wear-out), or is 2,000 just an average over failures scattered from day one (random), or are most failing early from bad installs (infant mortality)? Each demands a completely different response, yet all three can share the same average life. The Weibull distribution exists to separate them — it is flexible enough to model all three patterns with one equation, by changing a single shape parameter.
The three functions
Everything in life-data analysis is built from the reliability function R(t) — the probability a unit survives past age t:
The most diagnostic view is the hazard rate h(t) — the instantaneous failure rate given survival so far (an item that hasn’t failed yet). Its slope is the whole story:
β — the number that decides your strategy
β is the most actionable parameter in reliability engineering because it tells you whether age matters:
| β | Pattern | Hazard | What to do |
|---|---|---|---|
| < 1 | Infant mortality | Decreasing | Find the cause (installation, manufacturing, commissioning). Do NOT time-replace — new parts restart the infancy risk. |
| = 1 | Random | Constant | Age tells you nothing. Time-based PM is wasted — use condition monitoring or run-to-failure. |
| 1–3 | Early wear-out | Gently rising | Wear-out is starting. Age-based PM begins to pay; find the optimal interval. |
| > 3 | Rapid wear-out | Steeply rising | Strong, predictable wear-out (≈ normal distribution near β=3.4). Scheduled replacement works well. |
This is the quantitative backbone of the RCM decision and the reason a famous finding of RCM studies — that a large share of components show random or infant-mortality patterns — matters so much: for those, the traditional “overhaul every X hours” does nothing or backfires. β is how you prove which case you’re in. Watch the hazard curve flip from falling to rising as you cross β = 1:
Interactive — Weibull explorer
Live modelReliability R(t)
Hazard rate h(t)
R(t)=e^(−(t/η)^β), h(t)=(β/η)(t/η)^(β−1), B10=η(−ln0.9)^(1/β), MTTF=η·Γ(1+1/β) (Γ via Lanczos approximation). A real analysis fits β and η to censored field data by median-rank regression or maximum likelihood, with confidence bounds — this explorer shows the shapes those fits produce.η, B10 and MTTF
Once β has told you the pattern, the other numbers quantify the life:
- η — characteristic life. Always the 63.2%-failed age, regardless of β. It anchors the time axis: doubling η doubles every life number.
- B-life (B10, B5…). The age at which a given fraction has failed —
B10 = η·(−ln0.9)^(1/β). Bearing makers quote L10 (= B10) life; warranty and spares planning lean on the low-percentile B-lives, not the average. - MTTF / mean life.
MTTF = η·Γ(1+1/β). This is the mean of the fitted time-to-failure distribution, which ISO 14224 calls mean time to failure (3.62) — not MTBF. The between-failures measure is a different quantity for a repairable item: the same standard gives it as METBF = MTTF + MTTRes (C.11), longer by the mean time to restoration. Note it is not η except near β≈1, and for wear-out items the mean hides the spread — which is exactly why the average alone misleads.
A subtle but vital point: a higher MTTF is not automatically better if β is low. A part with a huge mean life but β<1 is still throwing early failures; you fix that by hunting the infant-mortality cause, not by replacing parts on a schedule.
The bathtub curve is three Weibulls
The classic bathtub curve — a hazard rate that falls, then flattens, then rises — is simply three Weibull regimes laid end to end: an early β<1 infant-mortality phase, a long β≈1 useful-life phase of random failures, and a final β>1 wear-out phase. Real components rarely show all three cleanly; the value of fitting Weibull to your data is discovering which phase a given failure mode actually lives in — and therefore which maintenance strategy fits.
Where the data comes from. Weibull is only as good as the failure history fed to it — which is why disciplined work-order close-out with proper failure coding matters, and why OREDA and ISO 14224 failure-rate libraries exist for when your own data is thin. The β you find then drives the PM interval decision and feeds the availability model.
Key takeaways
- β tells you the failure pattern — <1 infant, =1 random, >1 wear-out — and therefore whether age-based PM helps, does nothing, or backfires.
- η is the characteristic life (63.2% failed); it scales every other life number.
- The hazard rate’s slope is β−1 — falling, flat, or rising — which is the whole bathtub curve in one parameter.
- Read B10 and MTTF from the fit, but never trust the mean without knowing β.