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), MTBF = 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.
1 · 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.
2 · 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:
3 · 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), MTBF=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.4 · η, B10 and MTBF
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. - MTBF / mean life.
MTBF = η·Γ(1+1/β). 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 MTBF 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.
5 · 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 MTBF from the fit, but never trust the mean without knowing β.