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Weibull Reliability R(t)

Survival probability, failure fraction, and hazard rate at a given time for a two-parameter Weibull life model.

InputR(t) = exp[ −(t/η)^β ] , F(t) = 1 − R(t) , h(t) = (β/η)·(t/η)^(β−1)

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The engineering

The two-parameter Weibull is the workhorse of life-data analysis. β (shape) tells you the failure mode: β < 1 is infant mortality (decreasing hazard), β = 1 is random/constant hazard (exponential), and β > 1 is wear-out (increasing hazard). η (scale) is the characteristic life — the age at which 63.2% of the population has failed, regardless of β.

Reach for this card once you have fitted β and η (from a Weibull plot or median-rank regression) and need the survival probability at a specific service time. Sanity check: at t = η, R always equals exp(−1) ≈ 0.368, so 36.8% survive to the characteristic life no matter what β is.

Note the hazard rate versus the PDF: h(t) is the instantaneous failure rate conditional on having survived to t, which is what drives maintenance intervals — not the unconditional PDF f(t).

Where this math comes from

Waloddi Weibull, a Swedish engineer, presented the distribution in a 1951 ASME paper titled "A Statistical Distribution Function of Wide Applicability," showing it fit everything from cotton-fiber strength to the fatigue life of steel. He wasn't the first to write the function — Fréchet had it in 1927 and Rosin-Rammler used it for particle sizes in 1933 — but Weibull made the case that one flexible form could model wildly different failure data.

The distribution became the backbone of reliability engineering largely through Robert Abernethy, whose "The New Weibull Handbook" turned it into a practical bench discipline for aerospace and turbine-engine work — median-rank plotting, small-sample corrections, and the β-reading conventions that field engineers still use today.

  1. 1927Maurice FréchetDescribes the extreme-value form underlying the distribution.
  2. 1933Rosin & RammlerApply the same function to particle-size distributions.
  3. 1951Waloddi WeibullASME paper demonstrates 'wide applicability' across failure data.
  4. 1983Robert B. AbernethyThe New Weibull Handbook codifies practical life-data methods.

See the full timeline of the math behind every calculator →

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