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    Reliability engineering / life data

    Weibull reliability

    The Weibull distribution is the workhorse of reliability engineering because its shape parameter tells you which failure regime you are in: infant mortality, random failure, or wear-out. This engine computes survival, hazard, MTBF and B10 life from the two parameters.

    Failure modeWear-outExpected failures216.5Shape β1.80
    Reliability at mission
    56.693%
    Surviving 8760 hours
    Hazard rate
    116.61 /Mh
    Instantaneous failure rate at that time
    MTBF
    1.067e+4 h
    Mean time between failures
    B10 life
    3437 h
    Time at which 10% have failed
    Reliability over the mission
    Survival probability and hazard rate at multiples of the mission time.
    β = 1.80
    Reliability over the mission
    TimeReliabilityHazard rateExpected survivors
    2190 h95.428%38.468 /Mh477.1 of 500
    4380 h84.961%66.977 /Mh424.8 of 500
    8760 h56.693%116.614 /Mh283.5 of 500
    1.314e+4 h30.806%161.296 /Mh154.0 of 500
    1.752e+4 h13.859%203.036 /Mh69.3 of 500
    2.628e+4 h1.657%280.832 /Mh8.3 of 500

    The hazard rate rises, falls or stays flat depending on β: below 1 is infant mortality, exactly 1 is memoryless, above 1 is wear-out.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Survival curve
    Probability of surviving to each time, for this shape and scale.
    Survival curve0.00%22.2%44.4%66.6%88.8%111.0%8768762190219043804380876087601.314e+41.314e+41.752e+41.752e+42.628e+42.628e+43.504e+43.504e+4R(t)
    Hazard rate
    Rising for β > 1, falling for β < 1, flat at β = 1.
    Hazard rate0.0007.919e-50.0000.0000.0000.0008768762190219043804380876087601.314e+41.314e+41.752e+41.752e+42.628e+42.628e+43.504e+43.504e+4β = 1.8β = 1 (exponential)
    Canonical equations
    The exact relationships implemented by this engine.

    R(t) = exp[−(t/η)^β] · survival probability

    h(t) = (β/η)(t/η)^(β−1) · instantaneous failure rate

    η·Γ(1 + 1/β) · not equal to η unless β = 1

    η(−ln 0.9)^(1/β) · the 10th-percentile life

    engineering methodology

    A clear path from inputs to results.

    The Weibull distribution is the workhorse of reliability engineering because its shape parameter identifies the failure regime: below 1 is infant mortality, exactly 1 is memoryless random failure, above 1 is wear-out. Two parameters describe the whole life curve.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the shape parameter β from a fitted dataset.
    2. Enter the scale parameter η, the time by which 63.2% have failed.
    3. Set the mission time you care about.
    4. Enter the fleet size to convert probability into expected failures.
    5. Read reliability, hazard rate, MTBF and B10 life.

    The formula

    R(t)=e−(t/η)β,h(t)=βη(tη)β−1,MTBF=η Γ ⁣(1+1β)R(t) = e^{-(t/\eta)^\beta}, \qquad h(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta-1}, \qquad \mathrm{MTBF} = \eta\,\Gamma\!\left(1+\tfrac{1}{\beta}\right)

    Reference: Abernethy, "The New Weibull Handbook", 5th ed.

    Worked example

    β = 1.8, η = 12,000 h, mission 8,760 h

    Inputs
    • Shape 1.8, scale 12,000 h.
    • Mission 8,760 h.
    • Fleet of 500 units.
    Calculation path
    1. 1

      Scaled time: (8,760/12,000)^1.8 = 0.5744.

    2. 2

      Reliability: exp(−0.5744) = 56.7%.

    3. 3

      Hazard rate at mission: (1.8/12,000) × 0.7137^0.8 = 116.6 per million hours.

    4. 4

      MTBF: 12,000 × Γ(1.556) = 10,672 h.

    5. 5

      B10 life: 12,000 × (−ln 0.9)^(1/1.8) = 3,437 h.

    Result

    56.7% survive the mission · MTBF 10,672 h · B10 3,437 h · 217 expected failures in 500 units.

    Interpreting the result

    The B10 life of 3,437 hours is far shorter than the MTBF of 10,672, and both are shorter than η. That is normal: B10 is the 10th percentile, MTBF is the mean, and η is the 63.2nd percentile. Quoting MTBF alone hides the early tail, which is what a warranty actually exposes you to.

    Assumptions & limitations

    • A single failure mode with independent units and a homogeneous population.
    • Time-to-failure follows a Weibull distribution exactly; small samples make β uncertain.
    • Failed units are replaced with new ones, so the hazard applies to the population as modelled.
    • No censoring is applied — field data with suspended units needs a maximum-likelihood fit, not this closed form.

    Frequently asked questions

    Why is MTBF not equal to η?

    Because the mean of a Weibull is η·Γ(1 + 1/β), and the gamma factor is only 1 when β = 1. For β = 1.8 the factor is 0.889, so MTBF is about 11% below η.

    Should I quote MTBF or B10?

    B10 for anything warranty- or safety-related, because it is a percentile you can promise. MTBF is a mean and says nothing about the early tail. Many specifications ask for both.

    What does β < 1 mean?

    A falling hazard rate: failures are concentrated early, which points to manufacturing defects, assembly errors or burn-in problems. Screening or burn-in is the usual response.

    How many failures do I need to fit a Weibull?

    At least 10–15 failures for a rough fit, and more for confidence intervals on β. With fewer, use a known β from a similar product family rather than fitting.

    Methodology last reviewed 2026-10-18. Results are informational and not financial or engineering advice — see our disclaimer.