Weibull Reliability Calculator

Understanding the Weibull reliability calculator — concept and principle

The Weibull distribution is the most widely used model for time to failure of components such as bearings, electronics, seals and fatigue-loaded parts. With just two parameters it can describe early-life failures, random failures or wear-out. This calculator takes the shape β, the scale η and an operating time t, and returns the reliability, the probability of failure by that time, the instantaneous hazard rate, the mean time to failure (MTTF) and the B10 life.

About 91 of 100 units are still working at 2,000 hours; 10% have failed by 2,231 hours.

How it is calculated

Reliability: R(t) = exp(−(t/η)^β); failure probability F(t) = 1 − R(t).
Hazard rate: h(t) = (β/η) × (t/η)^(β − 1), in failures per hour.
MTTF = η × Γ(1 + 1/β), where Γ is the gamma function.
B10 life (time by which 10% have failed) = η × (−ln 0.9)^(1/β).

η and t share the same time unit, here hours.

The principle

The hazard rate is the chance of failing in the next instant for a unit that has survived so far. Weibull's model lets it change as a power of time. With β < 1 the hazard falls (infant mortality), with β = 1 it is constant and the model becomes the exponential distribution, and with β > 1 it rises (wear-out). η is the characteristic life: at t = η, R = e^(−1), so about 63.2% of units have failed whatever the value of β. MTTF and B10 follow directly from integrating and inverting R(t).

Worked example

Inputs: β = 1.5, η = 10,000 h, t = 2,000 h.
t/η = 0.2; 0.2^1.5 = 0.08944.
R = e^(−0.08944) = 0.9144 = 91.44%; failure probability 8.56%.
Hazard: (1.5/10,000) × 0.2^0.5 = 0.00015 × 0.4472 = 0.0000671 per hour.
MTTF = 10,000 × Γ(1.6667) = 10,000 × 0.90275 = 9,027.45 h.
B10 = 10,000 × (0.10536)^(1/1.5) = 2,230.76 h.

Each point is the calculator re-run with only “Scale η (hours)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Scale η (hours)	Reliability R(t)
2,500	48.89%
5,000	77.65%
7,500	87.14%
10,000	91.44%
12,500	93.8%
15,000	95.25%
20,000	96.89%

How to read the result

Here 91.44% of units are expected to survive 2,000 hours. A β of 1.5 indicates mild wear-out, so the hazard keeps rising and preventive replacement can make sense. MTTF is an average and, for β > 1, is below η; B10 is the more conservative figure and is a common rating for bearings. The right β depends on the failure mode and should come from your own failure data.

Common mistakes

Using MTTF as a "guaranteed" life; many units fail well before it.
Assuming a constant failure rate (β = 1) for parts that clearly wear out.
Ignoring suspended (still running) units when fitting β and η, which biases the estimates.
Mixing hours and cycles between η and t.

Keep in mind: Results are only as good as the fitted β and η. Small samples give wide uncertainty, and mixed failure modes may need separate Weibull fits or a three-parameter model with a failure-free period.

Frequently asked questions

What does β = 1 mean?

A constant hazard rate: the Weibull becomes exponential, MTTF equals η and failures are random in time.

What is the difference between MTTF and MTBF?

MTTF applies to non-repairable items; MTBF is used for repairable systems, the mean time between successive failures.

How are β and η estimated?

Typically by plotting failure data on Weibull probability paper (median-rank regression) or by maximum likelihood estimation.

Sources: W. Weibull, "A Statistical Distribution Function of Wide Applicability," Journal of Applied Mechanics, 1951 · IEC 61649, Weibull analysis · R. B. Abernethy, The New Weibull Handbook

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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