Library / Probability distributions / PRD-13
Probability distributions
Weibull life: mean time to failure
Average lifetime of a single non-repairable item with a Weibull life.
- Ref
- PRD-13
- Rev
- v1
- Field
- Probability distributions
- Result
- E[X] [h]
The mean time to failure is s times the gamma function of 1 + 1/k. For k = 1 it equals s (the exponential case); for larger k it is slightly below s, about 0.89 s at k = 1.8. It is the average life of a single non-repairable item, not a guaranteed life: when k > 1 many units fail well before it.
Average lifetime of a single non-repairable item with a Weibull life. It equals the scale s only when k = 1.
Calculator
- Median life, s(ln 2)^(1/k)81.58 h
- Ratio of mean to scale, Γ(1 + 1/k)0.8893
- Fraction failed by the mean time0.555
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The mean life is 88.93 h, against a scale life of 100 h.
- By the mean time to failure, 55.5 % of units have already failed, so the mean is not a safe design life.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (h) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale life s | 10 to 300 h | 8.893 | 266.8 | 257.9 | +10% | +10 % |
| k Shape k | 0.5 to 5 | 88.56 | 200 | 111.4 | +10% | −0.325 % |
Sweep of Shape k
| Shape k | Mean time to failure (h) |
|---|---|
| 0.5 | 200 |
| 1.063 | 97.65 |
| 1.625 | 89.53 |
| 2.188 | 88.56 |
| 2.75 | 88.99 |
| 3.313 | 89.72 |
| 3.875 | 90.48 |
| 4.438 | 91.18 |
| 5 | 91.82 |
Sweep one variable
Variables and how each one contributes
Mean time to failure
What the equation gives you. Everything else feeds it.
Shape k
The shape parameter (a plain number). Below 1 the failure rate falls with age (early failures), at 1 it is constant, and above 1 it rises (wear-out). Typical bearings have k of about 1.5 to 3. Controls how quickly the failure rate changes with age. Raising it by 10 % (to 1.98) moves the result by −0.325 %, from 88.93 to 88.64 h.
Scale life s
The scale life, in hours: the age by which 63.2 % of units have failed, whatever the shape. Stretches every time in the model in proportion. Raising it by 10 % (to 110) moves the result by +10 %, from 88.93 to 97.82 h.
Worked example with the values above
- Gamma argument: 1 + 1/k = 1 + 1/1.8 = 1.5556.
- Gamma value: Γ(1.5556) = 0.88929.
- E[X] = 100 × 0.88929 = 88.929 h.
Assumptions and limits
- A two-parameter Weibull with zero location: nothing fails before it is put into service.
- Failure times are independent and the units are non-repairable.
- The shape and scale come from a fit to data for the same failure mode and operating conditions.
Common mistakes
- Mixing several failure modes in one Weibull fit, which flattens the shape and hides wear-out.
- Entering the scale in a different time unit from x, such as years against hours.
- Reading the density at one age as the probability of failing at that age: it is a rate per hour.
- Treating the mean time to failure as a guaranteed life: when k > 1 many units fail well before it.
Statistics and uncertainty
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
Where it is used
- Reliability analysis of bearings, motors, capacitors, fasteners and other components.
- Warranty planning: how many units fail by the end of the warranty period.
- Choosing a replacement age from the shape of the failure rate.
Maths you need
- Exponential function: e raised to a power
- Gamma function: the generalised factorial
- Reliability: chance of surviving to a given time