Library / Probability distributions / PRD-14
Probability distributions
Weibull life: lifetime variance
Variance of the lifetime, in squared hours when the scale is in hours.
- Ref
- PRD-14
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [h²]
The variance s²{Γ(1 + 2/k) − Γ(1 + 1/k)²} measures how widely lifetimes are spread, in squared hours when the scale is in hours. It falls as the shape k grows: a high k means a tight cluster of lifetimes around the mean.
Variance of the lifetime, in squared hours when the scale is in hours. A larger shape k gives a narrower spread of lifetimes.
Calculator
- Standard deviation51.12 h
- Mean time to failure88.93 h
- Coefficient of variation (standard deviation over mean)0.5749
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 standard deviation is 51.12 h against a mean of 88.93 h.
- Raising the shape from 1.8 to 2.8 would narrow the spread of lifetimes sharply, which is what a tighter manufacturing process aims for.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (h²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Shape k | 0.5 to 5 | 442.3 | 200,000 | 199,600 | +10% | −16.4 % |
| s Scale life s | 10 to 300 h | 26.14 | 23,520 | 23,500 | +10% | +21 % |
Sweep of Shape k
| Shape k | Lifetime variance (h²) |
|---|---|
| 0.5 | 200,000 |
| 1.063 | 8,456 |
| 1.625 | 3,191 |
| 2.188 | 1,824 |
| 2.75 | 1,222 |
| 3.313 | 890 |
| 3.875 | 682.4 |
| 4.438 | 542.2 |
| 5 | 442.3 |
Sweep one variable
Variables and how each one contributes
Lifetime variance
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 −16.4 %, from 2,614 to 2,186 h².
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 +21 %, from 2,614 to 3,162 h².
Worked example with the values above
- Second gamma term: Γ(1 + 2/k) = Γ(2.1111) = 1.0522.
- Squared first term: Γ(1 + 1/k)² = 0.79083.
- Var = 100² × (1.0522 − 0.79083) = 2,613.5 h².
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