Weibull Reliability Calculator
Reliability, failure probability, hazard rate, B10 life and mean life from a Weibull shape and scale
Inputs
Reliability at t
89.4%
10.6% probability of failure by 5,000 h
Hazard rate at t
0.056per 1,000 h
Shape above 2: wear-out. A time-based or condition task can be justified
B10 life
4,878h
Age by which 10 per cent of a population has failed
Mean life (MTTF)
10,647h
η × Γ(1 + 1/β), with Γ(1 + 1/β) = 0.8873
Conditional reliability
71.8%
Surviving the next 5,000 h from age 3,000 h. Reliability at age 3,000 h is 96.9%
How it's calculated
Shape and scale
β and η from a fitted population
Reliability
exp(−(t/η)^β)
Hazard and B10
Failure rate now, and the 10% life
Decision
Interval, condition task or run to failure
Reliability R(t) = exp(−(t ÷ η)β), the probability an item from new is still working at time t.
Probability of failure F(t) = 1 − R(t).
Hazard rate h(t) = (β ÷ η) × (t ÷ η)β − 1, the instantaneous failure rate of the survivors at time t.
B10 life = η × (−ln 0.9)1/β, the age by which ten per cent have failed. Any other percentile uses the same form.
Mean life = η × Γ(1 + 1/β), where Γ is the gamma function.
Conditional reliability = R(a + t) ÷ R(a), the chance an item that has reached age a survives a further t.
The two-parameter Weibull form is used here. A location parameter (failure-free period) shifts time by a constant and is rarely justified by field data. The parameters themselves come from fitting the failure and suspension times of a population, which the calculator does not do; see the guide below for how.
How to calculate reliability with the Weibull distribution: a worked example
Take a population of pump bearings whose failure times have been fitted, using the figures the calculator opens with: a shape parameter of 2.5 and a scale parameter, or characteristic life, of 12,000 hours. The question is how reliable a new bearing is over the next 5,000 hours, and what the figures say about a replacement interval.
| Step | Calculation | Result |
|---|---|---|
| Reliability at 5,000 h | exp(−(5000 ÷ 12000)^2.5) | 89.4% |
| Probability of failure by 5,000 h | 1 − 0.894 | 10.6% |
| Hazard rate at 5,000 h | (2.5 ÷ 12000) × (5000 ÷ 12000)^1.5 × 1000 | 0.056 per 1,000 h |
| B10 life | 12000 × (−ln 0.9)^(1 ÷ 2.5) | 4,878 h |
| Mean life | 12000 × Γ(1 + 1 ÷ 2.5) = 12000 × 0.8873 | 10,647 h |
| Conditional reliability, next 5,000 h from 3,000 h | R(8000) ÷ R(3000) = 0.696 ÷ 0.969 | 71.8% |
A shape of 2.5 is wear-out. Ten per cent of these bearings fail before 4,900 hours, and a bearing that has already run 3,000 hours has less chance of lasting the next 5,000 than a new one. That is the case for a time-based change-out, or for condition monitoring that catches the wear before it becomes a failure.
What the shape parameter tells you
The shape parameter, β, is the reason to use Weibull at all. MTBF gives one number for a population and says nothing about whether failures are becoming more or less likely with age. The shape does, and it maps directly to the kind of maintenance task that can help.
| Shape (β) | Failure rate with age | Typical causes | What helps |
|---|---|---|---|
| Below 1 | Falling | Installation and manufacturing faults, poor commissioning | Better installation and burn-in. Scheduled replacement makes it worse |
| About 1 | Constant (random) | External events, overloads, operator damage, electronics | Condition monitoring if there is warning, otherwise redundancy or run to failure. Time-based replacement does not help |
| 1 to 2 | Rising slowly | Early wear, corrosion, fatigue starting | Condition monitoring, inspection |
| Above 2 | Rising steeply (wear-out) | Bearings, seals, liners, belts, wear parts | Time-based replacement at a percentile life, or condition monitoring where it gives warning |
Where the parameters come from
The calculator takes the shape and scale as inputs because fitting them is a separate job. The parameters come from the failure times of a population of like items, including the items that have not failed yet, which are recorded as suspensions and matter as much as the failures. Fitting is done by rank regression or maximum likelihood in reliability software or a spreadsheet, and a fit on fewer than about ten failures is a hypothesis rather than a result.
Three things make the fit honest. Failures are grouped by failure mode, because a bearing that fails from fatigue and one that fails from contamination have different shapes and should not be mixed. Times are operating hours or cycles rather than calendar time. And the data come from a CMMS with failure codes people use, which is where most Weibull analyses quietly fail.
- One failure mode per fit. Mixed modes flatten the shape towards 1 and hide the wear-out.
- Include suspensions. Leaving out the survivors makes every population look shorter-lived than it is.
- Use the age at failure, not the date. The CMMS has to hold install dates and runtime for that.
- Read the shape before the scale. The shape decides the kind of task; the scale and the percentile set the interval.
Using the results
For a wear-out population, the replacement interval is read from a percentile life, often B10 or lower for a critical item, rather than from the mean. The mean is where half the population has failed, which is far too late. For a random-failure population, the hazard rate is the number to plan around: it is the same whether the item is new or old, so the money goes into detection and spares rather than scheduled changes.
The conditional reliability is the figure for an item already in service: the chance it survives the next period from its current age. On a wear-out item it falls with age, which is what justifies changing a part at a shutdown even though it is still running. The MTBF calculator gives the population average that this calculator takes apart.
Why it matters
MTBF describes a population with one number and hides the thing a maintenance decision needs most: whether failures are getting more likely as the equipment ages. The Weibull shape parameter answers that. A bearing with a shape of 2.5 wears out, and a replacement interval can be set from its B10 life. A control card with a shape of 0.8 fails early or not at all, and replacing it on a schedule introduces new failures rather than preventing old ones.
That one distinction decides between a time-based task, a condition task and leaving an item alone, which is the heart of reliability centred maintenance. Reading the shape and the B10 life off a fitted curve is quicker, and more honest, than arguing about an interval.
Common questions
What is the Weibull distribution used for in maintenance?
To describe how the failures of a population of like items spread over their life. Its shape parameter says whether the failure rate falls, stays constant or rises with age, which decides whether a time-based replacement, a condition monitoring task or run to failure is the right strategy. Its scale parameter sets the life, and percentiles such as B10 set replacement intervals.
How do you calculate reliability from Weibull parameters?
Reliability at time t is R(t) = exp(−(t ÷ η)^β), where η is the scale parameter and β the shape. With a shape of 2.5 and a scale of 12,000 hours, the reliability at 5,000 hours is exp(−(5000 ÷ 12000)^2.5), which is 89.4 per cent. The probability of failure by then is one minus that.
What does the Weibull shape parameter mean?
A shape below 1 means the failure rate falls with age, typical of installation faults. A shape of 1 means random failures at a constant rate. A shape above 1 means the failure rate rises with age, and above about 2 is clear wear-out, typical of bearings, seals and wear parts. Time-based replacement only helps when the shape is well above 1.
What is B10 life?
The age by which ten per cent of a population has failed, calculated as η × (−ln 0.9)^(1 ÷ β). Bearings are rated on it, and it is a common basis for a replacement interval on a critical wear-out item. Other percentiles use the same formula with a different probability.
What is the difference between MTBF and Weibull mean life?
MTBF is total operating time divided by the number of failures over a period, a single average for a repairable asset. The Weibull mean life is η × Γ(1 + 1 ÷ β), the mean of a fitted life distribution for a population of items. They can be similar numbers, but only Weibull tells you whether the failures are random or age related.
How many failures do you need for a Weibull analysis?
Enough to fit a line with some confidence, which in practice means ten or more failures of the same failure mode, with the surviving items recorded as suspensions. Fewer failures give a shape with very wide confidence bounds, and the analysis is better used to form a hypothesis than to set an interval.
Related reading
- MTBF, MTTR and Availability Calculator
- Reliability centred maintenance, and when criticality-led beats it
- Condition monitoring: techniques, intervals and building a program that works
- Fixed plant maintenance on a mine site: strategies, shutdowns, systems and the work that keeps it running
- Reliability engineering and asset strategy program
- Maintenance KPIs: fourteen measures, defined once and read together
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