Weibull Distribution Calculator | Reliability, Failure Rate & MTTF
Compute the PDF, CDF, reliability function, failure (hazard) rate, and mean time to failure (MTTF) for the Weibull distribution with shape parameter k and scale parameter λ. Evaluates survival probability at any time t, computes B10 life, and classifies the failure mode: infant mortality, random, or wear-out.
Weibull Parameters
What Is the Weibull Distribution Calculator | Reliability, Failure Rate & MTTF?
The Weibull distribution is the workhorse of reliability engineering and survival analysis. The shape parameter k governs the failure mechanism: k<1 models infant mortality (defect-driven early failures), k=1 reduces to the memoryless exponential distribution for random failures, and k>1 models wear-out failures where the hazard rate increases with age. The scale parameter λ sets the characteristic life — the time at which approximately 63.2% of units have failed.
Formula
f(t) = (k/λ)(t/λ)^(k−1)·exp(−(t/λ)^k), R(t) = exp(−(t/λ)^k), h(t) = (k/λ)(t/λ)^(k−1)
How to Use
- 1
Enter the shape parameter k. If you have Weibull-fitted data from a test, use those values. Otherwise use the preset buttons for typical component classes.
- 2
Enter the scale parameter λ (characteristic life). This is the time at which R(t) = e^−1 ≈ 36.8%, or equivalently F(t) ≈ 63.2%.
- 3
Enter the evaluation time t — the operating age at which you want to know reliability, failure probability, and hazard rate.
- 4
Click Calculate to see all reliability metrics: PDF, CDF (failure probability), reliability R(t), and hazard rate h(t).
- 5
Read the failure mode badge: it classifies your k value as Infant Mortality, Random, or Wear-out with a description.
- 6
Check the statistics cards for MTTF (Mean Time to Failure), B10 life (time when 10% fail), and B50 life (median lifetime).
- 7
Review the reliability table at multiples of λ to plan maintenance intervals or warranty periods.
Enter the shape k and scale λ for your component, set the evaluation time t, and click Calculate.
Example Calculation
A mechanical bearing has k=2.5 and λ=20,000 hours. MTTF = 20000·Γ(1.4) ≈ 17,834 hours. B10 life = 20000·(−ln 0.9)^(1/2.5) ≈ 6,825 hours — meaning only 10% of bearings will fail before 6,825 hours. At t=15,000 hours, R(t) = exp(−(0.75)^2.5) ≈ 66%, so about 34% have failed by then. This is wear-out behaviour with an increasing hazard rate.
Understanding Weibull Distribution | Reliability, Failure Rate & MTTF
Shape Parameter k Interpretation
The shape parameter k is the most important parameter in reliability analysis — it tells you the failure mechanism governing your component or system.
| k range | Failure type | Hazard rate h(t) | Engineering example |
|---|---|---|---|
| k < 1 | Infant mortality | Decreasing over time | Electronics with manufacturing defects, solder joints |
| k = 1 | Random (chance) | Constant — memoryless | Cosmic ray damage, random software errors |
| 1 < k < 2 | Early wear-out | Mildly increasing | Fatigue crack initiation in metals |
| k = 2 | Rayleigh (linear hazard) | Linearly increasing | Corrosion, surface wear in bearings |
| 2 < k < 4 | Normal wear-out | Increasing (concave) | Mechanical components, automotive parts |
| k ≈ 3.44 | Near-Normal | Close to symmetric bell | Many mechanical failure modes at end of life |
| k > 4 | Rapid wear-out | Steeply increasing | Light bulb filaments, O-rings past design life |
Weibull vs Exponential Distribution
The exponential distribution (k=1) is a special case of Weibull. Understanding the differences helps select the right model for your data.
| Property | Exponential (k=1) | Weibull (general k) |
|---|---|---|
| Hazard rate | Constant λ = 1/scale | (k/λ)(t/λ)^(k−1) — time-varying |
| Memoryless property | Yes — P(T>s+t|T>s) = P(T>t) | No (except at k=1) |
| MTTF | λ (the scale parameter) | λ·Γ(1 + 1/k) |
| CDF | 1 − e^(−t/λ) | 1 − e^(−(t/λ)^k) |
| Suitable for | Electronic components in useful life | Mechanical parts, any life phase |
| Parameter estimation | One parameter: λ = mean lifetime | Two parameters: k and λ (use MLE or Weibull plot) |
Reliability Engineering Applications
- ▸Bearing life prediction: Roller bearings follow approximately Weibull(2–3) distributions. B10 life (10% failure point) is the industry standard specification for bearing selection.
- ▸Battery degradation: Lithium-ion cell capacity fade follows a Weibull model with k typically between 1.5 and 3, depending on charge/discharge cycles and temperature.
- ▸Warranty analysis: Manufacturers fit Weibull parameters to warranty claim data to predict future claim rates and budget warranty reserves.
- ▸Preventive maintenance scheduling: When k > 1, scheduled replacement before the B10 life significantly reduces unplanned failures compared to a run-to-failure strategy.
- ▸Wind power generation: Wind speed distributions are well modeled by Weibull(2, λ) — the Rayleigh distribution — enabling accurate energy yield predictions.
- ▸Accelerated life testing: Products are tested under elevated stress (temperature, voltage) and Weibull parameters are fitted to project field reliability under normal conditions.
Frequently Asked Questions
What are the Weibull B10 and B50 life values?
B10 life (also written L10) is the time at which 10% of a population of identical components will have failed — equivalently, R(t) = 0.90. It is the standard bearing life rating in ISO 281. B50 life is the median life where 50% have failed (R = 0.50), which equals λ·(ln 2)^(1/k). B10 < MTTF < B50 is typical for wear-out distributions (k > 1).
How do I estimate Weibull parameters from test data?
The two main methods are: (1) Weibull probability plot — plot ranked failure times on Weibull probability paper (ln(−ln(1−F)) vs ln(t)); the slope is k and the intercept gives λ. (2) Maximum Likelihood Estimation (MLE) — numerically maximise the log-likelihood of the observed failure times. MLE handles censored data (units that have not yet failed) better than the probability plot.
What is the difference between scale λ and MTTF?
λ is the scale parameter of the Weibull distribution — it is the 63.2nd percentile of the lifetime distribution, not the mean. MTTF = λ·Γ(1+1/k). When k=1, MTTF = λ (scale equals mean). When k>1, MTTF < λ (failures cluster before the scale parameter). The scale λ is sometimes called the characteristic life or η.
Why does the hazard rate matter for maintenance?
The hazard rate h(t) (also called the instantaneous failure rate or force of mortality) tells you how quickly failures are accumulating at a given age. For k>1 (wear-out), h(t) increases with age, meaning older components fail faster than younger ones. This justifies time-based preventive replacement: replace before h(t) becomes unacceptably high. For k=1, h(t) is constant and age-based replacement provides no benefit over run-to-failure.
How does the Weibull bathtub curve work?
The classic reliability "bathtub curve" has three phases: (1) Infant mortality — high initial failure rate that decreases over time (model with a Weibull k<1 or an early-life Exponential). (2) Useful life — constant (random) failure rate (k=1 Exponential). (3) Wear-out — increasing failure rate (Weibull k>1). Real products often follow a mixture of several Weibull distributions to capture all three phases simultaneously.
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