Calculator Inputs
This tool supports two-parameter and three-parameter Weibull analysis. Enter x for density and probability outputs, then enter p for quantile estimation.
Example Data Table
| Scenario | Shape k | Scale λ | Location γ | x | F(x) | S(x) |
|---|---|---|---|---|---|---|
| Electronic component life | 1.80 | 450 | 0 | 300 | 0.4628 | 0.5372 |
| Ball bearing wear | 2.50 | 120 | 0 | 85 | 0.4568 | 0.5432 |
| Coated surface failure | 3.20 | 60 | 5 | 40 | 0.1918 | 0.8082 |
Formula Used
Probability density: f(x) = (k / λ) × ((x − γ) / λ)k−1 × e−((x−γ)/λ)k, for x ≥ γ.
Cumulative probability: F(x) = 1 − e−((x−γ)/λ)k.
Reliability: S(x) = e−((x−γ)/λ)k.
Hazard rate: h(x) = f(x) / S(x).
Cumulative hazard: H(x) = ((x − γ) / λ)k.
Quantile: Q(p) = γ + λ × [−ln(1 − p)]1/k.
Here, k is the shape parameter, λ is the scale parameter, γ is the location threshold, x is the observation, and p is the target cumulative probability.
How to Use This Calculator
- Enter the Weibull shape parameter to define failure behavior.
- Enter the scale parameter in the same units as x.
- Use location when failures start after a threshold.
- Enter x to evaluate density, cumulative probability, and reliability.
- Enter probability p when you need a percentile life.
- Choose whether the primary probability output is F(x) or S(x).
- Press the submit button to show results above the form.
- Use the export buttons to download a CSV or PDF report.
Frequently Asked Questions
1. What does the Weibull shape parameter mean?
The shape parameter controls failure trend. Values below 1 suggest early failures, near 1 suggest random failures, and above 1 suggest wear-out behavior.
2. What is the difference between F(x) and S(x)?
F(x) gives the probability of failure by x. S(x) gives the probability of surviving beyond x. They always sum to 1.
3. When should I use a location parameter?
Use the location parameter when failures cannot occur before a minimum threshold. It shifts the distribution right and models delayed failure onset.
4. What does the hazard rate show?
The hazard rate shows instantaneous risk at x, assuming the item has survived until x. It is useful in reliability and maintenance planning.
5. What is a Weibull quantile?
A quantile converts a cumulative probability into the corresponding life value. For example, the 0.90 quantile marks when 90% failure probability is reached.
6. Can this calculator handle two-parameter Weibull analysis?
Yes. Set the location parameter to zero. The calculator then behaves like the common two-parameter Weibull model used in engineering and quality analysis.
7. Why does the density become very large near the threshold?
When shape is below 1, the density can rise sharply near the threshold. That reflects strong infant mortality or high early-life failure concentration.
8. Which industries use Weibull probability modeling?
Weibull models are widely used in manufacturing, reliability engineering, energy systems, materials science, warranty studies, and maintenance optimization.