Calculator inputs
Example data table
This sample shows a common reliability-style setup for the three-parameter Weibull model.
| β | η | γ | x | p | t1 | t2 | F(x) | S(x) | Mean |
|---|---|---|---|---|---|---|---|---|---|
| 1.8 | 1200 | 0 | 900 | 0.90 | 500 | 1500 | 0.448886 | 0.551114 | 1067.144079 |
Formula used
Adjusted variable: z = (x - γ) / η, for x ≥ γ
Probability density: f(x) = (β / η) · zβ-1 · e-zβ
Cumulative distribution: F(x) = 1 - e-zβ
Reliability or survival: S(x) = e-zβ
Hazard rate: h(x) = f(x) / S(x) = (β / η) · zβ-1
Cumulative hazard: H(x) = zβ
Mean: μ = γ + η · Γ(1 + 1/β)
Variance: σ² = η² · [Γ(1 + 2/β) - Γ(1 + 1/β)²]
Median: γ + η · (ln 2)1/β
Mode: γ + η · ((β - 1)/β)1/β, when β > 1
Quantile: Q(p) = γ + η · [-ln(1 - p)]1/β
The shape parameter β determines failure behavior. Values below 1 indicate early-life failures, β = 1 produces the exponential case, and values above 1 show increasing wear-out risk.
How to use this calculator
- Enter the Weibull shape β, scale η, and optional location γ.
- Provide an evaluation point x to compute density, probability, survival, and hazard.
- Enter a probability p between 0 and 1 for the quantile result.
- Fill interval bounds t1 and t2 to estimate failure probability across that range.
- Choose a unit label so time-based outputs read clearly.
- Click the calculate button to display the result block above the form.
- Use the CSV or PDF buttons to export the current output table.
- Review the graph to compare density, cumulative probability, and reliability behavior.
FAQs
1) What does the Weibull distribution model describe?
It models lifetimes, strengths, waiting times, and failure behavior. It is especially useful when data can show infant mortality, random failure, or wear-out trends.
2) What does the shape parameter β mean?
β controls how the failure rate changes. Below 1 means decreasing risk, 1 means constant risk, and above 1 means increasing risk over time.
3) What does the scale parameter η mean?
η stretches the distribution horizontally. In reliability work, it is often called characteristic life because about 63.2% of items fail by that point when γ = 0.
4) Why use the location parameter γ?
γ shifts the model rightward. It helps when failures cannot happen before a threshold time, load, or stress level.
5) What is the difference between CDF and reliability?
The CDF gives failure probability by x. Reliability is the survival probability beyond x. They always add to 1.
6) When is the mode formula valid?
The interior mode formula applies only when β is greater than 1. For β less than or equal to 1, the curve peaks at the lower boundary.
7) What does the hazard rate show?
It shows the instantaneous failure tendency at x, assuming the item has survived to that point. It is important in maintenance and reliability studies.
8) Can this calculator be used outside reliability?
Yes. The Weibull model also appears in hydrology, material strength, wind studies, survival analysis, and quality engineering whenever skewed lifetime-like behavior matters.