Interactive module guide

Weibull Distribution Calculator & Guide

This three-parameter Weibull calculator lets you calculate lifetime probabilities and explore how shape, scale, and location change the PDF, CDF, and support threshold.

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The Weibull distribution is a continuous threshold model widely used for lifetimes and failure times because its shape parameter can represent decreasing, constant, or increasing hazard.

Scale sets a characteristic lifetime above the threshold, while location γ shifts the start of support.

Location defaults to zero, preserving the familiar two-parameter Weibull.

Use it when this hazard flexibility and threshold match the physical process.

Is this the right module for my question?

Use it when

  • Use a Weibull model for nonnegative time-to-event or strength measurements where the event rate may change with age. A shape k below one represents decreasing hazard, k=1 gives the constant-hazard exponential case, and k above one represents increasing hazard.
  • It is useful in reliability planning, survival analysis, wind-speed modeling, and materials work when a Weibull threshold model is defensible. The cumulative and survival functions answer questions such as failing before a warranty time or surviving beyond a mission duration.
  • The explorer helps distinguish shape, scale, and location. Shape k changes failure dynamics, scale λ stretches time above the threshold, and γ shifts the support without changing variance.

Choose another method when

  • Avoid using an uncensored Weibull calculation as a substitute for survival analysis when some units have not yet failed. Ignoring right-censored observations biases lifetime conclusions because surviving units contain information.
  • Do not use the model for hard upper bounds or multiple failure modes mixed without separation. A nonzero location can represent a defensible threshold, but it must not be added merely to force a convenient fit.

Interactive tool

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Example preview

Reliability at a mission time

Inputs
X ~ Weibull(shape=2, scale=1,000 hours); evaluate survival beyond 500 hours.
Representative result
P(X>500) = e⁻⁰·²⁵ ≈ 0.7788.

Illustrative only. Load the interactive tool to enter your own values and review assumptions.

Watch the explanation

2:57 min

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Read the complete transcript

The Weibull Distribution. What fraction of components survive a 500-hour mission?

The Weibull distribution is a continuous positive model widely used for lifetimes and failure times because its shape parameter can represent decreasing, constant, or increasing hazard. The scale parameter sets a characteristic lifetime. Use it when this hazard flexibility matches the physical process; do not interpret a visually good lifetime curve without accounting for censoring, operating conditions, and how failures were sampled.

The cumulative hazard is x divided by lambda to the power of k. Its power form creates the Weibull model’s flexible hazard pattern while retaining a simple survival function. A straight line in an appropriate Weibull probability plot can support the family, but deviations may reveal mixtures or changing conditions. The mean involves lambda times gamma of one plus one divided by k, so it changes with both parameters. Holding lambda fixed while changing k does not hold average lifetime fixed. Controlled comparisons should decide whether scale, mean, median, or a reliability percentile is the quantity to align. A CDF result at a warranty threshold is the modeled fraction failing by that time. Its complement is modeled reliability beyond the threshold. Report the parameter values, units, and whether the estimate represents a fitted population or a hypothetical scenario.

A component lifetime is modeled with Weibull shape k equals 2 and scale lambda equals 1,000 hours. An engineer wants the modeled probability that a new component survives a 500-hour mission, assuming operating conditions match those used for the model. Set k equals 2 and lambda equals 1000 hours. The survival probability at 500 hours equals the exponential of negative one half squared, which is the exponential of negative 0.25, approximately 0.7788. The corresponding cumulative failure probability is about 0.2212. Confirm that CDF and survival sum to one and that the threshold and scale use the same hour unit. Because k equals 2 exceeds one, the model has increasing hazard. A mission started with an already-aged component would require conditioning on its current age rather than reusing the new-component survival probability.

Under this Weibull model, about 77.9 percent of new components survive beyond 500 hours. The number is conditional on the chosen population and operating regime. It is not a confidence bound and does not incorporate parameter uncertainty, censoring uncertainty, maintenance, or competing failure modes.

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How to read the result

A CDF result at a warranty threshold is the modeled fraction failing by that time. Its complement is modeled reliability beyond the threshold. Report the parameter values, units, and whether the estimate represents a fitted population or a hypothetical scenario.

Hazard is a conditional instantaneous rate among units that have survived, not the probability of failure at an exact time. A rising hazard does not mean every older unit will fail; it means risk per small interval increases within the model.

Inputs and parameter meaning

Enter k as a strictly positive shape. For k<1 the density can be high near zero and the hazard decreases; this can describe early failures among a heterogeneous population. At k=1 the model is exponential. For k>1 the hazard increases, often associated with wear or accumulated damage.

Enter λ as a positive scale in the same unit as the lifetime. At x=γ+λ the CDF equals 1−e^(−1), about 63.2%, regardless of k. Thus λ is the characteristic distance above the threshold, not generally the mean or median.

Enter location γ as the support threshold. The density and CDF are zero below γ; at x=γ+λ the CDF equals about 63.2%. Setting γ=0 exactly reproduces the two-parameter model.

Use F(x)=1−exp[−((x−γ)/λ)^k] for x≥γ and zero below γ. Density height is not the failure probability at an exact instant.

Formula and available parameters

Density or probability mass

f(x)=kλ(x−γλ)k−1e−((x−γ)/λ)k,  x≥γf(x) = \frac{k}{\lambda}\left(\frac{x-\gamma}{\lambda}\right)^{k-1} e^{-((x-\gamma)/\lambda)^k},\;x\ge\gamma

Cumulative distribution

F(x)=1−e−((x−γ)/λ)k,  x≥γF(x) = 1 - e^{-((x-\gamma)/\lambda)^k},\;x\ge\gamma

Mean

γ+λΓ(1+1k)\gamma+\lambda\Gamma\left(1 + \frac{1}{k}\right)

Variance

λ2[Γ(1+2k)−Γ2(1+1k)]\lambda^2\left[\Gamma\left(1 + \frac{2}{k}\right) - \Gamma^2\left(1 + \frac{1}{k}\right)\right]
  • Shape parameter (k): interactive range 0.1 to 5; default 2.
  • Scale parameter (λ): interactive range 0.1 to 5; default 1.
  • Location / threshold (γ): interactive range -5 to 5; default 0.

What the method is doing

Above the threshold, cumulative hazard is ((x−γ)/λ)^k. Its power form creates the Weibull model’s flexible hazard pattern while retaining a simple survival function.

The mean is γ+λΓ(1+1/k). Changing γ shifts every quantile and the mean by the same amount while leaving variance and hazard shape above the threshold unchanged.

Worked example: reliability at a mission time

A component lifetime is modeled with Weibull shape k=2 and scale λ=1,000 hours. An engineer wants the modeled probability that a new component survives a 500-hour mission, assuming operating conditions match those used for the model.

  1. 1Set k=2 and λ=1000 hours. The survival probability is S(500)=exp[−(500/1000)^2]=exp(−0.25), approximately 0.7788.
  2. 2The corresponding cumulative failure probability is about 0.2212. Confirm that CDF and survival sum to one and that the threshold and scale use the same hour unit.
  3. 3Because k=2 exceeds one, the model has increasing hazard. A mission started with an already-aged component would require conditioning on its current age rather than reusing the new-component survival probability.

Interpretation

Under this Weibull model, about 77.9% of new components survive beyond 500 hours. The number is conditional on the chosen population and operating regime. It is not a confidence bound and does not incorporate parameter uncertainty, censoring uncertainty, maintenance, or competing failure modes.

Common mistakes

  • Software may label k and λ as beta, eta, shape, scale, or use an inverse scale. Verify the displayed 63.2% property at x=λ before moving parameters across tools.
  • Treating λ as the mean is incorrect except in special parameter settings. Use the gamma-function expression when an expected lifetime is required.
  • Combining different failure mechanisms can produce curvature that no single Weibull captures. A seemingly acceptable central fit may still misstate early-life or late-life risk.

Limits and independent validation

This explorer does not estimate k, λ, or γ, handle censored records, calculate confidence intervals, or model covariates and competing risks. It is a distribution calculator, not a complete reliability analysis.

Tail reliability is sensitive to parameter uncertainty and extrapolation beyond observed ages. Do not use a long-horizon result solely because the formula returns a precise decimal.

Before using the result

  • Review failure definitions, censoring, operating conditions, and age origin. Plot data using methods appropriate for censored lifetimes and look for evidence of multiple modes before choosing one Weibull family.
  • Verify a probability using both CDF and survival formulas. Document whether the item is new or has survived to a current age, because conditional remaining-life calculations differ when k is not one.

Related modules and resources

exponential distribution →Compare the k=1 constant-hazard special case.gamma distribution →Inspect another positive family used for waiting and lifetime quantities.data fitting →Compare supported candidate models with uncensored observed data while recognizing fitting limits.
← Beta DistributionLog-Normal Distribution →

Explore Other Probability Distributions

See every option in the Probability Distribution Explorer or review the DistriScope methodology. Educational information; last reviewed 2026-08-09. Verify consequential calculations independently.