Weibull Distribution Calculator

Distributions and Statistical Models

Enter the shape (k) and scale (λ) parameters, along with a value (x), to analyze the Weibull distribution.

Practical Examples

Explore different scenarios to understand how the Weibull distribution works in practice.

Bearing Failure Analysis

Component Reliability

Engineers are analyzing the reliability of a new type of bearing. From test data, they estimate a shape parameter (k) of 2.1 and a scale parameter (λ) of 8500 hours. They want to find the probability of failure by 7000 hours.

k: 2.1, λ: 8500, x: 7000

Wind Speed Modeling

Wind Speed

A meteorologist models daily average wind speeds using a Weibull distribution with k=1.8 and λ=12 mph. They need to calculate the probability of wind speed being exactly 15 mph (PDF) and less than or equal to 15 mph (CDF).

k: 1.8, λ: 12, x: 15

Decreasing Failure Rate

Infant Mortality Rate

When modeling certain phenomena like infant mortality or software bugs after a patch, the failure rate decreases over time. This can be modeled with k < 1. Let's use k=0.8 and λ=5 (months) to analyze the probability characteristics at month 3.

k: 0.8, λ: 5, x: 3

Exponential Distribution Case

Constant Failure Rate

When k=1, the Weibull distribution simplifies to the exponential distribution, which models events with a constant failure rate (e.g., random hardware failures). Let's see the metrics for k=1 and λ=500 hours, evaluated at 500 hours.

k: 1, λ: 500, x: 500

Other Titles
Understanding the Weibull Distribution: A Comprehensive Guide
An in-depth look into one of the most versatile and widely used lifetime data analysis distributions.

What is the Weibull Distribution?

  • Core Concepts
  • Key Parameters
  • Versatility
The Weibull distribution is a continuous probability distribution that is extremely flexible and widely used in reliability engineering, survival analysis, and industrial engineering to model failure times. Its ability to mimic the characteristics of other types of distributions (such as the normal and exponential) makes it a powerful tool for analyzing life data.
The Role of Parameters: Shape and Scale
The distribution is defined by two primary parameters: the shape parameter (k) and the scale parameter (λ). A third parameter, the location parameter (γ), is sometimes included to represent a failure-free period, but it is often assumed to be zero. The shape parameter, k, dictates the shape of the distribution curve and the nature of the failure rate. The scale parameter, λ, stretches or compresses the distribution along the time axis.
Why is it so Widely Used?
Its main advantage is its ability to model a wide range of failure rate behaviors. By adjusting the shape parameter (k), one can model systems where the failure rate is decreasing (k < 1), constant (k = 1, the exponential distribution), or increasing (k > 1). This makes it suitable for describing different phases of a product's life, from infant mortality to wear-out failures.

Step-by-Step Guide to Using the Weibull Distribution Calculator

  • Inputting Parameters
  • Executing Calculation
  • Interpreting the Results
Our calculator simplifies the process of analyzing the Weibull distribution. Follow these steps to get your results.
1. Enter the Shape Parameter (k)
Input the shape parameter (also known as the Weibull modulus). This must be a positive value. This value tells you about the failure mechanism.
2. Enter the Scale Parameter (λ)
Input the scale parameter (also known as the characteristic life). This must also be a positive value. It represents the time at which 63.2% of the population is expected to have failed.
3. Enter the Value (x)
This is the specific point in time or value for which you want to calculate the probabilities. It must be a non-negative number.
4. Click 'Calculate'
The calculator will instantly provide a full suite of results, including the PDF, CDF, survival probability, and key statistical metrics like mean, median, and variance.

Real-World Applications of the Weibull Distribution

  • Reliability Engineering
  • Wind Speed Analysis
  • Biological and Medical Modeling
The versatility of the Weibull distribution lends it to a vast array of applications across different fields.
Product Life and Warranty Analysis
Companies use Weibull analysis to predict how many units will fail over a certain period. This helps in setting warranty periods, planning for spare parts inventory, and assessing the cost of repairs.
Weather Forecasting
Meteorologists use the Weibull distribution to model wind speed distributions. This is crucial for wind farm planning, as it helps in estimating the potential power generation of a site.
Survival Analysis in Medicine
In medical research, it's used to model the survival times of patients after a particular treatment. The shape parameter can indicate whether the risk of death increases or decreases over time following the treatment.

Common Misconceptions and Correct Methods

  • Confusing Shape and Scale
  • Assuming a Normal Distribution
  • Ignoring the Location Parameter
While powerful, the Weibull distribution is sometimes misunderstood. Clarifying these points ensures accurate analysis.
Misconception: The Shape Parameter is Just an Abstract Number
Correction: The shape parameter (k) has a direct physical interpretation. k < 1 suggests 'infant mortality' or early-life failures. k = 1 indicates random failures (constant rate). k > 1 suggests wear-out failures. Choosing the right k is critical for a valid model.
Misconception: All Lifetime Data is Normally Distributed
Correction: Unlike normal distributions, which are always symmetric, lifetime data is often skewed. The Weibull distribution is inherently flexible and can model right-skewed data effectively, which is common for failure times.
Misconception: The Location Parameter is Always Zero
Correction: While our calculator assumes a location parameter (γ) of zero for simplicity (meaning failures can start at time t=0), real-world scenarios might have a failure-free period. In such cases, a 3-parameter Weibull distribution is needed, where γ > 0.

Mathematical Derivation and Formulas

  • Probability Density Function (PDF)
  • Cumulative Distribution Function (CDF)
  • Key Statistical Metrics
Here are the core formulas that power the Weibull distribution calculator.
Probability Density Function (PDF)
The PDF, f(x), describes the relative likelihood of a random variable being equal to a specific value x. The formula is: f(x; k, λ) = (k/λ) (x/λ)^(k-1) e^(-(x/λ)^k)
Cumulative Distribution Function (CDF)
The CDF, F(x), gives the probability that the random variable will take a value less than or equal to x. The formula is: F(x; k, λ) = 1 - e^(-(x/λ)^k)
Mean and Variance
The mean (expected lifetime) and variance require the gamma function (Γ). Mean = λ Γ(1 + 1/k). Variance = λ^2 [Γ(1 + 2/k) - (Γ(1 + 1/k))^2].