Free Online Weibull Analysis for Failure Prediction

Estimate reliability and predict failures using the Weibull distribution.

What is Weibull Analysis?

Weibull Analysis is a powerful statistical method used in reliability engineering to model time-to-failure data. It is highly versatile and can model the failure characteristics of complex systems, from infant mortality to wear-out failures.

The analysis yields two key parameters:

  • Beta (β) - The Shape Parameter: Describes the failure mode.
    • β < 1: Decreasing failure rate (infant mortality). Failures are more likely at the beginning.
    • β = 1: Constant failure rate (random failures). The process follows an exponential distribution.
    • β > 1: Increasing failure rate (wear-out). Failures become more likely as the product ages.
  • Eta (η) - The Scale Parameter: Also known as the characteristic life. It's the time at which 63.2% of the population is expected to have failed.

Why is Weibull Analysis Useful?

Originally developed by Waloddi Weibull to model material strength, this analysis is now a cornerstone of modern reliability engineering. It allows engineers and managers to make data-driven decisions about:

  • Maintenance Planning: Predict when parts are likely to fail to schedule preventive maintenance.
  • Warranty Analysis: Forecast warranty claims and costs.
  • Design Improvement: Identify failure modes (e.g., wear-out vs. early failures) to guide product improvements.
  • Risk Assessment: Quantify the reliability of a component or system over its operational life.

History & Origin

The Weibull distribution was first described by Swedish engineer and mathematician Waloddi Weibull in 1937. Initially developed to model material strength and fatigue life, Weibull presented his distribution to the American Society of Mechanical Engineers in 1951 in a famous paper titled "A Statistical Distribution Function of Wide Applicability."

What made Weibull's approach revolutionary was its flexibility to model various failure patterns through its shape parameter (β). Unlike the normal distribution, which assumes symmetrical variation around a mean, the Weibull distribution can effectively model:

  • Infant Mortality (β < 1): Early failures due to manufacturing defects
  • Random Failures (β = 1): Constant failure rate, equivalent to exponential distribution
  • Wear-Out Failures (β > 1): Age-related failures that increase over time

This versatility made Weibull analysis particularly valuable in reliability engineering, maintenance planning, and failure analysis across industries from aerospace to manufacturing.

How to Use This Tool

  1. Enter Failure Data: In the first field, type the times at which failures occurred, separated by commas. These can be hours, cycles, kilometers, etc. You can also press "Load Example" to see a sample dataset.
  2. Analyze: Press the "Analyze Failures" button.
  3. Review Results:
    • Numerical Results: Review the calculated Beta (β), Eta (η), and Mean Time To Failure (MTTF). Interpret the Beta value to understand the failure mode of your system.
    • Weibull Probability Plot: This special chart visualizes your failure data. If the points form a reasonably straight line, it confirms that the Weibull distribution is a good fit for your data. The blue line is the best-fit regression line used to calculate the parameters.
    • Make a Prediction (Optional): Enter a specific time (in the same units as your failure data) into the prediction field and press "Analyze Failures" again to see the cumulative probability of failure by that time.
    • Add and Export: You can add multiple analyses to a list and then export all results to an Excel file for your reports.
Weibull Analysis Calculator

Once you click "Analyze Failures", you will be automatically redirected to the Results tab to see the chart and parameters.

Results

Shape Parameter (β): -

Scale Parameter / Characteristic Life (η): -

Mean Time To Failure (MTTF): -

Probability of failure by time -: -%

Sigma Exacta Watermark
Aggregated Results

Mathematical Formulas

Weibull Probability Density Function (PDF)

$$ f(t) = \frac{\beta}{\eta} \left( \frac{t}{\eta} \right)^{\beta-1} e^{-\left( \frac{t}{\eta} \right)^{\beta}} $$

Where $t$ is time, $\beta$ is the shape parameter, and $\eta$ is the scale parameter.

Cumulative Distribution Function (CDF)

$$ F(t) = 1 - e^{-\left( \frac{t}{\eta} \right)^{\beta}} $$

Probability of failure by time $t$.

Reliability Function

$$ R(t) = e^{-\left( \frac{t}{\eta} \right)^{\beta}} $$

Probability of survival beyond time $t$.

Mean Time To Failure (MTTF)

$$ MTTF = \eta \cdot \Gamma\left(1 + \frac{1}{\beta}\right) $$

Where $\Gamma$ is the Gamma function.

Failure Rate (Hazard Function)

$$ h(t) = \frac{\beta}{\eta} \left( \frac{t}{\eta} \right)^{\beta-1} $$

Instantaneous failure rate at time $t$.

Median Life

$$ t_{0.5} = \eta (\ln 2)^{1/\beta} $$

Time when 50% of units have failed.

Weibull Parameters Legend

  • β (Beta): Shape parameter - determines failure pattern
  • η (Eta): Scale parameter - characteristic life (63.2% failure point)
  • t: Time or cycles
  • Γ: Gamma function
  • f(t): Probability density at time t
  • F(t): Cumulative probability of failure by time t
  • R(t): Reliability (survival probability) at time t
  • h(t): Failure rate at time t