What Is Weibull Analysis and How Does It Predict Product Life?

Weibull analysis is one of the most powerful statistical tools in reliability engineering. It lets you model failure data, estimate product life, and predict future failures—even with small sample sizes. If you have ever asked, “When will this component fail?” or “What is the B10 life of this product?”, Weibull analysis gives you a practical, data-driven answer.

What It Is

The Weibull distribution, introduced by Waloddi Weibull in 1951, is a flexible probability distribution used to describe life data (time-to-failure). It is widely applied in reliability engineering, warranty analysis, and maintenance planning.

Its popularity comes from its versatility: by adjusting two parameters, it can model early-life failures (decreasing failure rate), random failures (constant failure rate), and wear-out failures (increasing failure rate).

How It Works / Formula or Steps

The two-parameter Weibull cumulative distribution function (CDF) is:

\[
F(t) = 1 - \exp\left(-\left(\frac{t}{\eta}\right)^\beta\right)
\]

And the reliability function is:

\[
R(t) = \exp\left(-\left(\frac{t}{\eta}\right)^\beta\right)
\]

Where:

• \( t \) = time to failure
• \( \beta \) = shape parameter (slope). If \( \beta < 1 \), failure rate decreases over time (infant mortality). If \( \beta = 1 \), failure rate is constant (random failures). If \( \beta > 1 \), failure rate increases (wear-out).
• \( \eta \) = scale parameter (characteristic life). This is the time at which 63.2% of the population has failed.

Key derived metrics:

• MTTF (Mean Time To Failure): \( \eta \cdot \Gamma(1 + 1/\beta) \), where \( \Gamma \) is the gamma function.
• B10 life: The time by which 10% of the population is expected to fail. Solve \( F(t) = 0.10 \).
• Reliability at time t: directly from \( R(t) \).

Steps to perform Weibull analysis:

1. Collect failure times (and censored data—units that have not failed).
2. Rank the failure data (e.g., median rank method).
3. Plot the data on Weibull probability paper or use software to fit the distribution.
4. Estimate \( \beta \) and \( \eta \) (typically via maximum likelihood estimation or least squares).
5. Validate the fit, then calculate MTTF, B10, and reliability predictions.

A Worked Illustrative Example

Example data (illustrative only): Suppose you test 10 bearings and record the following failure times in hours: 120, 180, 240, 310, 380, 460, 550, 660, 800, 950.

Using standard Weibull fitting software, you obtain:

• Shape parameter \( \beta = 2.1 \)
• Scale parameter \( \eta = 620 \) hours

Now calculate:

• Reliability at 500 hours:

\[
R(500) = \exp\left(-\left(\frac{500}{620}\right)^{2.1}\right) \approx \exp(-0.634) \approx 0.53
\]
So about 53% of bearings are expected to survive past 500 hours.

• B10 life (time at which 10% fail):

Set \( F(t) = 0.10 \):
\[
0.10 = 1 - \exp\left(-\left(\frac{t}{620}\right)^{2.1}\right)
\]
Solving gives \( t \approx 220 \) hours. So 10% of bearings are expected to fail by 220 hours.

• MTTF:

\[
MTTF = 620 \cdot \Gamma(1 + 1/2.1) \approx 620 \cdot 0.886 \approx 549 \text{ hours}
\]

These numbers show how Weibull analysis turns raw failure times into actionable life predictions.

Common Pitfalls

• Ignoring censored data: Units that have not failed still carry information. Excluding them biases the estimates.
• Small samples: With fewer than 5 failures, parameter estimates are unstable. Use caution and confidence bounds.
• Forcing a 2-parameter fit: If your data shows a threshold (no failures before a certain time), consider the 3-parameter Weibull.
• Overlooking goodness-of-fit: Always check whether the Weibull model actually fits your data before trusting its predictions.

---

Weibull analysis turns messy failure data into clear predictions about product life, failure modes, and maintenance needs. To apply it quickly to your own data, try the free interactive Weibull analysis tool at 6SQ Weibull Analysis.