- Question it answers
- How long will it last, and how does the failure rate change with age?
- Data needed
- Failure times, plus suspension times for units that did not fail
- Key output
- Shape, scale, B10 life, reliability at a time, with intervals
- Shape
- Below 1 early failures; 1 random; above 1 wear-out
- Assumptions
- One failure mode; the Weibull model fits; independent units
- Excel
- WEIBULL.DIST, GAMMALN, SLOPE; Solver for censored fits
- Minitab
- Stat > Reliability/Survival > Parametric Distribution Analysis
- Why it matters
- It tells you whether to improve, screen, or replace
The Idea in Plain Language
Reliability is the probability that something still works at a given time. Life data have two features that ordinary statistics handle poorly: the times are skewed (nothing can fail before time zero), and the test usually ends before every unit has failed. Units still working when the test stops are censored: you know they lasted at least that long, and that is real information you must not throw away.
The Weibull distribution is the workhorse of life data because one flexible family can describe early failures, random failures, and wear-out. Two numbers do the work:
| Parameter | Symbol | Meaning |
|---|---|---|
| Shape | β | How the failure rate changes with age: below 1 it falls, at 1 it is constant, above 1 it rises |
| Scale (characteristic life) | η | The age by which 63.2% of units have failed, whatever the shape |
What the Shape Parameter Means
| Shape β | Failure rate | Typical cause | Action |
|---|---|---|---|
| Below 1 | Decreasing | Defects, infant mortality, poor assembly | Improve the process; burn-in or screening |
| About 1 | Constant | Random external events, electronics in useful life | Age-based replacement does not help; use condition monitoring or redundancy |
| Above 1 | Increasing | Wear, fatigue, corrosion | Planned replacement before wear-out |
| Above 4 | Rapidly increasing | Brittle, tightly controlled failure mode | Replace just before the characteristic life |
Useful quantities: reliability R(t) = exp[−(t/η)β]; the median life = η(ln 2)1/β; the B10 life (when 10% have failed) = η(−ln 0.9)1/β; the mean time to failure = η Γ(1 + 1/β).
Worked Example: A Life Test With Suspensions
12 bearings were run in a test stopped at 1,500 hours. 9 failed, at 236, 392, 470, 756, 762, 949, 1,288, 1,327, 1,330 hours. The other 3 were still running and are suspended (censored) at 1,500 hours.
Step 1: rank the failures, adjusting for the suspensions. Because suspended units were still alive, the failure ranks cannot simply be 1, 2, 3, …. Johnson’s adjusted rank handles it, and Benard’s formula converts it to a median rank:
| Order | Failure time (h) | Reverse rank | Increment | Adjusted rank | Median rank F |
|---|---|---|---|---|---|
| 1 | 236 | 12 | 1.000 | 1.000 | 5.6% |
| 2 | 392 | 11 | 1.000 | 2.000 | 13.7% |
| 3 | 470 | 10 | 1.000 | 3.000 | 21.8% |
| 4 | 756 | 9 | 1.000 | 4.000 | 29.8% |
| 5 | 762 | 8 | 1.000 | 5.000 | 37.9% |
| 6 | 949 | 7 | 1.000 | 6.000 | 46.0% |
| 7 | 1,288 | 6 | 1.000 | 7.000 | 54.0% |
| 8 | 1,327 | 5 | 1.000 | 8.000 | 62.1% |
| 9 | 1,330 | 4 | 1.000 | 9.000 | 70.2% |
Step 2: plot. On Weibull paper (log time, double-log probability), a Weibull sample falls on a straight line. Fitting the line gives the parameters: the slope is the shape and the time at 63.2% is the scale.
- Median rank regression (a line through the points) gives shape = 1.62, scale = 1,295 h, with correlation r = 0.988 on the plotted scale.
- Maximum likelihood (the usual software method; it uses all 12 units including the suspensions) gives shape 1.83 and scale 1,267 h.
- Mean time to failure = 1,267 × Γ(1 + 1/1.83) = 1,126 h. Median life = 1,037 h.
- B10 life = 1,267 × (−ln 0.9)1/1.83 = 371 h.
- Reliability at 500 h = exp[−(500/1,267)1.83] = 83.4%.
| Quantity | Estimate | 95% interval (parametric bootstrap) |
|---|---|---|
| Shape β | 1.83 | 1.11 to 3.72 |
| Scale η (h) | 1,267 | 867 to 1,874 |
| B10 life (h) | 371 | 168 to 742 |
| Mean life (h) | 1,126 | 781 to 1,734 |
Run It in Excel and Minitab
ExcelStep by step
- Probabilities: gives the fraction failed by x; reliability is (0.834). Without TRUE it gives the density.
- Mean life: (1,126).
- Percentile: ; for B10 use p = 0.1 (371).
- Fitting from complete data: compute median ranks , then regress on with and : the shape is the slope and the scale is exp(−intercept/shape).
- Censored data: Excel has no direct tool. Write the log-likelihood in a cell and maximize it with by changing the shape and scale, or use Minitab.
MinitabStep by step
- Put the times in one column and a censoring column in another (for example 1 = failed, 0 = suspended).
- . Enter the time column as Variable; under Censor, choose the censoring column and the value that marks a censored unit.
- Under Assumed distribution choose Weibull. Under Estimate choose Maximum Likelihood.
- Under Estimate you can ask for percentiles (such as 10) and reliability at specific times.
- Under Graphs select the probability plot, the survival plot, and the hazard plot.
- Which distribution? compares Weibull, lognormal, exponential, and others on probability plots.
Distribution Analysis: Hours Variable: Hours Censoring Column in Censor Censoring Value: 0 Estimation Method: Maximum Likelihood Distribution: Weibull Parameter Estimates Parameter Estimate Shape 1.8322 Scale 1,266.66 Log-Likelihood = -72.047 Characteristics of Distribution Mean(MTTF) 1,125.51 Median 1,037.01 Table of Percentiles Percent Percentile 10 370.89 50 1,037.01 Table of Survival Probabilities (time = 500) Time Probability 500 0.83350 (Minitab also reports standard errors and confidence intervals; they are omitted here.)
Reading and Reporting
- Check the fit first: the points should follow the line on the probability plot. Curvature suggests another distribution or a mixture of failure modes.
- Report the shape with its interval, and say what it implies about the failure mechanism.
- Give a decision-ready number: B10 life, or the reliability at the mission time, with an interval.
- State how many failures and suspensions the estimate rests on. Fewer than about 10 failures gives very wide intervals.
- Analyze one failure mode at a time. Mixing modes bends the plot and hides the shape.
Common Mistakes
| Mistake | Why it misleads | Better |
|---|---|---|
| Dropping the suspended units | Throws away real information and biases life estimates low | Include them as censored |
| Treating suspensions as failures | Makes the product look worse than it is | Mark them censored |
| Mean life as the only summary | A wear-out product can have a long mean and still fail before it | Use B10 or reliability at a mission time |
| Combining different failure modes | The plot bends; the shape is meaningless | Analyze each mode separately |
| Trusting a fit from three or four failures | The intervals are enormous | Plan the test for enough failures, or report the interval |
| Using the normal distribution for life data | Gives impossible negative times and wrong tails | Use Weibull or lognormal |
Try It Yourself
A component has a Weibull shape of 2.5 and a scale of 8,000 hours.
- What fraction fail by 8,000 hours?
- What is the reliability at 4,000 hours?
- Is the failure rate rising or falling?
Show the answer
By the scale, 63.2% have failed (1 − e−1 = 0.632), whatever the shape.
R(4,000) = exp[−(4,000/8,000)2.5] = exp[−0.1768] = 0.838.
The shape is above 1, so the failure rate rises with age: the component wears out, and planned replacement is worthwhile.
Reliability and Weibull Analysis: Frequently Asked Questions
What does the Weibull shape parameter tell me?
It tells you how the failure rate changes with age. Below 1 the failure rate falls (early failures), at 1 it is constant (random failures), and above 1 it rises (wear-out).
What is censored data?
Units that had not failed when the test ended, or that were removed for another reason. Their lifetimes are at least as long as the time observed, and including them as censored avoids a serious bias.
What is the B10 life?
The age by which 10% of units are expected to have failed. It is widely used for bearings and similar parts, and it is more useful than the average for wear-out products.
When should I use lognormal instead of Weibull?
Both fit many life data sets. Compare their probability plots and fit statistics, and use engineering knowledge about the failure mechanism. Lognormal is common for fatigue and some electronics wear.
How many failures do I need?
For a rough estimate of the shape, ten or more failures. With five or fewer the intervals are extremely wide. Tests often are designed to run to a target number of failures.
Is reliability the same as MTBF?
No. MTBF is an average. For a constant failure rate (shape 1) reliability at time t is exp(−t/MTBF), but for wear-out products the average badly overstates the early life.
Sources and Further Reading
- Robert B. Abernethy, The New Weibull Handbook, 5th ed.
- William Q. Meeker and Luis A. Escobar, Statistical Methods for Reliability Data, Wiley.
- NIST/SEMATECH, e-Handbook of Statistical Methods, Reliability (itl.nist.gov/div898/handbook).
- Leonard G. Johnson, The Statistical Treatment of Fatigue Experiments, Elsevier, 1964 (adjusted ranks).
- Minitab Support, “Methods and formulas for Parametric Distribution Analysis” (support.minitab.com).
This content is educational. Worked examples use made-up data. Menu names for Minitab follow recent versions of Minitab Statistical Software and can differ slightly in older releases; Excel steps use Microsoft 365 and the Analysis ToolPak.