Asset owners will frequently have some failure information, but not a full history of asset-by-asset failures. If we know how many failures have occurred in the past few years it is possible to combine this information with the subjective judgment of the subject-matter experts to create a calibrated probability of failure curve.

**Example calibration of subject-matter experts' curve**

The Lawn King of Tacoma won the lottery and moved to Cayman. You purchased his lawn-mowing business, including all his mowers, one year ago. You had four mower failures in that year, and you are interested in estimating future failures. At the Tuesday afternoon meeting, you ask the mower-operators how long does a mower last? After much discussion the group estimates that half of all mowers fail by age seven, and that 90 percent make it past age four. Based on this, you select a Weibull curve with scale (beta) of eight and shape of three.

You have only one year of experience with the mowers - not enough to fit a failure curve rigorously. But how well does your experience match the subjective expertise of the mowers? To answer this, let's compare how many failures you had in the year you have owned the population of mowers with what would be expected based on the judgment of your operators. The figure below shows the age distribution of the mower population when you bought the business and the failure probability curve based on the information your operators gave you.

If we add up the failure probabilities for all ten mowers in the population, we will have the number of failures we should have expected based on the operators' curve.

The table shows that the expected number of failures was 2.7, whereas the actual number was four. You now have two options. First, you can do a statistical test to determine whether re-calibration is needed. In this case, the test suggests that there is no statistically significant difference between the predicted number of failures and the observed number (i.e., p=0.13), so you may decide to accept the experts' curve. However, you may still decide to adjust the curve.

The predicted number of failures (2.7) is lower than the actual (4), which suggests that the operators' curve may be based on too long a mean life. If we adjust the scale factor from eight to seven, then the expected number of failures goes to 4.1, which is very close to the actual. You could of course keep adjusting the scale factor to 7.039, which makes the expected number of failures exactly 4.0, but this level of precision does not seem warranted.

**Other comments**

*What about adjusting the shape factor?*Most of us do not have a good intuitive understanding of what the shape factor means. A high shape factor means that failures begin suddenly and the rate increases quickly. A lower factor means the failure probability is flatter with age. Lacking anything else to go on, our rule of thumb is to use a shape factor of 3 for most mechanical and electrical equipment. (This produces a failure probability curve that increases with the square of age.) Civil or structural assets may have smaller shape factors, say 2.5. And some electronic assets have higher shape factors.

*Why Weibull instead of normal?*The advantage of the Weibull is that its hazard rate is a simpler function. It is easier to calculate and it is better behaved at extreme ages. The advantage of the normal curve is that we all have a reasonable understanding of what its parameters, mean and standard deviation, mean. If you want to use a normal curve instead of a Weibull curve we're not going to talk you out of it.

*What about assets that can fail more than once?*Failures can be destructive (leading to replacement of the asset) or non-destructive (leading to repair. See destructive vs non-destructive failures for a more detailed explanation. In the case of non-destructive failures, we think of the mean time to failure as the mean time to*first*failure. Otherwise, the approach is the same. As always, it is important that the failure scenarios (sometimes referred to as failure modes) and failure probability curve be in coordination.