Calculating reliability from data

A short example on calculating reliability from data. And, a comparison to the calculation assuming the failure rate is constant.

In the last note, we calculated MTBF using some test data. Now let’s start with the same situation and calculate reliability instead. As before: There are occasions when we have either field or test data that includes the duration of operation and whether or not the unit failed.

This can be, say, 10 large motors. For sake of argument, the test ran each motor for 1,000 hours and when a motor failed it was repaired quickly and returned to the test.

There were 3 failures. And, if that is all the information we have available, then at most we can calculate MTBF as in the previous note.

Being a professional, we ask a few questions, dig into the test data and attempt to learn a little more about the three failures. If three motors failed within an hour of starting and nothing failed afterwards, we may have an issue with early life failures. If the three failures were all near the ned of the 1,000 hours of operation, we might have a wear out issue.

Knowing the time to failure would help us sort out what kind of issue(s) may be causing the motor failures.

Also, product testing is rarely done in isolation. How were the motors expected to fail? How  were the motors operated, under load, power cycling, elevated temperature, etc? How were the motors monitored? What readings were made to determine if the motors were operating as expected? Plenty of questions and most likely there are answers related to any product test and often associated with customer use.

Let’s say for the sake of argument we find the motors have an expected annual operating time of 1,000 hours. The test was done at nominal conditions with an 80% constant load (again a nominal use condition.)

The test did not cycle the load whereas in use the motors would operate about 10% of the time. The test also did not cycle the power or load as would occur when the motor starts or stops during normal operation.

We also find out that primary concern was the insulation on the windings in the motor would decay over time leading to a loss of efficiency of the motor. The testing measured motor efficiency daily and deemed a motor as failed when it fell by 10% of initial efficiency.

The three failures were three different motors at 890 hours, 951 hours, and 973 hours.

The failures were only due to loss of efficiency and the motors were repaired (restored to near original efficiency) and returned to the test to complete the 1,000 hours. The remaining 7 motors operated within specifications for 1,000 each.

The repaired motors are pretty much as good as new.

A first pass estimate at the reliability needs one other piece of information. When (duration) do we want to know the probability of success (reliability) of the motors? 1,000 hours seems reasonable and may coincide with warranty terms or customer expectations. Since 7 of the 10 units operated without failure for at least 1,000 hours, a first pass estimate of 70% reliable at 1,000 hours seems reasonable.

This roughly based on binomial distribution properties and one could use the binomial distribution to calculate a lower confidence bound. This is rough and maybe good enough for the decisions at hand.

Since the windings are replaced when this type of failure occurs, we could model the winding failure rate using a Weibull analysis and treat the repaired units as new.  We then include the three ‘new’ units as suspended with 110, 49 and 27 hours. Using Weibull++ from Reliasoft, I enter 7 suspended at 1000 hours, the times to failure and the three ‘new’ motors and their respective suspended (didn’t fail yet) times.

A quick fit with a two parameter Weibull results in a beta of 15.4 and eta of 1055. This means the reliability based on this data is estimated at 64.7% will survive 1,000 hours and the lower 90% confidence bound is 42.6% reliable.

The rough guess of 70% is pretty good after all.

The very high beta is concerning though. If the failure analysis reveals the failure are due to the expected failure mechanism, I would want to check the literature to see if our testing method or something else caused such a high beta value.

It’s possible, and it’s worth checking. Comparing the result to the MTBF value of 3,333 hours which suggests 74% would survive 1,000 hours. 64% vs. 74% may not make a difference with your decision, or it might.

Be conscious of how to determine reliability from data and ask the questions to challenge any assumptions being made. What has been your experience? How do you estimate product reliability? How’s it working for you?

Author: Fred Schenkelberg

I am an experienced reliability engineering and management consultant with my firm FMS Reliability. My passion is working with teams to create cost-effective reliability programs that solve problems, create durable and reliable products, increase customer satisfaction, and reduce warranty costs.

4 thoughts on “Calculating reliability from data”

    1. I used Reliasoft’s Weibull++ software which fit the curve for a Weibull distribution to the data. There are many methods for the regression and handling censored data, in this case with just a few failures, one should use the maximum likelihood estimator method.



  1. Dear Fred.
    I am a mechanical engineer and am interested in pursuing my mtech in reliability engineering. I have been regularly going through your posts and must say have been really helpful. I would like to know about curve fitting and estimating parameters from it. I am a beginner and would like to learn more on it. Hope you would help.


    1. Hi Paresh,

      Thanks for the kind words, that is much appreciated.

      To learn regression analysis will take a bit more than an article or two here. I would recommend taking a course on data analysis that includes regression. Also, a good place to read about the basics is the NIST Engineering Statistics Handbook. One entry is on linear least squares regression and another is a tooled to develop a regression model,

      It’s a good set of skills to have as an engineer.



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