Accendo Reliability

Your Reliability Engineering Professional Development Site

  • Home
  • About
    • Contributors
  • Reliability.fm
    • Speaking Of Reliability
    • Rooted in Reliability: The Plant Performance Podcast
    • Quality during Design
    • Way of the Quality Warrior
    • Critical Talks
    • Dare to Know
    • Maintenance Disrupted
    • Metal Conversations
    • The Leadership Connection
    • Practical Reliability Podcast
    • Reliability Matters
    • Reliability it Matters
    • Maintenance Mavericks Podcast
    • Women in Maintenance
    • Accendo Reliability Webinar Series
  • Articles
    • CRE Preparation Notes
    • on Leadership & Career
      • Advanced Engineering Culture
      • Engineering Leadership
      • Managing in the 2000s
      • Product Development and Process Improvement
    • on Maintenance Reliability
      • Aasan Asset Management
      • AI & Predictive Maintenance
      • Asset Management in the Mining Industry
      • CMMS and Reliability
      • Conscious Asset
      • EAM & CMMS
      • Everyday RCM
      • History of Maintenance Management
      • Life Cycle Asset Management
      • Maintenance and Reliability
      • Maintenance Management
      • Plant Maintenance
      • Process Plant Reliability Engineering
      • ReliabilityXperience
      • RCM Blitz®
      • Rob’s Reliability Project
      • The Intelligent Transformer Blog
      • The People Side of Maintenance
      • The Reliability Mindset
    • on Product Reliability
      • Accelerated Reliability
      • Achieving the Benefits of Reliability
      • Apex Ridge
      • Metals Engineering and Product Reliability
      • Musings on Reliability and Maintenance Topics
      • Product Validation
      • Reliability Engineering Insights
      • Reliability in Emerging Technology
    • on Risk & Safety
      • CERM® Risk Insights
      • Equipment Risk and Reliability in Downhole Applications
      • Operational Risk Process Safety
    • on Systems Thinking
      • Communicating with FINESSE
      • The RCA
    • on Tools & Techniques
      • Big Data & Analytics
      • Experimental Design for NPD
      • Innovative Thinking in Reliability and Durability
      • Inside and Beyond HALT
      • Inside FMEA
      • Integral Concepts
      • Learning from Failures
      • Progress in Field Reliability?
      • R for Engineering
      • Reliability Engineering Using Python
      • Reliability Reflections
      • Testing 1 2 3
      • The Manufacturing Academy
  • eBooks
  • Resources
    • Accendo Authors
    • FMEA Resources
    • Feed Forward Publications
    • Openings
    • Books
    • Webinars
    • Journals
    • Higher Education
    • Podcasts
  • Courses
    • 14 Ways to Acquire Reliability Engineering Knowledge
    • Reliability Analysis Methods online course
    • Measurement System Assessment
    • SPC-Process Capability Course
    • Design of Experiments
    • Foundations of RCM online course
    • Quality during Design Journey
    • Reliability Engineering Statistics
    • Quality Engineering Statistics
    • An Introduction to Reliability Engineering
    • Reliability Engineering for Heavy Industry
    • An Introduction to Quality Engineering
    • Process Capability Analysis course
    • Root Cause Analysis and the 8D Corrective Action Process course
    • Return on Investment online course
    • CRE Preparation Online Course
    • Quondam Courses
  • Webinars
    • Upcoming Live Events
  • Calendar
    • Call for Papers Listing
    • Upcoming Webinars
    • Webinar Calendar
  • Login
    • Member Home

by nomtbf Leave a Comment

Lower Confidence

Let’s say we have a population and we are interested in the mean (average) of that population’s life. We select a sample (at random if at all possible) and measure a value, like time to failure, for each selected item in the sample.

We calculate the mean life of the sample by summing the sample values and dividing by the number of items in the sample.

Because we are only using a subset of the population it is possible the sample items are from one part of the population, say the tall part only. It may not be likely, yet it is possible to have selected samples that do  not represent the range of values in the population.

It is this possibility that the sample statistic expected to represent the population parameter   doesn’t actually even come close is the notion of statistical confidence. In a positive manner, we say there is a 95% confidence that the true unknown population parameter falls within a range of values, also called the confidence interval or bounds. That means there is a 5% chance that the actual and unknown population parameter is outside that range. In other words we are 95% confident that the sample is ‘this’ close to the actual value.

For MTBF confidence intervals we are often only interested in the lower limit (one-sided). We expect with said confidence that the true unknown value is above this lower confidence value, with Type II – predetermined number of failures terminates the test.

$latex \displaystyle \theta \ge \frac{2T}{\chi _{\left( \alpha ,2r \right)}^{2}}$

Where,

  • θ is the calculated mean life (MTBF)
  • T is the total time the samples operated before failing (or the test was ended)
  • χ2 is the Chi-squared distribution
  • α is the level of risk (1 – confidence)
  • r is the number of failures, 2r is then the degrees of freedom for the chi-squared

Now an example. Given the MTBF for a test with 2 failures is 1525 hours. The total time, T, is 3050 hours and there were 2 failures, r. Calculate the 90% lower confidence interval for the estimated MTBF.

$latex \displaystyle \theta \ge \frac{2(3050)}{7.779}=784\text{hours}$

This means there is a 90% chance that the true and unknown population MTBF is greater than 784 hours. And, there is a 10% chance that it is less. Unless we determine the population mean (measure ever unit in the population) we won’t know.

For fun, consider we are willing to take more risk of the sample not representing the population. The same sample, just change the confidence. Let’s go from 90% to 60% and we find

$latex \displaystyle \theta \ge \frac{2(3050)}{4.045}=1,508\text{hours}$

Which has a higher value. Interesting. Same data, more risk, smaller confidence range. In other words by accepting more risk, we are saying there is now a greater chance that the true unknown parameter falls outside the range described by the confidence interval. The true value doesn’t change, the sample statistic doesn’t change. And, we’re saying the lower confidence value is higher than before.

Without careful consideration it appears the population lasts twice as long, 784 to 1,508 hours. Nothing actually has changed, just the increase in risk that the sample represents the true value.

Check out the next datasheet that crosses your desk – too many use a 60% confidence – why is that?

Filed Under: Uncategorized

« Statistical Confidence
Mann Reverse Arrangement Test »

Comments

  1. nm says

    May 8, 2017 at 12:28 AM

    thank you for the useful artcile. I’m working in reliability and failure analysis field. as you mentioned, usually in datasheets they use 60% of confidence level. would you please answer the question “too many use a 60% confidence – why is that?” and should I use the data related to 60% of CL or 90% for example for sensitive projects? thank you

    Reply
    • Fred Schenkelberg says

      May 8, 2017 at 7:19 AM

      Hi NM,

      Keep in mind that the confidence interval is an indication the result reported is from a sample used to represent the population. We often do not have the information such as sample size or testing conditions to properly judge the values.

      A single test with a fixed sample size will report different value at different confidence levels. The 60% level will appear ‘better’ then a 90% level yet the underlying data hasn’t changed. The 60% level shifts more of the risk to you, as they are saying there is a 40% chance the actual product will not perform within the stated range.

      If the item under consideration is important to the performance of your product, do not accept vendor data that is poorly reported. Get the details or do the testing/characterization yourself. You need to understand how the item will fail and how the testing evaluates that failure mechanism.

      Cheers,

      Fred

      Reply
  2. SJ says

    August 3, 2017 at 6:38 AM

    Hi Fred
    How would you go about interpreting exactly what the following statement means
    “Reliability required = 98% to 95% confidence of mission success”
    The only other data I can see given is that
    “Availability required = 95% based on 10 days operation in 2 month deployment”
    and
    “Assume 500 running hours per year” over a 15 year life
    Thanks in advance

    Reply
    • Fred Schenkelberg says

      August 3, 2017 at 9:39 PM

      Not a whole lot to go on here. If someone want a 98% chance of surviving they really should tell us over what duration. This may be very easy to accomplish over an hour and nearly impossible over 20 years.

      The 95% confidence is only useful for planning for testing… it is not part of the goal. The goal is for the population and what chance of success you, they, really want to have happen.

      For test planning and in particular sample size selection we need to know things like confidence. This helps us know what risk of the sample not representing the population we’re willing to accept.

      If you change the confidence you may need more or fewer samples yet the actual reliability does’t change.

      What is the mission duration, environment, set of stresses, profile of use/stress, etc.

      500 hrs per year over 15 years is great, is the mission time?

      During a mission can we have failures and quick repairs, or is that not an option?

      The little information provides means you could pretty much provide any product or system you want at any reliability level, and you’re probably good (or could look good under specific circumstances.)

      I would be asking a lot more questions.

      Cheers,

      Fred

      Reply
      • SJ says

        August 7, 2017 at 3:37 AM

        Thanks Fred I suspected something like that. For mission I think we take 10 days in 2 months so mission time = approx 85 hours. During mission no repairs.

        Reply

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

[popup type="" link_text="Get Weekly Email Updates" link_class="button" ]

[/popup]

The Accendo Reliablity logo of a sun face in circuit

Please login to have full access.




Lost Password? Click here to have it emailed to you.

Not already a member? It's free and takes only a moment to create an account with your email only.

Join

Your membership brings you all these free resources:

  • Live, monthly reliability webinars & recordings
  • eBooks: Finding Value and Reliability Maturity
  • How To articles & insights
  • Podcasts & additional information within podcast show notes
  • Podcast suggestion box to send us a question or topic for a future episode
  • Course (some with a fee)
  • Largest reliability events calendar
  • Course on a range of topics - coming soon
  • Master reliability classes - coming soon
  • Basic tutorial articles - coming soon
  • With more in the works just for members
Speaking of Reliability podcast logo

Subscribe and enjoy every episode

RSS
iTunes
Stitcher

Join Accendo

Receive information and updates about podcasts and many other resources offered by Accendo Reliability by becoming a member.

It’s free and only takes a minute.

Join Today

Dare to Know podcast logo

Subscribe and enjoy every episode

RSS
iTunes
Stitcher

Join Accendo

Receive information and updates about podcasts and many other resources offered by Accendo Reliability by becoming a member.

It’s free and only takes a minute.

Join Today

Accendo Reliability Webinar Series podcast logo

Subscribe and enjoy every episode

RSS
iTunes
Stitcher

Join Accendo

Receive information and updates about podcasts and many other resources offered by Accendo Reliability by becoming a member.

It’s free and only takes a minute.

Join Today

Recent Articles

  • test
  • test
  • test
  • Your Most Important Business Equation
  • Your Suppliers Can Be a Risk to Your Project

© 2025 FMS Reliability · Privacy Policy · Terms of Service · Cookies Policy