Accendo Reliability

Your Reliability Engineering Professional Development Site

  • Home
  • About
    • Contributors
    • About Us
    • Colophon
    • Survey
  • Reliability.fm
  • Articles
    • CRE Preparation Notes
    • NoMTBF
    • on Leadership & Career
      • Advanced Engineering Culture
      • ASQR&R
      • 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 Maintenance Management
      • 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
      • RCM Blitz®
      • ReliabilityXperience
      • 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
      • Field Reliability Data Analysis
      • Metals Engineering and Product Reliability
      • Musings on Reliability and Maintenance Topics
      • Product Validation
      • Reliability by Design
      • Reliability Competence
      • Reliability Engineering Insights
      • Reliability in Emerging Technology
      • Reliability Knowledge
    • 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
      • Institute of Quality & Reliability
      • Integral Concepts
      • Learning from Failures
      • Progress in Field Reliability?
      • R for Engineering
      • Reliability Engineering Using Python
      • Reliability Reflections
      • Statistical Methods for Failure-Time Data
      • Testing 1 2 3
      • The Manufacturing Academy
  • eBooks
  • Resources
    • Accendo Authors
    • FMEA Resources
    • Glossary
    • Feed Forward Publications
    • Openings
    • Books
    • Webinar Sources
    • Podcasts
  • Courses
    • Your Courses
    • Live Courses
      • Introduction to Reliability Engineering & Accelerated Testings Course Landing Page
      • Advanced Accelerated Testing Course Landing Page
    • Integral Concepts Courses
      • Reliability Analysis Methods Course Landing Page
      • Applied Reliability Analysis Course Landing Page
      • Statistics, Hypothesis Testing, & Regression Modeling Course Landing Page
      • Measurement System Assessment Course Landing Page
      • SPC & Process Capability Course Landing Page
      • Design of Experiments Course Landing Page
    • The Manufacturing Academy Courses
      • An Introduction to Reliability Engineering
      • Reliability Engineering Statistics
      • An Introduction to Quality Engineering
      • Quality Engineering Statistics
      • FMEA in Practice
      • Process Capability Analysis course
      • Root Cause Analysis and the 8D Corrective Action Process course
      • Return on Investment online course
    • Industrial Metallurgist Courses
    • FMEA courses Powered by The Luminous Group
    • Foundations of RCM online course
    • Reliability Engineering for Heavy Industry
    • How to be an Online Student
    • Quondam Courses
  • Calendar
    • Call for Papers Listing
    • Upcoming Webinars
    • Webinar Calendar
  • Login
    • Member Home
  • Barringer Process Reliability Introduction Course Landing Page
  • Upcoming Live Events
You are here: Home / Articles / Equipment Failure Probability Density Functions

by Mike Sondalini Leave a Comment

Equipment Failure Probability Density Functions

Equipment Failure Probability Density Functions

Charting historic equipment failure events visually shows the Failure Probability Density Function curve for that equipment. It is also known as a Failure Density Distribution Curve.

VALUE OF A DISTRIBUTION CURVE

Failure Probability Density Function curves, Failure Density Distribution Curves contain useful information about what has happened to equipment. These curves illustrate the chance of failure of an equipment over a period. Here, our fixed variables are the equipment’s components’ material-of-construction and design. As such, the only changing variable is the operating and maintenance strategies, including work processes, used by the company.

This means that when looking at a Failure Probability Density Function graph it is possible to identify the success of the operating and maintenance procedures in use. Below is an example activity used in the Plant Wellness Way EAM Training Courses that illustrates how to use these graphs to identify and improve machinery reliability.

PROCEDURE AS A VARIABLE

The activity has two parts. Firstly, the participants to break a paperclip in any way they wish. This creates a situation where random stress events occur. Each person chooses how their paperclip fails – be it by bending, twisting, or a combination of those two actions. Participants count the cycles to failure, and this information is plotted on the graph (Figure 1). The spread of point forms a sample Failure Probability Density Function curve of a paper clip.

As the material-of-construction and design of the paper clip are know and consistent across this example, they are not the variables. This means that the only variable in the activity is the way people broke their paperclips. The individual procedures used by the participants are what produced this Failure Probability Density Function curve.

This is a hugely important understanding in equipment reliability improvement: the procedure used is a variable. That is a foundational insight into the Plant Wellness Way EAM methodology. 

dot plot showing spread of cycles to failure of paper clips - from 4 cycles to 32 cycles
Figure 1: Sample activity instructions and the Failure Probability Density Function Curve for breaking a paperclip with no standard instructions.

STANDARD OPERATING PROCEDURES (SOPS) FOR CONTROLLING QUALITY OF WORK OUTCOMES

For the second part of this activity, our participants are asked to, again, break a paperclip but this time they must follow specific instructions about how to break the paperclip. In this case, the paper clip is placed on the corner of a table and cycled from zero to 180° opposite position until it fails. Again, the cycles to failure at plotted and create a new Failure Probably Density Function curve.

paperclip bending standard operating procedure
Figure 2: Standard Operating Procedure instructions for how to break a paperclip.

dot plot of cycles to failure with less variability, ranging from 7 to 24 cycles
Figure 3: Sample activity instructions and the Failure Probability Density Function Curve for breaking a paperclip with standard instructions

When the new graph, Figure 3, is compared to the previous (Figure 1), the influence of standard operating procedures on the shape of the curve is obvious. The shape of the failure density distribution curve is now narrowed. The randomness of failure counts caused by the various procedures’ participants used is removed. By using a standard operating procedure, the outcomes are controlled to within a tight performance range (11 to 18 cycles to failure).

By controlling the method used to cause this failure, we can predict, within 90% confidence, when a paper clip will fail. If this paperclip were a component, then we could plan preventative maintenance to replace the part before it fails. The graph also allows us to identify outliers – which indicates that there were two occurrences where the SOP was not followed correctly.

One person had only 7 cycles to failure. Had they done the SOP properly they would have produced a failure event somewhere between 11 to 18 cycles. In this instance, the person who produced this result should be retrained in the procedure. Once retrained they should complete a series of trial runs to demonstrate that they now are able to produce results fall within the required distribution.

Another person experienced 23 cycles to failure. While they did not follow the procedure, their approach generated far greater reliability than all the other outcomes. The aim of Standard Operating Procedures should be to both control the quality of work and deliver the greatest reliability. As this person achieved the greatest reliability, it necessary to interview and watch this person to learn how they reached such high reliability.

The information collected from the interview and work observation is used to revise the SOP. This new Standard Operating Procedure will become the company-wide specification. With every revision or creation of a new SOP, everyone is trained in the new procedure and tested to ensure they can perform to the new standard. By doing this process every user will get higher reliability, within a known range of results.

Standard Operating Procedures allow you to control the distribution of task quality, producing repeatable, known results and guaranteed job quality. They should be updated with the latest best practices and users trained to ensure that everyone is able to perform the task with consistency.

Filed Under: Articles, Life Cycle Asset Management, on Maintenance Reliability

About Mike Sondalini

In engineering and maintenance since 1974, Mike’s career extends across original equipment manufacturing, beverage processing and packaging, steel fabrication, chemical processing and manufacturing, quality management, project management, enterprise asset management, plant and equipment maintenance, and maintenance training. His specialty is helping companies build highly effective operational risk management processes, develop enterprise asset management systems for ultra-high reliable assets, and instil the precision maintenance skills needed for world class equipment reliability.

« The Two Types of Agility You Need
How to Define Proper Product Reliability Goal »

Leave a Reply Cancel reply

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

Headshot of Mike SondaliniArticles by Mike Sondalini
in the Life Cycle Asset Management article series

Join Accendo

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

It’s free and only takes a minute.

Join Today

Recent Posts

  • Gremlins today
  • The Power of Vision in Leadership and Organizational Success
  • 3 Types of MTBF Stories
  • ALT: An in Depth Description
  • Project Email Economics

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