Basic QA Statistics Series(Part 4)- Interquartile Range-IQR
REFLECTION: FOR STUDENTS: A good rule in organizational analysis is that no meeting of the minds is really reached until we talk of specific actions or decisions. We can talk of who is responsible for budgets, or inventory, or quality, but little is settled. It is only when we get down to the action words-measure, compute, prepare, check, endorse, recommend, approve-that we can make clear who is to do what. -Joseph M. Juran
FOR ACADEMICS: Without a standard there is no logical basis for making a decision or taking action. -Joseph M. Juran
FOR PROFESSIONALS/PRACTITIONERS: Both pure and applied science have gradually pushed further and further the requirements for accuracy and precision. However, applied science, particularly in the mass production of interchangeable parts, is even more exacting than pure science in certain matters of accuracy and precision. -Walter A. Shewhart
Foundation
When we left this small series on basic QA statistics, we had just discussed basic measures of Dispersion- Range, Variance, and Standard Deviation. As promised, we are now covering the basics of Interquartile Range (IQR for short). IQR is also a measure of dispersion, but as I’m sure you will be exposed to IQR in the future, I thought it best to give it a separate post.
The IQR range, like the other measures of dispersion, is used to measure the spread of the data points in a data set. IQR is best used with different measurements like median and total range to build a complete picture of a data set’s tendency to cluster around its mean. IQR is also a very useful tool to use to identify outliers (values abnormally far from the mean of a data set), but do not worry about the more in-depth math.
First, to Define all of the aspects of IQR
-First Quartile (Q1)- The value at which 25% of the data are less than or equal to this value (does not have to be a value in the data set).
-Second Quartile (Q2)- The value at which 50% of the data are less than or equal to this value. It is also known as the median. The second quartile or median does not have to be a value in the data set.
-Third Quartile (Q3)- This is the point at which 75% of the data are less than or equal to this value. It also does not have to be in the data set.
-Fourth Quartile (Q4)- This value is the maximum value in the data set (100% of the data are less than or equal to this value).
-Interquartile Range (IQR)- IQR is the Third Quartile minus the First Quartile and considered a measure of dispersion.
(Kubiak, 2017)
Calculating Quartiles
There are several methods for calculating quartiles, so the technique I am going to use is just what I consider the most basic without delving into any more in-depth math.
Steps:
- Order the data set from smallest to largest.
- Determine the median (reference my post: Basic QA Statistics Series(Part 2)- Basic Measures of Central Tendency and Measurement Scales).
- This determination separates the data into two sets (an upper half and lower half). This Median is Q2
- The First Quartile (Q1) is found by determining the median of the lower half of the data (not including the Median from the previous step when calculating the lower half data set median).
- Q3 is the median of the upper half of the data set, not including the value for Q1 in the top half median determination
- Q4 is the maximum in the data set.
(Kubiak, 2017)
Data Set: 22,26,24,29,25,24, 23,26,28,30,35,40,56,56,65,57,57,75,76,77,74,74,76,75,72,71,70,79,78, 1000,10,12,13,15,16,12,11,64, 65,35, 25,28, 21,44,46,55,77, 79,85,84,86,15,25,35, 101,12,25,35,65,75
Conclusion
As you can see, I stacked the data deck with a massive outlier in the data set. 1000 is far from the mean, but the IQR is not affected by this enormous outlier, as it only takes into account Q1 and Q3.
This property of IQR helps prevent outliers from convincing you the mean is just fine, when in fact, the entire system may be out of whack but compensated for by outliers in your data. The little chart you see is called a Box and Whisker plot, and we will give it a separate post later after we discuss Histograms in the nest post.
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Bibliography
Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Contingency Planning- What Has Covid-19 Taught Us?
REFLECTION: FOR STUDENTS: Would you choose a college that could lose all service to you at any time? In the future, customers will be much more aware of supply chain impact on the customer, and past failures of supply chain maintenance.
FOR ACADEMICS: Covid-19 probably upended a lot of traditional models, though some may have been ahead of the game. Moving “Virtual Communication” from a side note to a centerpiece of future educational models will help traditional students integrate better, and understand virtual communication is not a fad, but an essential tool.
FOR PROFESSIONALS/PRACTITIONERS: Supply Chain and Management sometimes seek the cheapest vendors, to the point of shuttering internal production capability. Quality, as part of the APQP process should always present the risks of cheaper vendors and demand all possible failure modes have been mitigated before accepting the cheapest vendor. Supply chain should try to avoid committing to just one vendor because if you do not have the option to drop a vendor, the vendor holds the negotiation power.
Contingency Planning
ISO 9001-2015 8.2.1 e) includes the statement required: “establishing specific requirements for contingency actions, when relevant.”
A Contingency Plan is by definition, simple to craft. Yet effectively implementing a standing contingency plan requires you to have confidence the plan will be simple to follow and efficient. A contingency is simply a plan of action for when something goes wrong.
The contingency plan can be on a catastrophic scale like “what if a tornado hits us and disables production,” or a more common contingency plans can scale down and address individual machines, “what machine takes up the slack if machine x needs unscheduled maintenance?”. For maximum protection of your business, suppliers, and customers, all possible risks should be accounted for regardless of the probability. If the plan of action is in place, time is saved in deciding the next course of action.
Many ISO 9001-2015 certified companies have been certified for several years, yet only a handful of those companies had Pandemic Contingency Plans in place to adequately protect the supply chain. Businesses that had few or zero domestic suppliers/potential suppliers for essential goods were thinking along short term financial lines. Businesses concerned for continuity were concerned about their customers.
If every potential supplier of essential items for a pandemic had a robust and effective contingency plan in place, planning for a worst-case pandemic, the supply chain strain would have been much less severe. Hopefully, one of the lessons consistent “cut the costs” mentality bureaucrats will learn from the result of not planning for a risk (even when that risk is highly unlikely) is that having the mechanisms in place to save a business is worth the investment and the maintenance costs.
One of my old bosses made a contingency plan with all the usual disaster responses planned out. There was an unusual one on the list as well- Though it was done in jest, he provided a contingency for Alien Invasion. Highly unlikely…yet he always got a laugh at audits, but as a Marine, he always looked for the most unlikely risks and prepared for them, THANKS TOM 🙂 !
Conclusion
Creation and implementation of Covid-19 type contingency efforts for businesses will NOW likely start robustly, then as those efforts become bothersome will stagnate and fall away over time and be declared low risk and not needed because they are unlikely to occur. Please take a real-life lesson learned to the boardroom if anyone should try to say setting up a robust contingency plan is too expensive or takes too much time: Lesson: pandemic planning is necessary and it will occur.
What allows contingency planning to have any real meaning is not about getting the certification by having the plan on paper, but rather, about having an effective system in place the company can verify as capable of responding to the stressors on the business and continuing to serve your customers. Never just write the plan up and let it age. Be sure it is truly implemented and verified!
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Basic QA Statistics Series(Part 3)- Basic Measures of Dispersion and Statistical Notation
REFLECTION: FOR STUDENTS: “It is not possible to know what you need to learn.” -Philip Crosby
FOR ACADEMICS: “Quality is the result of a carefully constructed cultural environment. It has to be the fabric of the organization, not part of the fabric.”-Philip Crosby
FOR PROFESSIONALS/PRACTITIONERS: “Quality has to be caused, not controlled.”-Philip Crosby
Foundation
Before we go further, this post will give you the basic notation for simple statistics so we can communicate more efficiently. It will also make understanding instructions from textbooks much less challenging. Please don’t give up here. These notations are just a secret code mathematicians use. If you learn it, you will begin to see that statistics is quite accessible. After the code is passed on, we will move on to the Measures of Dispersion.
Review: Part 1 and 2 covered the definition of Population, Sample, and how the terms Parameter and Statistic relate to Population and Sample, respectively. Also, we covered the concept of what data is, as well as the different kinds of data that exist, and the measurement scales used to analyze measurement data.
STATISTICAL NOTATION
Typically, capital letters and Greek letters are used to refer to population parameters, and lower-case or Roman letters are used to note sample statistics.
I will be providing information in the table below specifically for this post. As posts are added in the series, more tables will be added to address any other notations referenced in the future. This post will become the notation reference page to allow any who are new to statistical notation an easy reference.
(Kubiak, 2017)
MEASURES OF DISPERSION
There are three primary Measures of Dispersion- Range, Variance, and Standard Deviation. I will address each and explain them plainly. If you are new to statistics, I will avoid mathematics as much as possible, but alas, you will find it inescapable.
First comes RANGE. Range is probably the most well known and most easily understood. Range is simply the difference between the largest (Maximum or MAX) value and the smallest (Minimum or MIN) value in a data set.
Example: 24, 36, 54, 89, 12, 14, 44, 55, 75, 86
Min 12, Max, 89
Range (R)= Max-Min = 77
Though Range is easy to use, it is not always as useful as the other measures of dispersion, because sometimes two separate data sets can have very similar ranges, with the other measures looking nothing alike. On that note, comes something a bit more complicated.
At first, it sounds pretty simple:
VARIANCE- This is the measure of how far off the data values are from the mean over-all. Obtaining this measurement by hand can be painful. You have to find the difference between the mean and each data point in the population or sample, square the differences, and then find the average of those squared differences.
Variance RoadMap
- Calculate the mean of all the data points Calculate the difference between the mean and each data point(Xi – μ or x ̅), Xi being a representation ith value of variable X.
- Square the calculated differences for all data points
- Add these Squared values together
- Divide that number by N if the data set is a population (N), or divide by n-1 if the data is a sample
Follow the underlined statements above, and the formula for Variance below is achieved, but most stat software will calculate Variance with minimal effort.
Sample
Population
Standard Deviation (SD)
A negative of Variance is though you can measure the relative spread of the data, it is not representative of the same scale because it has been squared. For example- data collected in inches or seconds and then checked for variance is effectively square inches or seconds squared.
Standard Deviation is more useful because the units of Standard Deviation end up on the same scale and are directly comparable to the mean of the population or sample. Standard Deviation is the Square Root of the Variance and can be described as the average distance from each data point to the mean. The lower the SD, the less spread out the data is. The larger the spread of data, the higher the SD. Once again, most Stats programs and calculators will provide SD with no problem. The SD helps you understand how much your data is varying from the mean.
Two Examples: (using sample sets)
Set 1: 35, 61, 15, 14, 1
Mean(Set 1): 25.2
Set 2: 45, 48, 50, 43, 40
Mean(Set 2): 45.2
S=√(((45-45.2)²+(48-45.2)²+(50-45.2)²+(43-45.2)²+(40-45.2)²)/4)=3.96
When you first glance at the small sample of data, set one looks like it has a much larger spread from the average than set two. When you run the numbers, the SD results back up your “gut feeling.” An analysis is always better than a “gut feeling,” no matter how intuitive you are. The larger the sample set you are looking at, the more the initial appearance of the data can mislead you, so always run those numbers!
Conclusion
To recap, Range is the most well-known and straightforward Measure of Dispersion, but only describes the dispersion of the extremes of the data, and therefore may not always provide much new information. Range is usually most useful with smaller data sets. I should also mention a term known as the Interquartile Range (IQR). I will be dedicating a separate post to IQR next post.
Variance is an overall measure of the variation occurring around the mean using the Sum of Squares methodology. Remember, variance does not relate directly to the mean, so you cannot evaluate a variance number directly, so you should use variance to see how individual numbers relate to each other within a data set. Outliers (data points far from the mean) gain added significance with variance as well. Standard Deviation tends to be the most useful Measure of Dispersion, as it relates directly to the mean, and can be used to compare the spreads of various data sets. Remember, your stats programs will help you, and many online resources will walk you through any calculation. If you have any questions, shoot me a comment, and I will answer it for you. See you next time as we dig a bit deeper into IQR. 😊
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Bibliography
Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Basic QA Statistics Series(Part 2)- Basic Measures of Central Tendency and Measurement Scales
REFLECTION: FOR STUDENTS: Learning is not compulsory… neither is survival. – W. Edwards Deming
FOR ACADEMICS: Our schools must preserve and nurture the yearning for learning that everyone is born with. -W Edwards Deming
FOR PROFESSIONALS/PRACTITIONERS: Data are not taken for museum purposes; they are taken as a basis for doing something. If nothing is to be done with the data, then there is no use in collecting any. The ultimate purpose of taking data is to provide a basis for action or a recommendation for action. The step intermediate between the collection of data and the action is prediction. -W. Edwards Deming
Foundation
The previous post covered just the definition of Population and Sample and the descriptions of each using Parameters for Population and Statistics for a Sample. We also mentioned data. To be able to communicate about data, we first have to define data. Define should always be the first step for better understanding.
Data are characteristics or information (usually numerical) that are collected through observation. In a more technical sense, data consists of a set of values of qualitative or quantitative variables concerning one or more persons or objects.
The two broadest categories of Data are: Qualitative and Quantitative-
Qualitative data deals with characteristics and descriptors that cannot be easily measured but can be observed in terms of the attributes, properties, and of course, qualities of an object (such as color and shape). Quantitative data are data that can be measured, verified, and manipulated. Numerical data such as length and weight of objects are all Quantitative.
On the next level of Data are Discrete and Continuous Data.
Discrete Data– Pyzdek and Keller defined discrete data as such: “Data are said to be discrete when they take on only a finite number of points that can be represented by the non-negative integers” (Kubiak, 2017). Discrete data is count data and sometimes called categorical or attribute data. A count cannot be made more precise. You cannot have 2.2 fully functional cars.
Continuous Data– Pyzdek and Keller state- “ Data are said to be Continuous when they exist on an interval, or on several intervals.” Another term used is Variable data. Height, weight, and temperature are continuous data because between any two values on the measurement scale, there is an infinite number of other values (Kubiak, 2017).
Measurement Scales
- Nominal
- Classifies data into categories with no order implied
- Ordinal
- Refers to data positions within a set, where the order is essential, but precise differences between the values are not explicitly defined (example: poor, ok, excellent).
- Interval
- An Interval scale has meaningful differences but no absolute zero. (Ex: Temperature, excluding the Kelvin scale)
- Ratio
- Ratio scales have meaningful differences and an absolute zero. (Ex: Length, weight and age)
(Kubiak, 2017)
I know that it seems like a lot to digest, but recording data correctly is critical. Next, we will discuss the Central Limit Theorem: Per the central limit theorem, the mean of a sample of data will be closer to the mean of the overall population in question, as the sample size increases, notwithstanding the actual distribution of the data. In other words, the true form of the distribution does not have to be normally distributed (a bell curve) as long as the sample size is sufficiently large(Kubiak, 2017). There will eventually be a separate post(s) on sampling, distribution, and choosing the ideal sample size, but we are starting at the basics.
Note: Ordinal Data can be confusing. It depends on the how the ordinal scale is arranged. The Likert Scale would be considered quantitative ordinal, while the Movie rating scale would be considered qualitative ordinal.
(Kubiak, 2017)
Measures of Central Tendency
Three Common ways for quantifying the centrality of a population or sample include the
- Mean
- Arithmetic Average of a data set. This is the sum of the values divided by the number of individual values Ex: [1,3,5,10] Average is 4.75
- Median
- This is the middle value of an ordered data set. When the data are made up of an odd number of values, the median value is the central value of the ordered set. [1, 3, 5], so 3 is the median. When there is an even number of data points, the median is the average of the two middle values of the ordered set [1, 3, 5, 10]. In this case, the Median is the average of 3 and 5: (3+5)/2=4
- Mode
- The mode is the most frequently found value in a data set. It is possible for there to be more than one mode. EX: [1,2,3,5,1,6,8,1,8,1,3]- The Mode is 1
(Kubiak, 2017)
Conclusion
Correctly recording Data and using the proper scale to track your Data is the first step to understanding your process outputs.
A next baby step is knowing how to measure your process based upon your data scale. Being able to calculate the Measures of Central Tendency helps you, but Stats software will do much of this for you. Still, you need to know what you are seeing. It is always most helpful to know what those stat software programs are doing with your data so you can more robustly defend your decisions. Next time we will go a little deeper and talk about Measures of Dispersion (and it is precisely what it sounds like!).
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Bibliography
Kubiak, T. a. (2017). The Certified Six Sigma Black Belt Handbook Third Edition. Milwaukee: ASQ Quality Press.
Intro to Basic QA Statistics Series (Part 1)
REFLECTION: FOR STUDENTS: Don’t throw your stat’s books away. You will use them one day, and unlike most other textbooks, old statistics textbooks are usually just as valid as they were years ago.
FOR ACADEMICS: Yes, teach your students to use R and Minitab, but be sure they can do the calculations by hand
FOR PROFESSIONALS/PRACTITIONERS: I know you likely are old hands at this, but thank you for any advice you can provide those entering the QA field who have questions. Stats sometimes seems mystifying, so be sure to share the knowledge with mentorship.
Foundation
Most in the industrial world understand that tracking and analysis of data is a requirement to know if a process is producing waste or if your process is stable and everything is running as it should. The problem that arises from time to time is the “craftsman attitude” of a process owner, or perhaps even that same attitude in a plant manager.
What I am describing with the term -“craftsman attitude”- is the acquired belief of a knowledgeable Subject Matter Expert or Expert Operator, which leads a person to decide of their own accord they are wise enough to make a critical decision without proper Data Analysis.
At this point in any conversation with a person trying to argue they do not need Data, I would quote Deming- “In God we trust, all others must bring DATA.”
This phenomenon is likely a combination of two primary things:
Resistance to Change and a natural aversion (at least in Western Society) to developing a deep understanding of mathematics. Resistance to Change is inherent in many human beings and very hard to overcome. Awareness of the potential Resistance and how it may influence your decisions can help mitigate how you react to Quality Improvement issues.
Mathematics is something you must pursue on your own unless you were gifted with a natural aptitude (as the school systems are not very effective when it comes to producing mathematically educated high school graduates). Fear not.
Lack of mathematical prowess means very little in this day and age. You need only understand the basic concepts of statistics and how to apply them, and most stats software will walk you through the rest.
This post will be the first of a few on basic stats to help those who may feel like it is beyond them to help show that it is not that complicated.
I will be keeping the stats on a basic level, and breaking the posts up into digestible bits, so please continue reading 😊
Basic Statistical Terms
The first terms that should be understood first are Population, Sample, Statistic, and Parameter.
-A Population considers every member/unit of a group
-A Sample would be a random surveillance study of a portion of the Population
-A Parameter is Derived from Analysis of the Population
-A Statistic is Derived from Analysis of the Sample of the Population
Conclusion
A Parameter is a description of an entire group (Population). A Statistic is the description of a sample of the Population. Understanding the difference is critical. A Parameter indicates that the whole population has been evaluated in some way to obtain the result.
A Statistic depends upon randomness and sample size to adequately estimate the outcome of the population parameter.
The larger the random sample, the better!
If you have the time and resources, always choose a larger sample size!
-Next Time: Basic Measures of Central Tendency
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The Deming Chain Reaction
REFLECTION: FOR STUDENTS: How can you weave Quality into all Management endeavors to help sustain what you are managing?
FOR ACADEMICS: How can I incorporate MORE quality concepts into class/classroom management?
FOR PROFESSIONALS/PRACTITIONERS: Do you operate in a firefighting mode, preventive mode, or risk management mode? Are those quality decisions ever made by the front line operators, or is it managers who solve the problems?
The “Whats” of Quality Culture
During my 20 years of experience as a Quality Professional I have witnessed a similar pattern emerge constantly. One book I read (Angle, 2019) captured the essence of what happens with just three observations, that I can firmly attest to as the three most impactful “Whats” that I have seen.
-The First “What” was: Sustainable Corrective Actions were not properly enacted to address noted quality failures. Pencil whipping corrective actions to meet a deadline or going after the immediate surface problem only allows the eventual recurrence of the issue.
-The second “What” was how curiously common it was for a company to not understand the power of Quality from the highest ranks, deep into the culture, and how critical not understanding the need to monitor the existence and state of the Culture of Quality was to the company. Reactive measures are firefighting tasks brought on by the need for a quick fix. A culture that chooses to ignore quality and go for the non-sustainable path will imprint those same values across the entire organization. When the organization is short term focused, so are it’s employees, so quality and profit fall over the long term.
-The third and final “What” is far more common: There may be no true strategy for Quality. No strategy for Continuous Improvement, Change Management, or Strategic Alignment that is truly Quality Driven. Too often Quality is a department that measures and inspects, and any other decision is financially driven. Pure financial drive leads to non-measurement of quality costs, impacts, and ramifications (the infamous Hidden Factory), which tends exclude Quality from risk/based business decisions not related to regulated industries. ISO claims that “over a million” companies have been certified, so if I add, let us say, 9 million other companies with various other certified QMS systems (and I am being generous) then, (using the top 15 GDP countries for the low estimate and population extrapolation for high) out of the approximately 60-100 million companies in the world the % of companies with a certified QMS is 10% to 16%. **NOTE: [Not verified numbers, just estimates to convey the current state of quality in the business world based upon available data and the numbers are always in flux.] Certification only helps push a company toward a viable QMS. Though minding your P’s and Q’s helps, a viable QMS is in no way guaranteed by a certificate.
How a Culture of Quality Impacts the World
The end result of a company allowing any one of these “Whats” to manifest beyond the level of “I’m new to this quality thing” stage of an employee’s professional development is usually to the detriment of the company’s reputation, customer retention, and the company’s ability to obtain the best ROI. Those who argue for quick and dirty quality are selling snake oil. Quick and Dirty is the Challenger disaster management mentality of “I’m in charge! NASA expects this bird in orbit! I don’t care if all the engineers in the world are telling me that it might be too big a risk!”
Human nature is to break the rules, and it takes a true Quality Culture of excellence to hold an organization to a consistent level of excellence beyond that of normal human nature. Evan a well-designed system is useless if it can be overridden by customer demand, time constraints or public pressure that might jeopardize public safety, of the safety of even one human (Bombardier Business Aircraft, 2018).
Overcoming the tendency of humans to take the path of least resistance (especially in management) is a topic for another post, but The work of quality gurus such as W. Edwards Deming, Joseph M. Juran and Armand V. Feigenbaum helped enlighten Japan beginning in the 1950’s, and Japan took off with their teachings and enriched the concept, until Japan had risen from a country decimated by war to a major economic power. In the 1970’s America began to wake up to what was happening, and the work of Deming, Juran, Feigenbaum and then Philip B. Crosby, Taiichi Ohno, and Eiji Toyoda were recognized and the path was actively pursued in the West. All looked at the philosophy a little different, but they all understood how costly it was to ignore quality (especially from competitors). Eventually the economic result of a strong quality culture (that can still be seen in Toyota) was described by Deming as a Chain Reaction (ASQ, n.d.).
Basically: improve quality →decrease costs →improve productivity → increase market share with better quality, lower price → stay in business → provide more jobs, and now Toyota is (at this time) the most profitable automaker in the world (about twice GM’s profitability).
Improve quality can relate to every kind of buzzword, but the main goals are to reduce waste, reduce variation, and provide value to the customer (for which the customer would be willing to pay). Waste, Defects, Rework, Delays, all decrease, OEE goes up, Costs drop and productivity goes up, and you then have lots of higher quality and more desirable items on the market available at a lower price. Competitors can flood the market with cheap items, but quality lasts if it is not too expensive for the market, so your market share expands due to the ability to sell a higher quality product at a more affordable price. You stay in business and your company provides more jobs as they expand market share (Victor E. Sower, 2016).
Conclusion
Culture is king. Without a proper Quality Culture, the customer will never be properly served. The Voice of the Customer guides you, but the Culture is your company’s heart. If your heart is tainted with pure desire for profit, then you are not serving the customer, but yourself. The initial investment required to improve internally can be framed as non-customer focused by some, but always remember, that inward looking improvement of the company from management to the front line worker, across every department, and every process can only benefit the customer in the end. Due to the Pareto effect, the relatively vital few companies with a viable QMS and honest Culture of Quality do clearly have a significant economic impact on our world, and all companies should ask themselves: Do they want to be one of the Vital Few or Trivial Many?
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Bibliography
Angle, A. S. (2019). Unleash Quality. Milwaukee: ASQ Quality Press.
ASQ. (n.d.). The History of Quality. Retrieved from https://asq.org/quality-resources/history-of-quality
Victor E. Sower, K. W. (2016, 07). Retrieved from Quality Progress: http://asq.org/quality-progress/2016/07/basic-quality/dead-or-alive.pdf
Bombardier Business Aircraft. (2018, June 1). The Normalization of Excellence. Retrieved from https://safetystanddown.com/en/normalization-excellence
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Quality Concepts Matter 