Laws of nature

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Oct 12 2022

Musings on Large Numbers

Anyone who has taken an introductory course in probability, or even SPC, has heard of the law of large numbers. It’s a powerful result from probability theory, and, perhaps, the most widely used. Wikipedia starts the article on this topic with a statement that is free of any caveat or restrictions:

In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.

This is how the literature describes it and most professionals understand it. Buried in the fine print within the Wikipedia article, however, you find conditions for this law to apply. First, we discuss the differences between sample averages and expected values, both of which we often call “mean.” Then we consider applications of the law of large numbers in cases ranging from SPC to statistical physics. Finally, we zoom in on a simple case, the Cauchy distribution. It easily emerges from experimental data, and the Law of Large Numbers does not apply to it.

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Feb 11 2022

Scientific Thinking and Manufacturing Improvement

“Scientific thinking” appears more and more in discussions of Lean, Kaizen, or TPS. What is it? Well, it’s the way scientists think.  In reality, however, talk to actual scientists about PDCA, DMAIC, the 8D, A3 thinking, Why-Why analysis, TRIZ, or even statistical design of experiments,  and their eyes glaze over. Most will have no idea what these methods are. This is true for physicists, chemists, biologists, or even economists. If you elaborate, they will dismiss these tools as trivial or devoid of any connection with their work.

Improving how things are made does make the world a better place but it’s not science. By growing a body of knowledge that is our greatest asset as a species, scientists make another contribution, that we should recognize as different. 

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Apr 16 2021

Measuring QC Efficacy: A Proposal

[The featured image is of a Vietnamese satellite undergoing final test in Japan]

As Jay Bitsack pointed out in his comments on LinkedIn about my previous post, the portability of a method from epidemiology to manufacturing quality is not a foregone conclusion. Formally, the logic of validating a vaccine seems applicable to the solution of a quality problem. They look similar when you consider only outcomes in terms of infection rates or the proportion of defectives. 

There are differences between data sets from a clinical trial and tests run before and after a process change in production that may affect the applicability of a method. We examine the conditions for the approach developed by Carlo Graziani for vaccine efficacy to cross over to quality control. Then we work out the math of Graziani’s method and the means to apply it. 

 

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Jul 31 2017

Acceptance Sampling In The Age Of Low PPM Defectives

Today, some automotive parts manufacturers are able to deliver one million consecutive units without a single defective, and pondering quality management practices appropriate for this level of performance is not idle speculation. Of course, it is only achieved by outstanding suppliers using mature processes in mature industries.

You cannot expect it during new product introduction or in high-technology industries where, if your processes are mature, your products are obsolete. While still taught as part of the quality curriculum, acceptance sampling has been criticized by authors like W. E. Deming and is not part of the Lean approach to quality.

For qualified items from suppliers you trust, you accept shipments with no inspection; for new items or suppliers you do not trust, you inspect 100% of incoming units until the situation improves. Let us examine both what the math tells us about this and possible management actions, with the help of 21st century IT.

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Mar 22 2017

Saturation In Manufacturing Versus Service

In Capacity Planning For 1st Responders, we considered the problem of dimensioning a group so that there is at least one member available when needed. Not all service groups, however, are expected to respond immediately to all customers. Most, from supermarket check stands and airport check-in counters to clinics for non-emergency health care, allow some amount of queueing, giving rise to the question of how long the queues become when the servers get busy.

Patients waiting in Emergency Room

At one point in his latest book, Andy and Me And The Hospital, Pascal Dennis writes that the average number of patients in an emergency room is inversely proportional to the availability of the doctors. The busier the doctors are, the more dramatic the effect. For example, if they go from being busy 98% of the time to 99%, their availability drop by half from 2% to 1%, and the mean number of patients doubles. Conversely, any improvement in emergency room procedures that, to provide the same service, reduces the doctors’ utilization from 99% to 98%, which cuts the mean number of patients —  and their mean waiting time — in half.

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Feb 17 2017

Variability, Randomness, And Uncertainty in Operations

This elaborates on the topics of randomness versus uncertainty that I briefly touched on in a prior post. Always skittish about using dreaded words like “probability” or “randomness,” writers on manufacturing or service operations, even Deming, prefer to use “variability” or “variation” for the way both demand and performance change over time, but it doesn’t mean the same thing. For example, a hotel room that goes for $100/night in November through March and $200/night from April to October has a price that is variable but not random. The rates are published, and you know them ahead of time.

By contrast, to a passenger, the airfare from San Francisco to Chicago is not only variable but random. The airlines change tens of thousands of fares every day in ways you discover when you book a flight. Based on having flown this route four times in the past 12 months, however, you expect the fare to be in the range of $400 to $800, with $600 as the most likely. The information you have is not complete enough for you to know what the price will be but it does enable you to have a confidence interval for it.

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