Mar 17 2014

## Averages in Manufacturing Data

The first question we usually ask about lead times, inventory levels, critical dimensions, defective rates, or any other quantity that varies, is what it is “on the average.” The *second* question is how much it varies, but we only ask it if we get a satisfactory answer to the first one, and we rarely do.

When asked for a lead time, *people* usually give answers that are either evasive like “It depends,” or weasel-worded like “*Typically*, three weeks.” The beauty of a “typical value” is that no such technical term exists in data mining, statistics, or probability, and therefore the assertion that it is “three weeks” is immune to any confrontation with data. If the assertion had been that it was a mean or a median, you could have tested it, but, with “typical value,” you can’t.

For example, if the person had said “The median is three weeks,” it would have had the precise meaning that 50% of the orders are delivered in less than 3 weeks, and that 50% take longer. If the 3-week figure is true, then the probability of the next 20 orders all taking longer, is $latex 0.5^{20}= 9.6\,ppm$. This means that, if you do observe a run of 20 orders with lead times above 3 weeks, you know the answer was wrong.

In Out of the Crisis, Deming was chiding journalists for their statistical illiteracy when, for example, they bemoaned the fact that “50% of the teachers performed beneath the median.” In the US, today, the meaning of averages and medians is taught in Middle School, but the proper use of these tools does not seem to have been assimilated by adults.

One great feature of averages is that they add up: the average of the sum of two variables is the sum of their averages. If you take two operations performed in sequence in the route of a product, and consider the average time required to go through these operations by different units of product, then the average time to go through operations 1 and 2 is the sum of the average time through operation 1 and the average time through operation 2, as is obvious from the way an average is calculated. If you have n values $latex X_{1},…,X_{n}$

the average is just

$latex \bar{X}= \frac{X_{1}+…+X_{n}}{n}$

What is often forgotten is that most other statistics are *not* additive.

To obtain the median, first you need to sort the data so that $latex X_{\left(1\right)}\leq … \leq X_{\left(n\right)}$. For each point, the sequence number then tells you how many other points are under it, which you can express as a percentage and plot as in the following example:

Graphically, you see the median as the point on the x-axis where the curve crosses 50% on the y-axis. To calculate it, if n is odd, you take the middle value

$latex \tilde{X}= X_{_{\left (\frac{n}{2}+1\right )}}$

and, if n is even, you take the average of the two middle values, or

$latex \tilde{X}= \frac{\left[ X_{_{\left (\frac{n}{2}\right )}}+X_{_{\left (\frac{n}{2}+1\right )}}\right]}{2}$

and it is not generally additive, and neither are all the other statistics based on rank, like the minimum, the maximum, quartiles, percentiles, or stanines.

An ERP system, for example, will add operation times along a route to plan production, but the individual operation times input to the system are not averages but worst-case values, chosen so that they can reliably be achieved. The system therefore calculates the lead time for the route as the sum of extreme values at each operation, and this math is wrong because extreme values are not additive. The worst-case value for the whole route is *not* the sum of the worst-case values of each operation, and the result is an absurdly long lead time.

In project management, this is also the key difference between the traditional Critical Path Method (CPM) and Eli Goldratt’s Critical Chain. In CPM, task durations set by the individuals in charge of each task are set so that they can be confident of completing them. They represent a perceived worst-case value for each task, which means that the duration for the whole critical path is the sum of the worst-case values for the tasks on it. In Critical Chain, each task duration is what it is actually expected to require, with a time buffer added at the end to absorb delays and take advantage of early completions.

That medians and extreme values are not additive is experienced, if not proven, by a simple simulation in Excel. Using the formula “LOGNORM.INV(RAND(),0,1)” will give you in about a second, 5,000 instances of two highly skewed variables, X and Y, as well as their sum X+Y. On a logarithmic scale, their histograms look as follows:

And the summary statistics show the Median, Minimum and Maximum for the sum are not the sums of the values for each term:

Averages are not only additive but have many more desirable properties, so why do we ever consider medians? There are real problems with averages, when taken carelessly:

*Averages are affected by extreme values*. It is illustrated by the Bill Gates Walks Into a Bar story. Here we inserted him into a promotional picture of San Fancisco’s Terroir Bar:Attached to each patron other than Bill Gates is a modest yearly income. But his presence pushes the average yearly income above $100M, which is not a meaningful summary of the population. On the other hand, consider the median. Without Bill Gates, the middle person is Larry, and the median yearly income, $46K. Add Bill Gates, and the median is now the average of Larry and Randy, or $48K. The median barely budged! While, in this story, Bill Gates is a genuine outlier, manufacturing data often have outliers that are the result of malfunctions, as when wrong measurements are recorded as a result of a probe failing to touch the object it is measuring, or the instrument is calibrated in the wrong system of units, or a human operator puts a decimal point in the wrong place…Large differences between average and median are a telltale sign of this kind of phenomenon. Once the outliers are identified, assessed, and filtered, you can go back to using the average rather than the median.*Averages are meaningless over heterogeneous populations*. The statement that best explains this is “The average American has exactly one breast and one testicle.” It says nothing useful about the American population. In manufacturing, when you consider, say, a number of units produced, you need to make sure you are not commingling 32-oz bottles with minuscule free samples.*Averages are meaningless for multiplicative quantities*. If you data is the sequence $latex Y_{1}, …,Y_{n}$ of yields of the n operations in a route, then the overall yield is $latex Y= Y_{1}\times …\times Y_{n}$, and the plain average of the yields is irrelevant. Instead, you want the*geometric mean*$latex \bar{Y}=\sqrt[n]{Y_{1}\times …\times Y_{n}}$.

The same logic applies to the compounding of interest rates, and the plain average of rates over several years is irrelevant.*Sometimes, averages do not converge when the sample size grows*. It can happen even with a homogeneous population, it is not difficult to observe, and it is mind boggling. Let us say your product is a rectangular plate. On each one you make, you measure the differences between their actual lengths and widths and the specs, as in the following picture:

Assume then that, rather than the discrepancies in length and width, you are interested in the slope ΔW/ΔL and calculate its average over an increasing number of plates. You are then surprised to find that, no matter how many data points you add, the ratio keeps bouncing around instead of converging as the law of large numbers has led you to expect. So far, we have looked at the averages as just a formula applied to data. To go further, we must instead consider that they are estimators of the mean of an “underlying distribution” that we use as a model of the phenomenon at hand. Here, we assume that the lengths and widths of the plates are normally distributed around the specs. The slope ΔW/ΔL is then the ratio of two normal variables with 0 mean, and therefore follows the Cauchy distribution. This distribution has the nasty property of*not having a mean*, as a consequence of which the law of large numbers does not apply. But it has a median, which is 0.

The bottom line is that you should use averages whenever you can, because you can do more with them than with the alternatives, but you shouldn’t use them blindly. Instead, you should do the following:

- Clean your data.
- Identify and filter outliers.
- Make sure that the data represents a sufficiently homogeneous population.
- Use geometric means for multiplicative data.
- Make sure that averaging makes sense from a probability standpoint.

As Kaiser Fung would say, use your number sense.

Aug 8 2014

## The meaning(s) of “random”

“That was

random!” is my younger son’s response to the many things I say that sound strange to him, and my computer hasRandomAccess Memory (RAM), meaning that access to all memory locations is equally fast, as opposed tosequentialaccess, as on a tape, where you have to go through a sequence of locations to reach the one you want.In this sense, a side-loading truck provides random access to its load, while a back-loading truck provides sequential access.

While these uses of

randomare common, they have nothing to do with probability or statistics, and it’s no problem as long as the context is clear. In discussion of quality management or production control, on the other hand,randomnessis connected with the application of models from probability and statistics, and misunderstanding it as a technical term leads to mistakes.In factories, the only example I ever saw of Control Charts used as recommended in the literature was in a ceramics plant that was firing thin rectangular plates for use as electronic substrates in batches of 5,000 in a tunnel kiln. They took dimensional measurements on plates prior to firing, as a control on the stamping machine used to cut them, and they made adjustments to the machine settings if control limits were crossed. They did not measure every one of the 5,000 plates on a wagon. The operator explained to us that he took measurements on a “random sample.”

“And how do you take random samples?” I asked.

“Oh! I just pick here and there,” the operator said, pointing to a kiln wagon.

That was the end of the conversation. One of the first things I remember learning when studying statistics was that picking “here and there” did not generate a

randomsample. Arandom sampleis one in which every unit in the population has an equal probability of being selected, and it doesn’t happen with humans acting arbitrarily.A common human pattern, for example, is to refrain from picking two neighboring units in succession. A true random sampler does not know where the previous pick took place and selects the unit next to it with the same probability as any other. This is done by having a system select a location based on a random number generator, and direct the operator to it.

This meaning of the word “random” does not carry over to other uses even in probability theory. A mistake that is frequently encountered in discussions of quality is the idea that a

random variableis one for which all values are equally likely. What makes a variablerandomis that probabilities can be attached to values or sets of values insomefashion; it does not have to be uniform. One value can have a 90% probability while all other values share the remaining 10%, and it is still a random variable.When you say of a phenomenon that it is

random, technically, it means that it is amenable to modeling using probability theory. Some real phenomena do not need it, because they are deterministic: you insert the key into the lock and it opens, or you turn on a kettle and you have boiling water. Based on your input, you know what the outcome will be. There is no need to consider multiple outcomes and assign them probabilities.There are other phenomena that vary so much, or on which you know so little, that you can’t use probability theory. They are called by a variety of names; I use

uncertain. Earthquakes, financial crises, or wars can be generically expected to happen but cannot be specifically predicted. You apply earthquake engineering to construction in Japan or California, but you don’t leave Fukushima or San Francisco based on a prediction that an earthquake will hit tomorrow, because no one knows how to make such a prediction.Between the two extremes of deterministic and uncertain phenomena is the domain of

randomness, where you can apply probabilistic models to estimate the most likely outcome, predict a range of outcomes, or detect when a system has shifted. It includes fluctuations in the critical dimensions of a product or in its daily demand.The boundaries between the deterministic, random and uncertain domains are fuzzy. Which perspective you apply to a particular phenomenon is a judgement call, and depends on your needs. According to Nate Silver, over the past 20 years, daily weather has transitioned from uncertain to random, and forecasters could give you accurate probabilities that it will rain today. On the air, they overstate the probability of rain, because a wrong rain forecast elicits fewer viewer complaints than a wrong fair weather forecast. In manufacturing, the length of a rod is deterministic from the assembler’s point of view but random from the perspective of an engineer trying to improve the capability of a cutting machine.

This categorization suggests that that a phenomenon that is almost deterministic is, in some way, “less random” than one that is near uncertainty. But we need a metric of randomness to give a meaning to an expression like “less random.” Shannon’s entropy does the job. It is not defined for every probabilistic model but, where you can calculate it, it works. It is zero for a deterministic phenomenon, and rises to a maximum where all outcomes are equally likely. This brings us back to

random sampling.We could more accurately call it “maximum randomness sampling” or “maximum entropy sampling,” but it would take too long.By Michel Baudin • Data science, Technology • 2 • Tags: Quality, Quality Assurance, Randomness