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How about that sigma? Or, does anyone really understand what total quality management (TQM) is all about?
Let’s assume you want to design a three legged milking stool. Now since almost no one milks cows by hand anymore, there must be some other use for this milk-maiden device and the answer is that it seems to be showing up in homes as a decorative piece as early (prior to 1950) Americana.
The purpose of three legs was to permit one to sit a stool on uneven ground and still have a substantial seat. However, with the advent of concrete slabs for floors as well as traditional wood, we now have a level playing field for furniture. So a four legged chair sits quite steady (or almost so) in most houses. The three legged variety now suffers from a disadvantage not originally envisioned, that is, the eye judges parallel lines most critically and when a surface is not level, it is quickly discerned. We therefore must design our three-legged stool not only to sit firmly on the floor but also for the top seating surface to be parallel/perpendicular to our floors/walls. This means that the legs must screw in to the same depth and be firmly held in place.
Our statistical friends (that’s another way of saying that in a population, we have some acquaintances that are more friendly than others) have devised ways for us to measure the precision in which the threaded leg will screw into the tapped hole in the stools base. For this exercise they like to remind us that by randomly selecting legs and bases, they can come up with a likelihood that when assembled, the stool will actually stand on its own three legs. Their first focus of attention is on leg design and variation with the bored holes in the seat to be considered next.
So they begin with a statement that in a “binomial” distribution, or a population that is evenly distributed about a mean (call it an average if you like), some will be smaller in diameter and some will be larger in diameter than the others that roll off the production line (somehow, I guess, we already knew this). Now attacking the problem, they call forth an “empirical rule” without in most cases actually defining what they are talking about. As a reminder, nothing from the lips of a statistician can be trusted unless they at first assure you that the population’s distribution is bell-shaped. Which is another way of saying that there’s nothing strange going on here that causes “out-liers” or quirks representing a design flaw or human or mechanical screw-ups. (This is a fatal flaw in the TQM people’s approach to perfection.)
To permit us to use the empirical rule, or is it the other way about, to establish what the empirical rule is, they throw in another term, the standard deviation (they may use sigma in the new parlance but that’s wrong because in manufacturing you will never have access to the entire population before something shifts.) The standard deviation is a way of expressing how much individual values differ from the average.
In a bell shaped distribution, the mean and standard deviation can be related by determining the proportion of measurements falling within certain limits. The empirical rule states that: about 68 percent of our measurements will fall within one standard deviation from the average (that’s to say that about 34 percent will be larger and 34 percent smaller and still lie within one standard deviation of the average.)
Onward, what about those remaining stool legs we produced, those 32 percent that will be either larger or smaller than one standard deviation from the average. If you apply an additional standard deviation to our so called “standard” curve then an additional 27 percent will fall within. Or said differently, about 95% (68 plus 27) of all our legs will be within plus or minus two standard deviations from our average.
While that sounds pretty good, how about if you extend it one additional standard deviation unit on both sides of the average? That’s three standard deviations and the statistician will remind us that about 99.7 percent of all our legs will fall within this range. So if we want what some would consider to be near perfection, all we have to do is to engineer our leg production to a precision of plus or minus three standard deviations. Now we encounter the word “sigma” for the first time. If this were the whole world of milking stool production, then we could substitute the word sigma for standard deviation and everyone would understand that our legs are pretty good with only 0.3 percent (three out of one thousand) being outside three standard deviations. Of course only half of these offending legs would either be too large or too small by our selective criterion and this then becomes 1500 legs in a million on either side of the average. Suddenly when you express it in legs per million the number seems huge!
By now you may be able to guess that the further away from the average you go in accepting the production of your wayward machines or employees as “normal”, the more standard deviations (or sigmas) are required. At a stretch you come to six sigma as defined by Motorola way back in the 80's. Here they expect to have only three “defects” per million. There is no explanation of why it’s three per million and not 1 or so by the TQM people.
As any good consultant worth their salt and their five digit consulting fees will tell you; it’s not the acceptable performance you’re after, it’s spotting the defective stool legs before they get into the public’s hands. So they twist the idea to; let’s not accept more than “six sigma” defects or more than three per million. Sounds good, but it’s really not statistics.
At any rate if we return to our milk-maid’s stool with this understanding of what we want in allowable production variation we can begin to see just how ridiculous the idea is. We know that the individual leg has screw threads. Now, each leg can have a slightly different diameter, the curvature of the thread about the leg can vary, as well as the depth of the cut of the thread and the width of the thread as the part is machined. So we take three parts in a million for the diameter of the leg , three for the depth of the cut, three for the width of the cut and three for curvature and suddenly we have not three parts per million but; 3 x 3 x 3 x 3 = 81. Since the hole in the seat has the same potential for variation we’ve got to do the same math and come up with 81 as well. And then with three legs, have got to add that in too. Whoops, almost forgot that this leg screws in to the seat so have got to add another 3 parts per million for variation in drilling the hole in the seat. Our perfection begins to looks a little less that acceptable. 81 plus 81, times 3 (legs), times 3 (bore-holes) is 1,458 screw-ups per million. Of course the length of the individual legs is of concern, after all they really should be of the same length, which adds another factor of three to consider so now it’s 1,458 times three which is 4,374 miss-fits per million. In the the real world, that’s almost 4.4 out of every thousand or about one out of ever two hundred. In the eye of the customer, they are going to complain (or may return those defective stools). Hey, we got a problem!
It appears that statistics aren’t going to save us. What will? Technology! Just grab your glue gun and put a dab in the hole in the seat, there’s no need for precise fit so do away with threads altogether. If management insist on screw type construction, how about a length of Teflon™ tape to snug up the threads. That’s what engineers are for, to help us solve today’s problems. And of course the solution is actually built in to the design. As long as the leg goes into the hole and stays there, you have provided the customer a built in means for adjusting the table’s level. Just provide instructions to screw the legs in or out to get a proper fit.
If you believe this is a fictional case, take the four legged variety of kitchen, restaurant, or decorative table and note how many times it’s necessary to put a pad of wadded paper or such under one of the legs to make the damn thing stay free of wobble.
And, of course the table owner(s) didn’t depend on statistics to solve the problem. Now that’s total quality management.
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If you happen to read the December 15, 1999 Wall Street Journal article by Holman W. Jenkins Jr. you will find the statement that “U.S. industry as a whole operates around three sigma, or 66,000 mistakes per million, or an error rate of 6.6%.” Where he got his hodgepodge of “statistics” is unknown but they are simply wrong! Perhaps what is more disturbing is that a number from the medical profession wrote in protesting Mr. Jenkins proposal that the medical profession use TQM, but none complained about his error regarding statistics. And in fact one writer even repeated the numerical error in his letter. Which means not only Mr. Jenkins and the doctors but the editors of the Wall Street Journal all fell into the same trap. They simply believed the numbers because, after all, they are numbers and numbers are, of course, not suspect. Sorry Mr. Jenkins, medical professionals who complained and the journalist who should know better, you have to go back to class (if you ever had one on such matters.)
See Dave Barry’s commentary on numbers. His conclusion: if the number has a decimal point in it, it must be real.
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Visit your local drugstore and pick up an oral digital thermometer that announces your temperature, all for just $9.99. Now reflect on the product. The package is plastic, designed to protect the thermometer, advertising is displayed on a cut paperboard, instructions are included and inside the shelf pack is another plastic package for storing the thermometer. The device itself has a display screen, off and on switch, the probe and of course the electronics for detecting temperature, and reporting same. All this is powered by a battery which is included. As you hold this amazingly complex yet inexpensive medical device in you hand, it’s time to notice that it is made in China. Someone there had to design the thermometer, see it manufactured to specifications, assembled, packaged, shipped to the United States, and at that point the distribution system in this country took over, seeing that the package is delivered to the drugstore and shelved for display and hopefully sold at a profit. Nowhere in the product is there a role for United States labor. Even production of the equipment used to manufacture or package the thermometer is done off-shore. We can’t make this thermometer! And, there are countless other items, well made and sold cheap that can’t be made here as well! While management and countless consultants have worried about six-sigma, someone elsewhere has quietly gone about the job of making products that are needed in the United States. The three-legged stool analogy dies from lack of use. Robots and precise design and molding have made it a moot issue.
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Two additional points to be made: one, the three-legged stool was the stand on which notables pronounced their thoughts for the day in olden times.
And in more recent years, a three legged device was sometimes used to stabilize a hanging pole. The pole with a cross arm could be used to hoist an offending party to his embarrassment and sometimes death. (Woodcuts of the period of the American Revolution show an individual suspended by his belt by a rope so that he swam in the air.)
So we see there is actually a good use to be made of three legged stools and such; just rig a rope like in the olden days and hang the management people that try to persuade stockholders that they are doing a cracker-jack job (while at the same time exporting American jobs overseas.)
December 16, 1999/rev. September 14, 2002
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