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Variation, tolerances, probability, and specifications.

The Normal Distribution or “Bell Curve”.

In the last post I did a lot of talking about specifications and tolerances of components and the “worst case” situation. After the article was posted I realized I probably should explain some about all those terms.

When a manufacture marks a tolerance specification on a component, he is basically saying that he guarantees that the component will be within those limits. We have no idea how he obtains that.  It could be that he inspects each and every component and simply discards those that are out of tolerance.  It could be that he simply marks the ones out of tolerance with a greater tolerance value and sells those at a discount.  Ideally, he knows enough about his manufacturing process that he holds all of the inputs of the process at a tight value and only samples his product to ensure nothing unexpected is happening.

One of the things pushed in the last 20 years or so is something called six sigma.  Sigma is the Greek symbol (σ) used to symbolize a value called standard deviation.  If the process has variations following the normal distribution shown in the picture, then 99.73% of all the items produced will be within +/- 3 σ, hence the name 6 sigma. If the specification limits are at those values then 99.73% will be within limits.  Said another way, only 7 per 10,000 will be out of specification limits.

However, hidden within that statement was a bit “IF”.  Few processes follow that distribution.  For example, think of a machine boring a hole.  If the tool wears as it is being used, then the hole size can only go one way, smaller. This could be very bad news if we are intending to install round rod inside hole and the rod is being cut on a lath and when that tool wears the rod will get larger.

Again, hopefully the manufacture knows his process and compensates by starting the run with the inside hole large but within tolerance.  In any case, it is a safe assumption that the components have the highest probability of being toward the center of the specification.

In the last worst case analysis of the thermistor circuit in “One more step closer to completing the Thermistor Circuit”, I assumed the thermistor was at the very edge of the tolerance in one direction and the resistor in parallel was at the very edge in the opposite direction.  This worst possible case has about zero probability of happening.

To show why I am able to make this statement, lets take the example of throwing a pair of dice and getting boxcars (two 6’s).   The odds of getting a 6 on one die in a toss is 1/6.   The odds of getting two 6’s at the same time is 1/6 X 1/6 or 1/36.  The odds decrease quickly

If we imagine that both the thermistor and the resistor follow the bell curve show in the picture and that 5% = 3σ for the resistor, and 10% = 3α for the thermistor.  Then from the graph picture I only have about 2.2% probability that the resistor is less than 9.667K (10K – (5%x2σ/3σ)).  Likewise I only have a 2.2% probability that the thermistor is greater than 10.667K  (10K + (10%x2σ/3σ)).  This means I only have 0.022 X 0.022 or 0.0484% chance that both will be at opposite far extreme sides of their tolerance specification at the same time.

Should we worry about that?  The exercise was good and we know how bad things could possibly get.  People play the lottery all the time with much much smaller odds for winning.  Since it will add only one small step in our calibration process and one component I will add it just to feel all warm and fuzzy.   If I was going to produce 1000’s of these things then I would want to do a lot of testing before I committed to this because it would affect the profit margin and competitiveness of the product.  If it was 100,000’s of product to be made, we would have enough leverage to negotiate quality records from the component manufactures as well and we would know their variation distribution both in a run and from batch to batch and we could do even more to save money on calibration costs as well as ensure a high quality product.

We do not have those “IF’s” in our favor, so we play it conservative and spend the time to “do it right”.


The normal curve picture is covered by the creative commons share and share alike license and was created By Mwtoews [CC-BY-2.5 (http://creativecommons.org/licenses/by/2.5)], via Wikimedia Commons.


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