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What is the probability of having a component with the minimum and maximum specified value into the datasheet ?

I suppose this follow a normal law and the typical value is the mean :

enter image description here

Have a nice day !

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    \$\begingroup\$ If the vendor bins their products, you'll often find that the distribution is non-Gaussian since the better performing parts will be sold as a different product leaving the lower side or possibly the extreme ends of the distribution overrepresented. \$\endgroup\$ May 26, 2021 at 14:04
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    \$\begingroup\$ If the vendor bins their products, you may as well get two maximums and a hole in the middle of the distribution (the dip is because of parts binned as the higher accuracy product). \$\endgroup\$
    – fraxinus
    May 27, 2021 at 7:28
  • \$\begingroup\$ The actual distribution may look nothing like your curve. Take a transistor - perhaps the min/max values on the datasheet were defined 10 years ago. It is now produced in a new factory on a totally new production line at a totally different process and linewidth. The current distribution might all fall within 1/10th of the original range. They may well be selling product under a new part number with tighter limits, but there is still a market for the old part number (think 2N2222 transistors, still available 60 years later - not made the same way!). \$\endgroup\$
    – Jon Custer
    May 27, 2021 at 16:28
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    \$\begingroup\$ Additionally, they may trim the part asymmetrically. as in, for a resistor they laser away more material to increase the resistance until it is within the tolerance. In which case you can expect them to be predominantly on the low side so you can't even count on just the center being binned out. \$\endgroup\$ May 28, 2021 at 17:14

3 Answers 3

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As you can see beautifully from your figure, it's the integral over your probability density function of the parameter you're considering.

Your figure shows something that looks quite a bit like a normal distribution. So, the cumulative density function \$\Phi(x)\$ of your normalized parameter would answer your question.

Generally, very few things in life are actually normally distributed (it's just that it's often a good approximation, if the extreme values that don't make physical sense are sufficiently rare). So, you'll need to study what the distribution of the parameter you care about is. Unless something or someone specifically says what the distribution is, you can't assume anything, and you can't answer your question. It's as simple as that!

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One important thing to add is that if you have a 1% resistor, it is not going to be normally distributed around for example, 100ohm.

In fact, you are almost guaranteed that it will not be 100.0 ohm as those are often binned out and sold as a higher tolerance.

It it not as simple as that though; supply/demand might persuade factories to supply 0,1% resistors into the 1% resistor supply chain, etc.

So to answer your question; You don't know the distribution, and you should always assume the worst.

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Speaking as an application engineer with 20+ years at an analog semiconductor manufacturer...

What is the probability of having a component with the minimum and maximum specified value into the datasheet ?

For a shipping IC? Must be 100%. Even for a passive component it's 100%. (As a customer: if I buy a 1k 5% resistor that turns out to be 800 ohms, I'll demand a refund and never buy from that vendor again.) That's not to say that 100% of the manufactured products are good, just saying that the whole point of the Min/Max limits is that those limits are what defines 100% of the shippable product.

In your graph, you highlighted a normal distribution curve with test limits set at +/- 2 sigma (standard deviations), which gives a yield loss of about 5%. When you're manufacturing hundreds of thousands of parts, a yield loss of 5% is really a big problem. It's better for the vendor to just set the Min/Max limits to where 99.9..% parts meet the specification. Different manufacturers have different policies, but the industry norm is somewhere between +/- 3 sigma and +/- 6 sigma.

Min/Max limits are not arbitrarily selected. They are contract limits. If vendor ships product that provably does not meet published Min / Max specifications, vendor is liable for damages. Even without a formal lawsuit, this is a situation vendors try to avoid -- nobody wants to buy from an unreliable vendor.

Typical numbers are the only "not guaranteed" numbers in the datasheet, since typical values are measured on a representative sample of the initial production material.

Test and measurement equipment always has some level of measurement uncertainty, so there is a "guard band" between the published Min/Max values and the true test limits. We must not ship a bad part just because a piece of test equipment was near the edge of its calibration spec one day.

Yield loss is bad. Any product material that went through fabrication but did not pass final test, is waste that cannot be sold for profit. So manufacturers use Statistical Process Control to try to avoid manufacturing bad material in the first place.

Statistical Process Control (SPC) is generally about minding how accurately the fabrication is working, especially the mean and standard deviation trends. Every wafer includes multiple "test chip" coupons that are used to measure in detail the parameters of that particular wafer (strong N / weak P, oxide capacitance, sheet resistance, etc.). The precise measurement of the parameters of the most recently fabricated wafers are tracked as corrective feedback to keep the manufacturing process on-target. So there is an active effort to minimize the standard deviation of the manufactured product, to try to avoid manufacturing material that cannot be sold.

Even for a passive component like a resistor or capacitor, where the production testing is more straightforward, there will be emphasis on using Statistical Process Control to try to minimize production line variance. Since the cost of physically handling each component is going to be a significant part of the test cost for such a simple test, it's likely that a resistor manufacturer could test a representative sample of their end production units to confirm that the lot is within spec, instead of testing 100% of the units as we do with ICs.

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    \$\begingroup\$ That's all true, but I'm not sure it answers what the question is getting at. Yes, 100% of products sold will fall between max and min, but the title asks for the distribution (within that band) and the body asks for (in my interpretation) the probability of getting a part that lands on the edge of the permissible range. Both of these are unknowable except for the manufacturer \$\endgroup\$
    – Chris H
    May 27, 2021 at 8:25
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    \$\begingroup\$ The edge of the permissible range is the guard band I referred to above. The manufacturer estimates how much measurement error is likely, and sets the test limits so that they are slightly inside of the min and max values, to avoid anything going out of band. \$\endgroup\$
    – MarkU
    May 27, 2021 at 8:50
  • \$\begingroup\$ Yes, so you shouldn't get any part right on the edge of the band, just close to it. But wherever the exact edges are drawn, what distribution would you expect? Would you expect a Gaussian, an even random distribution, or as one of the other answers states, potentially a notched distribution if tight-spec parts are binned for a higher price. Your answer is closer to answering "what should I design for?" than the actual question; to that question, or "what can I assume?" this would be a good answer \$\endgroup\$
    – Chris H
    May 27, 2021 at 8:59
  • \$\begingroup\$ @MarkU Anecdotal. For the datasheet parameters which aren't tested in production, min and max are 4 sigmas away from the typical, typically. I once had a conversation with a test engineer from a major semiconductor company. (I don't remember which company it was. I don't remember what kind of IC we were speaking about.) \$\endgroup\$ May 28, 2021 at 16:22

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