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Is there any authoritative definition for what the tolerance value of a resistor actually specifies, or is it just a de-facto common understanding, or are the details left up to individual manufacturers?

To put it differently, If I very precisely measured a large number of nominally equivalent resistors with the same tolerance, what does that tolerance value actually tell me about the distribution of resistances I would actually see? Would I expect a bell curve with, say, a mean at the nominal value and a standard deviation corresponding to the tolerance, or a random distribution strictly within the tolerance range, or something else entirely? All I can find on any descriptive sites are generic statements like "the amount a resistor may vary from its stated value," but I realized there are multiple possible interpretations to that kind of statement. (For example, if they just reject resistors that fall outside the tolerance range, I might expect to see a flat distribution within that range and nothing outside of it, as opposed to maybe a normal distribution with or without some less accurate cases.) This question sounds very similar, but is concerned more with changes in observed resistance over time. I'm specifically asking about the definition of the tolerance value itself (if one exists!). This page mentions specs ("High accuracies also go hand-in-hand with closer tolerances, but the two specifications are actually different") but doesn't elaborate on the specifications. This answer, about the feasibility of averaging resistances to try to tighten the equivalent tolerance, very nearly answers my question (and suggests it's more the bounded unspecified distribution with no guaranteed mean) but I'm still curious about an authoritative definition, say from some IEEE or IEC standard or something. (Otherwise, how can I assume that interpretation for any particular batch of components?)

epiii2
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    It really just means "The resistance value will be somewhere in this range" and doesn't guarantee anything beyond that, especially not any kind of distribution. – Jonathan S. Mar 24 '23 at 18:14
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    In my experience, for modern wide tolerance thick film resistors (1% and 5%), the vast majority are tightly clustered very close to the specified value because manufacturing tolerances are very good. You'll only occasionally find one that is even half way to the tolerance, if even that far. If you're talking about 0.01% thin film resistors, then I have no idea since even measuring them accurately enough to know is tricky. – user1850479 Mar 24 '23 at 18:21
  • [100K 5% Carbon Film Resistor Value Distribution](http://www.kerrywong.com/2013/02/01/100k-5-carbon-film-resistor-value-distribution/) – StainlessSteelRat Mar 24 '23 at 18:32

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AFAIK, no there is not. This is done manufacturer to manufacturer, the distribution can vary considerably depending on the manufacturer. It's generally understood to be an upper and lower bound. A manufacturer could theoretically have 99% of their resistors at near the same value and then 1% at say - 5% tolerance, and they would list the resistors at a 5% tolerance.

The actual distribution varies considerably here is an example of actual measured tolerances of an SMT resistor:

enter image description here Source: https://lambdafox.com/resistor-tolerances/

Voltage Spike
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Most things in nature and manufacturing follow normal distributions, so I would suspect that the distributions of resistance values would follow a normal curve whose tails are cut off above/below the rated tolerance.

The manufacturer's specific curve would depend on how good their process is.

I imagine you could model the distribution of a good manufacturer by setting the mean and standard deviation to values that set the 99th and 1st percentile to be within your tolerance (2% of resistors would be tossed). Or for a bad manufacturing process set them such that the 5th and 95th percentiles are within your tolerance (10% of resistors would be tossed).

Elliott Winsted
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You are GREATLY overthinking it.
The tolerance simply means that any of the resistors from the batch will deviate no more than the specified amount (at room temperature).
Simply put, as you already know, ±5% tolerance means that, for example, a 100Ω resistor will at the most have 105Ω (5% above) or at least 95Ω (5% below its indicated value), at room temperature.

So far, I haven't come across a single resistor which was outside of its specified tolerance range, and I've checked quite a few over the years. Almost all of the 5% resistors I checked were within a 3-4% range, so most of the time their value is even more accurate than their rated tolerance would indicate. And I have checked both new and old resistors, mostly carbon film and some metal film types, from various manufacturers.

There are no "multiple possible interpretations" of such a plain and mathematically clearly defined standard; you are the first and only person I have come across in my 3 decades of experience with electricity and electronics to make such a statement about such an elementary example.

No manufacturer can just say "well, I feel that up to 10% can be counted as 5%"; they would quickly discover that people don't like buying components which are falsely specified.
The beautiful thing about the manufacturing standards, specifications and tolerances, especially those given in the manufacturers' own datasheets, is that they give the worst case scenario ranges which in practice play out significantly better or at least no worse than specified.
The manufacturers who make inferior (electronic) components also show that in their datasheets and sell those components cheaper, and they find a large market share of people needing nothing better while saving money, but they appreciate that they can trust the manufacturers' own specifications and tolerances which are clearly, plainly and openly indicated and held onto.

Edin Fifić
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    Sometimes manufacturers can for example take all the parts within 1% and sell them as 1%, so buying 5% parts also means that *none* of them are within 1% and you should be prepared for that. Sometimes. – user253751 Mar 24 '23 at 19:57
  • Buying 5% parts means I count on them possibly being that much off, and it also means my circuits will tolerate that. If I need a precise voltage divider, for example, I will buy anything from 0.1% to 1%, with a very low temperature dependence. Most of the time resistors are used as current limiters, loads, dampeners or "ballpark" voltage dividers, making the cheap 5% parts usually accurate more than enough. – Edin Fifić Mar 24 '23 at 20:05
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    To clarify about multiple interpretations, I could get different distributions of real-world values depending on what the tolerance value means. For example [Voltage Spike's answer](https://electronics.stackexchange.com/a/660149) shows a real world example with a roughly normal--but offset-- distribution, while others have noticed bimodal distributions (presumably where outliers are sold off as a lower tolerance) like user253751 says. This has real implications, e.g. with the (evidently flawed) idea of ["averaging" resistors](https://electronics.stackexchange.com/questions/530600/). – epiii2 Mar 24 '23 at 20:09
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    @epiii2 The "bell curve" should mostly be filled with values closest to the one indicated on the part, but not necessarily. Still, NO PART should fall OUTSIDE of the tolerance range, otherwise it should be binned with higher tolerance parts. If a 100Ω resistor got 106Ω, it would not be included in the 100Ω 5% "bell curve" because it would be binned with 10% resistors. Just like a commenter said in the old question you linked to, if 1 out of 1000 resistors got outside of that tolerance band, I would look for another manufacturer. – Edin Fifić Mar 24 '23 at 20:41
  • Is there any particular reason to use relative percentage values instead of absolute units (like ±5Ω)? Using percentages obviously means the greater the resistance, the greater the absolute tolerance. – Crazymoomin Mar 25 '23 at 21:53
  • @Crazymoomin The higher the resistance value, the more accurate and precise both the components and the testing equipment would have to be, for example ±5Ω on a 100Ω resistor is both easily achievable and measurable. Now try making a component at 100,000Ω or 1,000,000Ω. At those values, +5Ω would be 100.005kΩ and 1.000,005MΩ. The vast majority of DMMs can measure with a 3.5 digit resolution and 2000 values, or 199.9kΩ and 1.999MΩ, which means you would need 20 and 200 times higher resolution, as well as accuracy and precision; with 1Ω tolerances, you need 100x and 1000x higher resolution. – Edin Fifić Mar 26 '23 at 00:12
  • @Crazymoomin So, it would be much more demanding and expensive, and with higher resistance values you also have "diminishing returns" for the same amount of accuracy in ohms. In a voltage divider, for example, you would get the same precision whether you use 5% resistors of lower values (tens or hundreds of ohms) or 5% resistors of higher values (tens or hundreds of kilo-ohms). With a 5Ω tolerance, you would get increasingly higher precision and accuracy but it would be both unnecessary in most cases, increasingly more expensive for larger values and much more difficult to achieve and measure. – Edin Fifić Mar 26 '23 at 00:40
  • @Crazymoomin The percentage-based tolerances are much more realistic, and much easier and cheaper to achieve and measure. And for certain or special cases where one needs higher precision, there are much more accurate resistors, down to 0.1%. – Edin Fifić Mar 26 '23 at 00:49