Is there a specific definition for the tolerance value of a resistor?

Question

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?)

Answer 1

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:

Source: https://lambdafox.com/resistor-tolerances/

Answer 2

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).

Answer 3

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.