I’m not sure how extreme your general take on communication is, and I think at least I have a fairly similar view.
I agree that the kind of practical experiences you mention can be a good reason to be more careful with the use of some mathematical concepts but not others. I think I’ve seen fewer instances of people making fallacious inferences based on something being log-normal, but if I had I think I might have arrived at similar aspirations as you regarding how to frame things.
(An invalid type of argument I have seen frequently is actually the “things multiply, so we get a log-normal” part. But as you have pointed out in your top-level comment, if we multiply a small number of thin-tailed and low-variance factors we’ll get something that’s not exactly a ‘paradigmatic example’ of a log-normal distribution even though we could reasonably approximate it with one. On the other hand, if the conditions of the ‘multiplicative CLT’ aren’t fulfilled we can easily get something with heavier tails than a log-normal. See also fn26 in our doc:
We’ve sometimes encountered the misconception that products of light-tailed factors always converge to a log-normal distribution. However, in fact, depending on the details the limit can also be another type of heavy-tailed distribution, such as a power law (see, e.g., Mitzenmacher 2004, sc. 5-7 for an accessible discussion and examples). Relevant details include whether there is a strictly positive minimum value beyond which products can’t fall (ibid., sc. 5.1), random variation in the number of factors (ibid., sc. 7), and correlations between factors.
I’m not sure how extreme your general take on communication is, and I think at least I have a fairly similar view.
I agree that the kind of practical experiences you mention can be a good reason to be more careful with the use of some mathematical concepts but not others. I think I’ve seen fewer instances of people making fallacious inferences based on something being log-normal, but if I had I think I might have arrived at similar aspirations as you regarding how to frame things.
(An invalid type of argument I have seen frequently is actually the “things multiply, so we get a log-normal” part. But as you have pointed out in your top-level comment, if we multiply a small number of thin-tailed and low-variance factors we’ll get something that’s not exactly a ‘paradigmatic example’ of a log-normal distribution even though we could reasonably approximate it with one. On the other hand, if the conditions of the ‘multiplicative CLT’ aren’t fulfilled we can easily get something with heavier tails than a log-normal. See also fn26 in our doc:
)