Precisely. If you are looking at some third world nation, well, there's all those kids who have various nutritional deficiencies, their IQs are impaired. The mean is lowered considerably, but that's through introduction of extra variables into the (approximate) sum.
If you don't take that into account and assume that only the mean in the distribution has changed, you get entirely invalid results at the high range due to how rapidly the normal distribution falls off far from the mean (as exponent of a square). For example if you were to calculate number of some rare geniuses out of the reference population (say, 300 millions with mean of 100 and standard deviation of 15), and from the world population assuming some lower mean and same standard deviation, for sufficiently rare "genius" you'll get a smaller number of geniuses in the whole world than in that one reference population (which is ridiculous).
edit: which you can see by noting that this with c smaller than b grows as x grows (i.e. ratio of prevalences between two populations grows with distance from the mean).
The example I'd give here is India, where you have lots of mostly distinct ethnic groups, and so it's reasonable to expect that the true distribution is a mixture of Gaussians. Knowing the Indian average national IQ would totally mislead you on the number of Parsis with IQs of 120 or above, if all you knew about Parsis was that they lived in India.
(It's not clear to me that malnourishment leads to multiple modes, rather than just decreasing the mean while probably increasing the variance, because I think damage due to malnourishment is linear, and it's probably the case that many different levels of severity of malnourishment are roughly equally well represented.)
I've been wondering how useful it is for the typical academically strong high schooler to learn math deeply. Here by "learn deeply" I mean "understanding the concepts and their interrelations" as opposed to learning narrow technical procedures exclusively.
My experience learning math deeply
When I started high school, I wasn't interested in math and I wasn't good at my math coursework. I even got a D in high school geometry, and had to repeat a semester of math.
I subsequently became interested in chemistry, and I thought that I might become a chemist, and so figured that I should learn math better. During my junior year of high school, I supplemented the classes that I was taking by studying calculus on my own, and auditing a course on analytic geometry. I also took physics concurrently.
Through my studies, I started seeing the same concepts over and over again in different contexts, and I became versatile with them, capable of fluently applying them in conjunction with one another. This awakened a new sense of awareness in me, of the type that Bill Thurston described in his essay Mathematics Education:
I understood the physical world, the human world, and myself in a way that I had never before. Reality seemed full of limitless possibilities. Those months were the happiest of my life to date.
More prosaically, my academic performance improved a lot, and I found it much easier to understand technical content (physics, economics, statistics etc.) ever after.
So in my own case, learning math deeply had very high returns.
How generalizable is this?
I have an intuition that many other people would benefit a great deal from learning math deeply, but I know that I'm unusual, and I'm aware of the human tendency to implicitly assume that others are similar to us. So I would like to test my beliefs by soliciting feedback from others.
Some ways in which learning math deeply can help are:
Some arguments against learning math deeply being useful are:
I'd be grateful to anyone who's able to expand on these three considerations, or who offers additional considerations against the utility of learning math deeply. I would also be interested in any anecdotal evidence about benefits (or lack thereof) that readers have received from learning math deeply.