Are big people smarter than small people?

"I have no doubt that I am very stupid; but I must confess that I am unable to follow you. For example, how did you deduce that this man was intellectual?" For answer Holmes clapped the hat upon his head. It came right over the forehead and settled upon the bridge of his nose. "It is a question of cubic capacity," said he; "a man with so large a brain must have something in it."

> Some epidemiological research on the subject has shown that there is a small but statistically significant positive correlation between height and intelligence after controlling for socioeconomic class and parental education

[0]: https://en.wikipedia.org/wiki/Height_and_intelligence

General health is still a probable shared cause here, they're going to need to control for more than just socioeconomic class and parental education.

What happens, for instance, if they control for stamina (I'd predict the effect goes away)

As a rule of thumb, yes.

(children vs adults)