With "gives an idea" I would normally expect "of" + "how"...
It´s similar to the sentence "your college major is a pretty good indication of how smart you are", in sense.
More of it:
Actually, about 38 percent of the harvard undergrads who apply to harvard law get. This is an extraordinarily high percentage, probably the highest for any undergraduate school in the country, and it gives an idea of how smart harvard undergrads are.
As an aside, because one could read one of the (misleading) sentences, and make confusion, so here we go:
Even with these lags, everybody remarks on how smart he is, and I’ve had several people say they think he’s a genius.
Here we are talking about something that is seemingly a fact, or at least a statet/perceived fact by the author of the sentence, so the ON equals to ABOUT.
As you know, organizational performance does not just depend on how smart executives are. Similarly, the final grade in the course is not just the result of how smart a student is.
With result, we use on - it´s somehow similar to the initial sentence. The first sentence here, is on because we have a concordance going there, DEPENDS goes with ON.
With time you get the hang of it.