I read it initially as referring to to the whole family, not just the students in them (granted, missing full context), but it could possibly refer to the individuals, grammatically.
Here is
an example (unfortunate topic but good example)
from headline: 不登校の小中学生の家庭ほどしつけに厳しかった
from actual text: 不登校傾向にある小中学生は、小学校入学前の幼少期に保護者がしつけなどを厳しく意識してきた家庭に多い
Another where the topic is switched from "families" to "children" deliberately:
料理手作りの家庭ほど子どもはヘルシー食志向になるらしい
And another where it isn't: (
the topic is already about children's education and "grades" obviously refers to the kids not the family)
さらに、全国学力調査では、年収の低い家庭ほど成績が振るわない傾向が明らかになる
(Quite possible I'm wrong, so wait a bit to see if anybody else jumps in!)