As the first two posters noted, individual definitions of 'community' and 'great public school system' are going to determine whether there's a correlation.
Since you seem personally interested, I would suggest that what matters most is how *you* define community and how your community can support the districts near you/ of interest to you.
For me, I'm with the first poster in that my definition is one in which folks in the community care about improving schools, whether they have kids at a given school, or even have kids at all!
If you know the name of a district you want to check out you can always enter its name on this page www.greatschools.net/modperl/go/CA under 'Compare schools in a district' and then select your state to search. Good luck!

Not sure what official studies have been done on this topic but I just read a great book that compares middle class educational attitudes/success to working class/poor educational attitudes and success. It was sooo interesting! Unequal Childhoods by Annette Lareau. Different classes look at education in different ways. Schools have expectations based on middle class values so middle class kids fare better overall. I have my own ideas about what makes a school great and in my version it would be a community that is involved with the school. Where community members/professionals are involved and support the school even if they don't have kids at that school. It may be the pizzeria who donates pizza to a school event or a mechanic who assists in a project on model cars or a grocery store that lets kids take a behind the scenes tour.

We actually don't have an agreed upon definition of what a great school system is. If you ask people what they mean by a great school system, you might get some different answers.
We don't even know yet how to really measure student achievement - if students do well on their SATS, does that mean they've learned a lot? If they do well on the state tests, does that alone mean it's a great school system?
What we do know is that students from affluent homes do much better in school than students of poverty. A student's socioeconomic status correlates to school success. School districts in affluent communities show higher scores on standardized tests than schools in districts that are less well-off.

I see your question is categorized for Boston, and while I'm not from MA, I have heard of studies which show a correlation. I'm sorry I don't know of specifics off the top of my head. This isn't exactly what you're looking for, but it might give you some data: community.greatschools.net/advice/184/Helpful-websites-when-seeking-info-about-HOW-SCHOOLS-COMPARE-STATE-TO-STATE-