It's not just public/private school issue. For example here in Lithuania public schools may be specialised too. Science, arts, humanities... Quality does vary a

It's not just public/private school issue. For example here in Lithuania public schools may be specialised too. Science, arts, humanities... Quality does vary a bit too. Quality of classmates vary more than quality of education, but it's essentially part of the same package. E.g. some schools are notoriously bad for historical reasons. When gentrification just starts and better-off people move in, many of them try to send their kids to schools in other neighbourhoods.