I went to a completely ALL WHITE school from K-2nd grade. This is a private school (Desoto)in Helena, Arkansas. This is a K-12 school and my yearbook from 1983 when I was in Kindergarten shows no black students, faculty or staff. Even the janitorial crew was an older white couple. Until I was probably 12 years old... I thought “Private School” meant no blacks allowed.
In the South, it often did. Private schools sprang up overnight as Lyndon Johnson, working through his bullying satrap Leon Panetta, threatened school officials throughout the South with imprisonment right now if they didn't desegregate their schools right now.
The private schools were legal, often hired teachers who'd left the new Yankee-run school regime (Southern States and counties totally lost control of their school systems during this period, everything was federalized under a cafeteria-style intervention by USG), and were referred to by the sneering liberal mot du jour, "segregation academies".
Nobody has ever tested to see whether those "segregation academies" outperformed their captive public counterparts, which generally declined from the late 60's onward.