intermediate school vs middle school

intermediate school

noun
  • A school for children that serves grades between elementary school and high school, and occasionally between the former and middle school. These grades vary from anywhere from grade 4 to grade 8. 

middle school

noun
  • A school which crosses the traditional divide between primary school and secondary school. 

  • An educational level or stage between primary school and secondary school. 

How often have the words intermediate school and middle school occurred in a corpus of books? (source: Google Ngram Viewer )