Check again in the Add/Remove for anything Norton. If you don't find anything there then you definitely should do a file search on the computer for anything Norton related and delete them
Go to Start, Search, Files and Folders, also be sure to click Advanced Options and put checkmarks in Search System Files and Folders, Search Hidden Files and Folders and also in Search Sub Folders.
Have it search in "C" drive first for Norton. If anything is found, delete it. Then do the same for Symantec and delete all that is found. If you like AVG8 that is fine.
Judy