I was wondering how is the 6mth libor index arm calculated? I know a margin is added to the libor but my question is, how is the 6mth libor index calculated? Do they take the average of 12mths and get index?

The 6 month LIBOR is a published index. The values changes with market forces. You loan rate is some margin (value) over that index. Your mortgage note will define exactly how your rate is calculated. Usually it is the index value at some stipulated date plus the margin.
There is one index that does represent a 12 month average (12MT) but its not based on the LIBOR