Array formulas would help here. Basically the idea is that you multiply
each element of the temperatures array (assumed C1:C10) with a logical
expression evaluating to 0 or 1. However, the operation must be done
for each element of the array, hence they must be entered using
Shift+Ctrl+Enter

=AVERAGE((C1:C10)*--(A1:A10="My Site"))

You can use this technique for all your aggregate functions with the
exception of MIN. Here, multiplying positive temperatres with 0 would
make 0 the minimum. In this case you can use something like:

=MIN(C1:C10+IF(A1:A10="My Site", 0, 1000))

HTH
Kostis Vezerides