I noticed these types of problems occurred often, I wanted to come up with some properties, can someone confirm/disprove them?

Rules: All numbers in set are integers

1) For set of consecutive items, if number of item is odd => average is integer

1a) The reverse is not true, average is integer, number of item is odd, could be consecutive or not.

2) For set of evenly-spaced items, if number of item is odd=> average is integer (item 1 is a subset of 2)

2a) The reverse is not true.

3) For set of consecutive items, if number of item is even => average is NOT integer

3a) Reverse is not true

4) For set of evenly-spaced items, if number of item is even=> average could or could not be an integer

4a) For set of evenly-spaced items, if number of item is even=> average is an integer if the difference between the items are even.

4b) For set of evenly-spaced items, if number of item is even=> average is not an integer if the difference between the items are odd.