[Mrs. Don-o]

Not to be to-o-o-o picky, but I think there are three kinds of averages: mean, median, and mode. So if we agree that when "half are above, half are below" it's the "median", it's still a type of average.

Dictionary.com (LINK) defines "mean" as an average, and then says, "Two other common forms of averages are the median and mode."

Though a lot of people do use "average" to mean "arithmetic mean" only.

[justlurking]

Not to be to-o-o-o picky, but I think there are three kinds of averages: mean, median, and mode.

As a colloquial word, average has been overloaded to mean a lot of things. In the scientific, statistical, and mathematical world, it's the arithmetic mean.

Even your own citation says that "average is a synonym for arithmetic mean".