The best interpretation of the word "average" is "representive". This is the way it is used in the label, often written by the media, of "the average man". The word "average" derives from an ancient Lain word, "havaria". meaning "a share". In ancient times, merchants (who shipped goods over The Mediterranean) suffered losses due to rats or thieves when in port; due to pirates at sea; or due to goods being thrown overboard during storms to stabilize the ship. To compensate for such losses, groups of merchants shipping on a given ship would agree that any of their group who suffered losses would be given a share from each of the others. It was a primitive form of insurance.

Here is what is said about average on p. 58 of Theory and Problems of Statistics, by Murray R. Spiegel of Renssalaer Institute, in The Schaum's Outline Series, of McGraw-Hill, NYC:
"An average is a value that is typical, or representative, of a set of data. Since such typical values tend to lie centrally within a set of data arranged according to magnitude, averages are also called meaures of central tendency. Several types of averages can be defined, the most common being the arithmetic mean, the median, the mode, the geometric mean, and the harmonic mean. Each has advantages and disadvantages depending on the data and the intended purpose."

The word "mean" derives from a Greek word translating as "middle". In ethical discussions, the word "mean" -- as in "The Golden Mean" advocated by the Greek philosopher, Aristotle (384-322 BC) -- is associated with the idea of "being moderate in all things".

I have ONLINE a problem in which the average is not the average. A government office in which the average salary is $30,000 a year, but two-thirds of the employees earn below the average. The office has only three members: the office head earned $50,000 a year; a secretary earned $25,000 a year; the receptionist earned $15,000. The arithmetic mean (usually taken to be th average) is 1/3($50000 + $25000 + $15000) = 1/3($90,000) = $30,000, which places two-thirds of this population below this "average"! So, in this case, the arithmetic mean is not a proper average, since it does not represent this population.

However, the median salary is $25,000 a year, which places one-third of the population members above this average and one-third below, so it properly represents.

I constructed a small problem to make comprehension easy. But a case of millions of people in which a smal fraction have a disproportionately great income would allow th arithmetic mean of income to have the same distortioning effect.

This is why (as we note in the associated file at this Website on "Conseuqences") that we should use median as the average for income and other fiduciary data.