Mean vs Median
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The "average" or "mean" is a measure of central tendency. It's one of 3 measures of central tendency, and is the most popular one.
The "average" of a number of quanties, such as incomes, is usually taken to refer to their sum, divided by the number of incomes that have been summed.
In other words, the average of 5 numbers, N1, N2, N3 and N4 + N5 is:
(N + N2 + N3 + N4 + N5)/5.
That average is also called the "mean" of those numbers.
Another popular measure of centreal tendency is the "median".
The median of a set of numbers is a number such that half of the numbers are above it, and half are below it.
Of course that means that, if the number of numbers is odd, the median is the middle number. And if the number of numbers is even, then the median (unless I'm mistaken) is the average of the middle two numbers.
If the numbers are 1, 2, 3, 4, and 5, then the median is 3, because half of the numbers are more than 3, and half of the numbers are less than 3.
If the numbers are 1, 2, 3, 4, 5, and 6, theln the median is 3.5, because the middle two numbers are 3 and 4, and the average of 3 and 4 is 3.5
The median is popular, for descriptive purposes, because it's more stable, in the sense that it is less affected by large extreme numbers being added to the set of numbers. But that doesn't make it better for all purposes. For many purposes, the mean is more relevant. I claim that taxation is such a purpose.
It seems to me that I've read somewhere that it isn't incorrect to call the median a kind of "average". But that's probably unusual. When people say "average", they pretty much always mean "mean". Still, for precision, it's probably better to say "mean", when that's what you're referring to.
Another central tendency is "mode". In a set of numbers, the mode is the most frequently-occurring number. For example:
If the numbers are 1, 2, 3, 3, 4, and 5, then the mode is 3, because 3 occurs more times, in that set, than does any other number.
For tax purposes, I suggest that mean is the more meaningful of the centreal tendencies, because of its close relationship to the sum. Because the sum of the incomes has relevance, I suggest that the mean is the most relevant central tendency for tax purposes.
The mean, median and mode are the 3 main and familiar measures of central tendency.