Mean vs Median

This blog entry is related to the poll:Regarding taxes, what general policies do you favor? (Total: 3 posts)
It is discussing the poll topics:
  • No tax on income below median. Compensate in high brackets (1 post)
  • No tax on incomes below mean. Compendate in upper brackets (1 post)

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.

For example:

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.

Comments

Extreme examples

Thanks for the explanation. Here is a more extreme example, so make the point clearer.

Take the set with the following values:

1, 1, 1, 8, 9, 9, 9

The mean or avergage is 38/7 =~ 5.4.

The median is 8.

And I am not sure what the mode would be, because both 1 and 9 occur 3 times. But, hopefully, the mode is never used for tax purposes ;)

Mean vs Median for tax purposes

Now that the definition of those terms is clearly established, and since we are talking about taxes, we should now consider the implications on the tax distribution.

First of all, have the concepts of 'mean' and 'median' ever been brought up in the context of tax policy? Has any tax code (anywhere in the world) ever included such concepts within the law? If there is any example, it would be interesting to see what the law or the policy discussion says about using the mean or the median.

Within the context of the tax policy discussion, here at minguo, such concepts are interesting because it helps to set specific boundaries. But then, we need to have a better understanding of what those boundaries are. Earlier, you wrote:

Because of the very high incomes at the high end, the mean is higher than the median.

What would be interesting is to have actual figures (even if rough approximations) about each. Do you know of any sources that could enlighten us?

Mean, Median, Taxation

The California Peace & Freedom party had (may still have), in its platform, a provision that there shall be no tax on incomes below the average. "Average" almost always means "mean", and it's safe to assume that that is what the PFP platform referred to.

Calculations showed that such a tax system is entirely feasible, achieving contemporary revenue totals, without undue taxation of barely-over-mean incomes.

PFP, it seems to me, advocated a _greaduated_ system of marginal tax rates. I intended to calculate the revenue of such a system (with no tax on below-mean incomes). But the publishee information was insufficient

So I instead examined a flat-rate system, with the above-mean incomes in one bracket, with one rate.

I found that, with such a system, as I said above, with no tax on below-mean incomes, and a flat-rate on above-mean incomes, current revenue-levels are achievable without unduly taxing barely-above-mean people.

Jerry Brown, in his camaign, advocated a flat-tax, exempting incomes below some level that was less than the mean.

For U.S., 2008, 4th quarter, the mean weekly wage was listed as $918.

That's about $47,900 per year.

(On average, there are slightly more than 52 weeks per year).

For U.S., 2008, "National Income" is listed as $12, 635.20

I don't know the definition of National Income. Maybe total income divided by total population?

For U.S., 2008, average weekly earnings of a production worker is listed as $607.99

That's $31,174 per year.

For U.S., 2008, 2nd quarter, median weekly earnings of workers is listed as $734

That's $38299 per year.

Michael Ossipoff

Fascinating

Thanks for sharing all this.

I am starting to think that beyond the Manifesto, where we write the policies that we want, we'd need a place where we can list such useful background information.

Not the mean or median of _all_ incomes.

Looking this morning, I couldn't find anything about the mean or median earnings, over all the people who have income. All I could find were wages, and workers' incomes.

Maybe the much larger overall mean income, due to the high incomes at top, would be an embarrassment, if published.

Continued

Obviously the _overall_ national mean of incomes (not limited to wages or workers' incomes) is what would be used for a tax system with no tax on below-mean incomes.

I probably found such a figure when I calculated the revenue, but I can't find it right now.

Document sources!

Michael is providing some interesting information that needs to be verified and documented.

Reading this thread, it occurred to me that we'd soon need some kind of wiki to keep track of the most important data and to organize it. I have just created such a wiki:
http://minguo.info/usa/wiki

The purpose of this wiki is to collect any information on policy issues that the community might find useful or interesting. It is complementary to the Manifesto in the way that the Manifesto only contains policies that we have voted on and agree with, while this wiki can contain references to policy papers, policy decisions and proposals as well as supporting data from anywhere in the country (or the world).

The wiki is brand new and fairly empty for now but it will grow.

Now, back on the topic of mean vs. median, if you know of any interesting sources that can provide some insight and examples of policy proposals, you may add them to the following wiki page on taxes:
http://minguo.info/usa/wiki/taxes