I think you're reaching for the wrong statistics. You want averages of subgroups, rather than medians or percentiles which tell about about where boundaries lie. In my original Elbonia example, knowing "There are peasants and nobles and the poorest noble has an income of $90" wouldn't give you enough information to work with.
The first chart shows an ascending staircase of tax-rates as each group has a higher average income (not shown) than the prior group, indicating a progressive tax scheme.
With a flat tax, every bar would be the same moderate height. We don't automatically have enough information to say exactly where the horizontal line would be, but clearly it has to be somewhere between today's "Bottom 50%" and "Top 1%", meaning those groups would see a tax-hike and a tax-cut respectively.
________
Another approach is the next chart, "High-Income Taxpayers Paid the Highest Average Income Tax Rates".
Much like my "many peasants" and "few nobles" groups, this chart has 6 groups. It shows the smallest group, the "top 1% of people" took in 22% of the taxable income, and paid 40% of the taxes, and so on down the line for the other chunks.
Under a flat tax, any group with X% of the income would also pay X% of the taxes. In other words, the right-hand stack would shift to look like the left-hand stack.
Knowing this, you can tell which groups' "taxes paid" boxes would grow (tax hike, boo) and which which groups' would shrink (tax cut, yay).
hahaha, I forgot to mention I totally agree with what you said.
> You've gone off the rails somewhere.
I'm glad you noticed :)
Ill demonstrate the jedi mind trick one more time...
Imagine 100 boxes, we ignore 90 of them.
The 10 boxes left have 149 on average in them.
if they had exactly 149 each it would be 10x149=1490
However, we are told 5 of these boxes have 352 on average.
How much is in the remaining 5 boxes?
If the boxes told about had exactly 352 each it would be 5x252=1760
If the remaining 5 boxes are empty the average would be 1760/10=176
176 is more than 149
See?