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).
Anyway, what does "sometimes you can bullshit people with bad math" have to do with progressive-vs-flat taxation?
> The 10 boxes left have 149 on average in them
> 5 of these boxes have 352 on average
"A portion weight more than the whole." -> "Uh, no."