This is highly contentious depending which taxes (and benefits) are included. The wiki page only includes federal income tax offset by ("in-kind benefits from social insurance and other government assistance programs") from a CBO report fig 3 (nominal tax) vs fig 6 (offset by benefits). And they find a huge gap, with the lowest bracket having a negative effective federal tax after benefits are subtracted.
On the other hand, Piketty et al. aggregating multiple tax sources "federal, state, and local levies—including corporate, property, income, estate, sales, and payroll taxes" conclude that the gap is much smaller:
Also the category of "rich people" is pretty vague. The top 1% may be too broad for that. The effective tax rate actually goes down as we subdivide that:
From WaPo's Christopher Inghram (2015), who is critical of this state of affairs:
In other words, a person in the top 0.001 percent income bracket -- who would have an adjusted gross income of at least $62,000,000 -- pays the nearly same effective tax rate as somebody in the top 20 percent bracket who makes $85,000 in adjusted gross income.
That's not how federal income taxes were, at least originally, designed to work. The super-rich pay a relatively low rate for a variety of reasons. They benefit from a whole host of deductions -- like the mortgage on a yacht, for instance -- and other tax benefits that many people don't qualify for.
Chief among these is the lower tax rate on capital gains -- think investment income. That maxes out at about 24 percent when you factor in a Medicare surtax that applies to some investment income. But wages are taxed at a top rate of 39.6 percent. Since many of the super-rich get most of their earnings from investments, they disproportionately reap the benefits of that lower capital gains tax rate.
In the year this data was compiled, 2012, the top capital gains rate was lower still, at 15 percent.
That graph probably has changed somewhat post-2013 reform. For 2014:
The top 1 percent of taxpayers (AGI of $465,626 and above) paid the highest effective income tax rate, at 27.2 percent, 7.9 times the rate faced by the bottom 50 percent of taxpayers.
Taxpayers at the very top of the income distribution, the top 0.1 percent (with AGIs over $2.14 million), paid an even higher average tax rate, of 27.7 percent.
And taxes were slashed by Trump in 2017 with "pass-through" companies supposedly mostly benefiting the rich.
Individuals in the bottom 80 percent earn virtually no pass-through income. Moreover, those with higher incomes tend to receive a much greater share of their income from business compared to those with lower incomes, as the top 1 percent only earn about 11 percent of wage and salary income. Thus, any reductions cuts in the tax rate on pass-through businesses would largely benefit high-income taxpayers.
According to a Bloomberg article:
The new law lowers the top individual rate to 37 percent from 39.6 percent. With the new 20 percent deduction, pass-through owners taxed at the top rate can now get their rates as low as 29.6 percent.
Critics say this creates an incentive for top earners to recast themselves as independent contractors and funnel wages taxed at ordinary rates through a pass-through entity. [...]
The law starts phasing out the pass-through deduction once the net income of an owner in one of those professional fields hits $157,500, or $315,000 for joint filers. Once their income hits $207,500 -- $415,000 for joint filers -- the deduction disappears altogether.
So the calculations would change again. I haven't found an updated version considering this. In that March 2018 Bloomberg piece, various experts were chiming in how easy it would be for the rich to [ab]use the new law.
[Proposed] regulations accompanying the new law have only been recently announced as of Aug, 2018, which probably explains some uncertainty of their impact.
So it's difficult to answer this question in general, without fixing a year or deciding who is the "rich".