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I'm working a problem, attempting to find a income tax rate that will change depending on the gross paycheck amount. Some data points:

$800 gross = 11% taxed

$1500 gross = 16% taxed

$2000 gross = 20% taxed

How would I go about finding the relationship between the gross paycheck amount and the percentage taxed? Beyond that as well, how will I then be able to apply that relationship mathematically in order to obtain other data points (\$1000 gross paycheck, $500, etc...) and get an accurate tax percentage for them as well? I have a feeling it'll be an answer lying somewhere in the statistics field, but I'm no math expert.

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Your problem has no unique solution, there is an infinite amount of tax regulations that will generate your output. – Listing Nov 26 '11 at 15:32
Well I'm not looking for something that's exactly perfect in it's output, but plotting the data points it does seem to follow a trend. So hopefully from that I can ascertain a polynomial that will give me accurate answers within a given percentage. – Mike S Nov 27 '11 at 13:37
It is possible to find a polynomial that returns exact values for finite number of points. If the relationship is truly a polynomial of degree n (or less), then every thing is fine. However, if the polynomial is of a degree > n, then any interpolation or extrapolation, may not provide an accurate value. – NoChance Jan 26 '12 at 2:29

As Listing says, there is no unique solution. One approach would be to plot tax paid against gross income, draw a smooth curve and estimate any interpolated values.

enter image description here

So for a gross income of $\$1000$, tax might be about $\$130$ i.e. about 13%.

If you are feeling brave, you might try to extrapolate outside the points you know (the tax on a gross income of $\$0$ might be $\$0$), but you are more likely to be wrong.

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The curve is looking exponential, so I should have enough data to solve for a regular polynomial I would assume. That may give me an answer that will help me at least estimate within a given means of accuracy. – Mike S Nov 27 '11 at 13:35
@Mike: an exponential curve would result in tax paid being above gross income, which is rare and difficult to collect. – Henry Nov 27 '11 at 23:26
You have an excellent point. It must be painfully clear I really have no idea what I'm talking about. – Mike S Nov 28 '11 at 14:46

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