Greater precision with regression?

Question

I used my TI-83 to find the quadratic regression of two data columns. The accuracy wasn't close at all. So I tried cubic and then finally quartic regression. The accuracy still isn't close enough. Is there any way I can improve the accuracy? I can't add any more data, unfortunetly.

Edit 1:

To address Henry's comment, my plotted data looks similar to y=x^(1/3)

Answer 1

This may mean your data is poorly fit by a polynomial. Three ways it can fail is either to be exponential, to have poles, or be like a sine wave. Plotting the data with the fits you have overlayed may give you some ideas. If it is exponential, taking the logarithm of one will render it linear (or close to polynomial). If it has poles, using a rational function will help. Procedures are given in chapter 3.2 of Numerical Recipes. The obsolete versions are free. If it is a sine wave, an FFT can help-chapter 12 of NR.

Answer 2

If, as you say, it looks like $y=x^\frac{1}{3}$, then you might try a power regression, which will give you a formula of the form $y=ax^b$.