How Do I Find the Best Fit for Parameters in Equations with Experimental Errors?

[Hepth]

I have 3 equations with two unknowns of the form:

$$r_i == f_i (g,\beta)$$

Where g and $\beta$ are the independent unknown variables and $r_i$ are known experimentally, but have some error associated with them, say something like $4.3 \pm 0.3$ and there are other errors associated with constants inside the $f_i$.

My question is how do I solve for the objectively "best" fit to those two parameters. I remember doing something like this back in prob/stats, I believe I do something like a $\chi^2$ fit but I can't find anything online about it that doesn't concern bins of data points, but rather continuous functions. (Or do I have to simulate data, then fit it)

Can anyone lead me in the right direction? I don't have any of my undergrad books with me.

The equations are all something like :
$$r_1 =\frac{\left(g^2 x+g^2y-\frac{\beta }{3}\right)^2}{g^2 }$$
with some variations on the form. So nothing with unsolvable functions.

EDIT: Oh and I plan on using Mathematica to do this, but I'd rather understand the mathematics first.

 

[Stephen Tashi]

My question is how do I solve for the objectively "best" fit to those two parameters. I remember doing something like this back in prob/stats,

There is no mathematical definition for "objectively best fit", so you have to supply the details. (You aren't alone. If you browse the posts in this section of the forum you find person after person asking for the "best" solution to problems. Most are not interested enough to supply a definition of "best" and a few seem downright offended to be asked for it.)

If you remember something from statistics, it probably involved data. Do you have the data that was used to supply the "known" parameters in these equations? If you don't have the data, do you know something about its distribution?

I can imagine a solution based on an arbitrary definition for "best fit" and imagining that you have data. You can define the best value of the unknowns as the one gives the best least squares fit of the equations when they are plotted versus the data that was used to estimate the constants in the equations. However, you'll get better advice if you use all the details of the real world problem that you are trying to solve, rather than solving some abstract carciature of it.