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$$\sigma_f(x,y,z)=\sqrt{\left| \frac{\partial f}{\partial x} \right| \sigma_x^2 + \left| \frac{\partial f}{\partial y} \right| \sigma_y^2 + \left| \frac{\partial f}{\partial z} \right| \sigma_z^2}$$

But I don't know what to do when you have many values for each (x,y,z).

Suppose you measure m and x in a spring to calculate the elasticity constant via k(x,m)=mg/x.The variables m and x are measured with an intrinsic error of the apparatus. You repeat the measurement several times leaving the variables fixed. The error could be combined using the cuadratic sum.

Now, you change m, so x varies too. You repeat the same process above. This would introduce a dispersion in k.

How is the error of k in this case? It seems that possible solutions could be Demming regression, total least squares, Monter Carlo methods... but I'm quite lost, a simple example would be helpful.

To give some numbers (totally invented):

Measurement #1: m=10.0,10.1 and x=5.3,5.0.

Measurement #2: m=21.0,20.2 and x=10.4,10.2.