A Fix for the Rescaling Problem Using Transverse Least Squares

 One can eliminate the dependency of the transverse least squares fit on units of measurement by using z-scores defined as the deviation of the variables from mean values rescaled by the standard deviations, a measure of the spread of the data.

Here's some data to compare this modified method with standard least squares.

The fits can now be done using the z-scores ξ and η instead of x and y.

The comparison shows that the transverse least squares appears to give a better value for the slope of the line.

One can then use the formulas above to convert the fit lines back into the x,y plane.