# Chi-squared fit with errors on both x and y

• I
Malamala
Hello I have some data points which have errors on both x and y coordinates. I want to fit a straight line to them but I am not sure how to take the error on x into account. Normally, when I have just the error on y, I want to minimize $$\sum\frac{(y_{pred}(x)-y_{measured}(x))^2}{\sigma_y^2}$$
Can I just replace ##\sigma_y^2## with ##\sigma_x^2+\sigma_y^2##? The errors on x and y are not correlated. Thank you!

Staff Emeritus
• WWGD
Mentor
It is also called orthogonal distance regression.

Staff Emeritus
It is also called orthogonal distance regression.

Yes. You start with the obvious thing - a line y = mx + b, and you try and do a least-squares fit using the perpendicular distances between the points and the candidate line instead of the y-distances. Problem is that doesn't always get you a unique unbiased solution.

That's why you need to specify what you are looking for very carefully.

• WWGD and Dale
Staff Emeritus
Even though this appears to be a drive-by posting, I'll make one more comment.

If you minimize a function of Δy only, it's clear what you are doing. If you minimize something like Δx2 + Δy2 it's not even guaranteed that you have a number with consistent dimensions: suppose y is temperature and x is time. What units would Δx2 + Δy2 even be in?

To get a well-defined answer, one needs to pose a much, much better defined question. And even then it may not exist.

• WWGD
Gold Member
Even though this appears to be a drive-by posting, I'll make one more comment.

If you minimize a function of Δy only, it's clear what you are doing. If you minimize something like Δx2 + Δy2 it's not even guaranteed that you have a number with consistent dimensions: suppose y is temperature and x is time. What units would Δx2 + Δy2 even be in?

To get a well-defined answer, one needs to pose a much, much better defined question. And even then it may not exist.
Maybe if you standardize your variables you can avoid the issue with units? I understand that is one if the reasons for standardization.

Malamala
Maybe if you standardize your variables you can avoid the issue with units? I understand that is one if the reasons for standardization.
What do you mean by this?