# Error propagation in least squares

1. Jul 14, 2009

### refrigerator

I am doing a calculation involving taking three or more temperature measurements and then plotting them against another quantity (dependent). I get a relationship that is pretty linear, so I take the line of best fit to obtain an equation with a slope and an intercept.

Now, my question is: how do you calculate error/uncertainty in the resulting slope? I have looked around the Internet and found ways to calculate error purely on the distribution of points, but I am rather looking for error caused by uncertainties in my measurements. For instance, if my temperature readings are good to 0.1K, how would that factor into the uncertainty of the slope? (I have previously used software like IGOR Pro that I think calculated those values for me, but I want to know how it is done.)

Should I, for example, take the worst cases (highest and lowest possible slopes based on measurement uncertainty) and take the difference as the error? Or is this a bit pessimistic? (A formula would be great, I could understand it from there.)

Thank you.

2. Jul 14, 2009

### CFDFEAGURU

How do you know that you have the best fit? I recommend a Chi Sqaured Analysis. See "Probability and Statistics for Engineering and the Sciences" by Jay L. Devore. That should point you in the right direction.

Hope this helps.

3. Jul 14, 2009

### refrigerator

Hmm, I currently do not have access to any books, so an electronic resource would be preferable. But I will make sure to take a look at that book as soon as I get access to a library. Looks like it has some useful information.

I am looking for error propagation. I would like to determine a fit through the least squares technique, and then determine probable error in slope based on uncertainty in my readings. I have done further searching, and found this:

http://www.ghiweb.com/cap/Lab114115/App%20B%20-%20graphing/appB%20Slope.htm [Broken]

Here, they take the average of the worst possible slopes as the error in slope. This is sort of similar to what I originally thought I might do. I suppose this is a reasonable definition, but before I move on does anyone know of any other ways of estimating error?

Thank you!

Last edited by a moderator: May 4, 2017