Hall Effect Measurement Error Analysis

[Mindscrape]

I did an experiment on the Hall Effect, and found the voltage for the Hall "probe" (it was a strip of bismuth) as a function of a current magnetizing the B-field. Anyway, I did a least squares fit to find the regression line with A^T A = A^T b, and found four lines for each of my four data sets. I also found the uncertainties on the each of the four with $$\sigma = \sqrt{\frac{1}{N-2} \sum_{i=1}^N (y_i - A -Bx_i)^2}$$.

My question is that if I find the average regression line, using $$\sum_{i=1}^N \frac{y_i}{N}$$, would I simply take the mean of errors since they will all be dependent on the same factors? Should I add the errors in quadrature since they were all errors would be random, and dependent on different random factors?

 

[marcusl]

You are asking the right questions! The answer depends on how you made your measurements. Are the uncertainties uncorrelated? If one is a voltage and another is a dimension, for instance, then you can assume no correlation = rms. If they are related, for instance the same meter is used separately to measure current and voltage, and there's a systematic error (meter calibration is off), then you might need to sum the errors.

 

[Mindscrape]

They are correlated, though I haven't actually done a correlation test, because all I did was increase the magnetic field until the galvanometer measuring the voltage hit its sensitivity peak, and then I turned the current back to zero (to get rid of remanent magnetization) and did the test all over again.

At the same time, I could see how since any error on the measurements will be a random error, as any systematic error would repeatedly show in every test, why they would be uncorrelated. Random events can't really correlate together.

Ultimately, I guess that simply adding the uncertainties will make the most since though.