# Hall Effect Measurement Error Analysis

• Mindscrape

#### 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?

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.

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.