I never quite got the hang of this during my entry stat course and it has been bugging me for a long time now.
Lets assume I'm trying to find the focal length of a concave lens. Using a mockup formula for f, lets say:
f = D*x/y
where x , y, and D are measurements done with calipers with "precision" of 1mm each.
Now I continue my experiment and make 10 different measurements for x, and 10 for y and compute 10 values for my f.
I want to present one final value for f and an "error" associated with that value.
Do I use the mean value of my 10 f's as my final value? what do I use as a representation of "error"? Do I use the STD of mean for my 10 values or do I propagate the 1mm errors from my measurement and then use them to compute a weighed mean with associated error?
Best Answer
Right, at the first-year undergraduate level, this is how you typically do it.
where
Now, there is a better way to estimate the uncertainty of f. This is done by taking multiple measuremenets instead of only 1 measurement. By taking multiple measurements, we can obtain the standard deviation and subsequently the standard error (aka standard deviation of the mean). There is a caveat: the minimum number of measurements that should be carried out to utilize the standard error is N >= 5.
Now, there is still the uncertainty of the instrument in hand! (Precision of 1mm as you stated). In the first-year undergraduate level, this instrument uncertainty is typically ignored (when taking multiple readings). However, if there is a large discrepancy between experimental and theoretical value, this uncertainty can be taken into account. This is done by considering the instrumental uncertainty as a form of systematic error.
Therefore the best way to describe an experimental value in this case would be