The treatment of errors and uncertainties?

[alexgmcm]

I am confused over how to work with errors and uncertainties.

So far when dealing with a small number of measurements I have used the partial derivative method to calculate the final error in my result e.g. if my result is
$$E=mgh \text{ then assuming no error in g my uncertainty is } \Delta E = \sqrt{\left|\frac{\delta E}{\delta m} \right| ^2 \cdot \Delta m ^2 + \left|\frac{\delta E}{\delta h} \right| ^2 \cdot \Delta h^2}$$

and when dealing with a large number of measurements (normally when the experiment has computerised data acquisiton) I use the standard error:
$$SE = \frac{s}{\sqrt{n}}$$
where n is the number of measurements and s is the standard error (calculated using Bessel's correction which makes it work for smaller N by some mathematical trickery):
$$s=\sqrt{\frac{1}{N-1} \Sigma^{N}_{i=1} (x_i - \bar{x})^2}$$

This has always seemed strange to me as N is therefore usually the same as n which just seems weird. I guess I'm doing it wrong but I am not sure how?

How should errors be treated both in the case when you have a small number of measurements of each variable a statistical approach is impossible, and when you have a large number of results and a statistical approach is more attractive?

Any help or advice would be greatly appreciated,
Alex.

 

[mathman]

I can't quite understand what you are asking. However: s is an estimate of the standard deviation of a single measurement, while SE is an estimate of the error in the average. Note that for large n, s -> σ (the theoretical standard deviation), while SE -> 0.

 

[Andy Resnick]

I am confused over how to work with errors and uncertainties.

What you wrote is appropriate for independent errors; that is the error in 'm' is independent of the error in 'h'.

By far the best book to learn from (at least, the best book I have seen so far) is John Taylor's book

https://www.amazon.com/dp/093570275X/?tag=pfamazon01-20

 

[alexgmcm]

However: s is an estimate of the standard deviation of a single measurement, while SE is an estimate of the error in the average. Note that for large n, s -> σ (the theoretical standard deviation), while SE -> 0.

Thank you! This finally made me understand that confusion I had. Now it all makes sense, I think so far in my experiments I am justified in assuming uncorrelated (i.e. independent errors) and so therefore need not worry about covariance.

 

[f95toli]

The official way of dealing with errors/uncertainties in measurements is to use whatever method is recommended by the GUM (which is a free document published by the JCGM). Note that by "official" I really mean "The method you must use in order to comply with X" where X would be most organizations/regulations/guidelines regardless of where in the world you are.

See link 2 on the following wiki page
http://en.wikipedia.org/wiki/Measurement_uncertainty#cite_note-GUM-1