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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

[tex]

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}

[/tex]

and when dealing with a large number of measurements (normally when the experiment has computerised data acquisiton) I use the standard error:

[tex]SE = \frac{s}{\sqrt{n}} [/tex]

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):

[tex]s=\sqrt{\frac{1}{N-1} \Sigma^{N}_{i=1} (x_i - \bar{x})^2}[/tex]

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.