Suppose I have a voltage versus current graph with various "peaks" in current at given voltages. For example, this is what is seen in a Franck Hertz experiment.
I can measure the value of voltage at the center of each peak, $v_i,$ and assign some uncertainty to this $\delta v_i.$
Now I want to consider the spacings between the peaks. $$\Delta v_i \equiv v_{i+1} -v_{i}.$$ The associated uncertainty here is $$\delta(\Delta v_i) = \sqrt{(\delta v_{i+1})^2 + (\delta v_{i})^2}$$ Finally, I want to consider the average spacing between the voltages. $$\bar{\Delta v} = \frac{\sum{ \Delta v_i}}{N}$$ The uncertainty here would be $$\delta(\bar{\Delta v}) = \sqrt{ \Sigma (\delta(\Delta v_i))^2 }.$$
However, I can rewrite the equation for the average, $$\bar{\Delta v} = ((v_2 - v_1) + (v_3 - v_2) + ...)/N = (v_{N+1} - v_1)/N$$ This would imply that the uncertainty for the average is just $$\delta(\bar{\Delta v}) = \sqrt{ (\delta(v_1))^2 +(\delta(v)_{N+1})^2}/N.$$
How to reconcile these different results?
In addition, what is there to be said about using the standard deviation in $\Delta v_i$ as opposed to the error propagation formula? Why is the error propagation formula "better?"