I know that generally the uncertainty in the mean of a sample should be equal to:
$\frac{V_{max} - V_{min}}{2} $
where $V_{max}$ is the maximum value and $V_{min}$ the minimum value of the sample of data. However, what if each value has its own uncertainty? For example, I have to values:
$R1 = 12.8 \pm 0.2$ m
$R2 = 13.6 \pm 0.4$ m
The mean would be $13.2$ m, but what about the uncertainty? Will be it be the range $1.4/2$ or will it be the combined uncertainty of each measurment?
If you have two uncorrelated quantities $x$ and $y$ with uncertainties $\delta x$ and $\delta y$, then their sum $z=x+y$ has uncertainty
$$\delta z = \sqrt{(\delta x)^2 + (\delta y)^2}$$
The average would then have uncertainty $$\frac{\delta z}{2} = \frac{\sqrt{(\delta x)^2 + (\delta y)^2}}{2}$$
Intuitively, one might imagine that
$$\delta z = \delta x + \delta y$$
However, this overestimates the uncertainty in $z$. If $x$ and $y$ are uncorrelated, then it is very unlikely that their errors would constructively add in this way. It is of course possible that $x$ and $y$ are correlated, but then more complicated analysis is required.
