Weighted Average from Errors This should be a really easy problem, but I want to know the right way to do it instead of making something up.
Data:
Given the data set ( the unit is nano meters ):
$$
D = \{ 3386, 3290, 3372, 3450 \}
$$
The following is the respective errors:
$$
\sigma_D = \{ 50, 180, 42, 100 \}
$$
Question:
How do I find the weighted average?
 A: On the contrary, this is a hard problem.  You need to have a model for the distribution of the measurement errors.  If they are normally distributed, so your measurement of $3386 \pm 50$ represents a normal curve with mean $3386$ and standard deviation $50$ you should weight each measurement with the inverse of its error squared.  This will give you the most probable value.  Unfortunately, the normal distribution is often a terrible model as the tails are so small while measurements that are a ways off are rather common.  If the measurements are like a meter stick, so the correct value has a uniform distribution through the range mean-error to mean+error you can just find the interval consistent with all the measurements.  Here the minimum is $3350$ coming from the last measurement and the maximum is $3414$ coming from the third.
A: This problem is related to data reconciliation.
The most probable value $Y$ is given by the minimization, with respect to $Y$, of
$$\Phi=\sum_{i=1}^n \left(\frac {Y-D_i}{\sigma_i} \right)^2\implies \frac{d\Phi}{dY}=0\implies Y=\frac{\sum_{i=1}^n \frac{D_i} {\sigma_i^2} }{\sum_{i=1}^n \frac{1} {\sigma_i^2}}$$ Applied to the given numbers $Y=3382$.
You could be interested by this paper.
