Combining uncertain measurements I have a ball on a table located in position $x', y'$. 
I am using many different rulers to measure the coordinates $x_i, y_i$ of the ball. I do this with $N$ different rulers, so $i = 1\ldots N$. Each measurement comes with an uncertainty $\epsilon_{x,i}, \epsilon_{y,i}$ drawn with mean 0 and uncorrelated but known variances $\sigma_{x,i}^2, \sigma_{y,i}^2$. 
The ball hasn't changed in its location, but I have many noisy measurements for where it is located. How can I combine all of these measurements to give my best guess as to the true coordinates $x',y'$? 
 A: If I'm understanding your question properly, this sounds like you need Inverse variance weighting.
https://en.wikipedia.org/wiki/Inverse-variance_weighting
The estimate of your $x'$ that would minimize the variance (so giving you the "best guess") will be given by 
\begin{equation}
\hat{x} = \frac{\Sigma_ix_i/\sigma^2_{x,i}}{\Sigma_i1/\sigma^2_{x,i}}
\end{equation}
You stated that the uncertainties in your measurements were "iid". If they have different variances, then they are not identical, just independent.
For Inverse Variance Weighting to work, they only need to be independent.
A: And the inverse square of the error on the combined value is the sum of the inverse squares of the individual errors:
$$ \frac{1}{\sigma^2} = \sum_i \frac{1}{\sigma_{x,i}^2}$$
For a derivation, see the section on statistical methods of any experimental physics handbook.
(The fact that you have each measurement has an x and y value doesn't add any complexity; only the x values contribute to the combined x value and only the y values contribute to the combined y value.)
