Whenever you measure something you usually repeat these measurements, in order to determine not only the quantity you are looking for but also access its uncertainty. Given that you have measured different quantities and derive a new one from a set of these, one will use these formal errors to weight the impact of these quantities. A simple example would be this:
Three groups measure a length together with its formal error. If someone wants to get the average over all three groups, he would compute a weighted mean (based on the formal errors of each group) as a most reliable solution. The problem is only that the information on how the formal errors are computed is not preserved. E.g. one group could compute the standard deviation of their measurements from only three observations during a time when systematic effects were small, but the other groups had much more data to get their results and thus see a larger formal error coming from systematics. However, the results based on the three measurements will get a much higher weight in the combined solution although it is basically less valuable than the other two results.
In order to compensate for such effects one can add a certain amount of uncertainty, i.e.
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I am going to demonstrate this by a quite illustrative example. I took four randomly distributed 2D points and then computed the weighted mean of all points by applying formal errors which I took from a normal distribution generator (with zero mean, and a standard deviation of one). I.e.

For each Monte-Carlo run I computed 300,000 points and I changed the additive noise level between 0 and 2 for each run. Here is a small animation looping through the different noise levels.

For low additive noise the points tend to be inside the area spanned by the four points (if you look at the density plot on the right you see that it is slightly more likely that a point is on one of the diagonals than on another place). However if you start adding noise, the original data weights become less important and the point are coming closer to the “truth”, i.e. the unweighted average of all four points.
This small post does not show anything surprisingly new, but I think it demonstrates quite clearly how one can “tune” a solution based on different data weights into totally different states.
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