We have n values of and error ,
With a weighting , the uncorrelated weighted mean and error is
when combining data, the weighting is
and the weighted error becomes
we measured a quantity n times, we can assume the intrinsic error of the data is fixed. Thus,
Therefore, when we take more and more data, the error is proportional to .
In normal distribution, the sample of size n, the estimator of the sample mean and sample variance are
Don’t mix up the sample variance and intrinsic error, although they are very similar.
To explain the formula of the weighted variance, we have to go to the foundation of the algebra of distribution.
For a random variable follow a distribution with mean and variance ,
Another random variable built on it,
The derivation is very simple, in this page.
The adding of two independent random variables is
But there is a catch, when the and , The rule does not apply. But lets look back, if the mean and variance are the same, the two distribution does not really independent.