Editing Weighted arithmetic mean (section)

==== Simple i.i.d. case ====

When treating the weights as constants, and having a sample of ''n'' observations from [[Uncorrelatedness (probability theory)|uncorrelated]] [[random variables]], all with the same [[variance]] and [[Expected value|expectation]] (as is the case for [[Independent and identically distributed random variables|i.i.d]] random variables), then the variance of the weighted mean can be estimated as the multiplication of the unweighted variance by [[Design effect#Unequal selection probabilities|Kish's design effect]] (see [[Design effect#Assumptions and proofs|proof]]):

: <math>\operatorname{Var}(\bar y_w) = \hat \sigma_y^2 \frac{\overline{w^2}}{ \bar{w}^2 } </math>

With <math>\hat \sigma_y^2 = \frac{\sum_{i=1}^n (y_i - \bar y)^2}{n-1} </math>, <math>\bar{w} = \frac{\sum_{i=1}^n w_i}{n} </math>, and <math>\overline{w^2} = \frac{\sum_{i=1}^n w_i^2}{n} </math>

However, this estimation is rather limited due to the strong assumption about the ''y'' observations. This has led to the development of alternative, more general, estimators.