selection probability (i.e.: the inflation factor). I.e.: . Variance of the weighted sum (pwr-estimator for totals) If the population size N is known we can estimate the population mean using . If the sampling design is one that results in a fixed sample size n (such as in pps sampling), then the variance of this estimator is: An alternative term, for when the sampling has a random sample size (as in Poisson sampling), is presented in Sarndal et al. (1992) as: With . Also, where is the probability of selecting both i and j. And , and for i=j: . If the selection probability are uncorrelated (i.e.: ), and when assuming the probability of each element is very small, then: Variance of the weighted mean (-estimator for ratio-mean) The previous section dealt with estimating the population mean as a ratio of an estimated population total () with a known population size (), and the variance was estimated in that context. Another common case is that the population size itself () is unknown and is estimated using the sample (i.e.: ). The estimation of can be described as the sum of weights. So when we get . When using notation from previous sections, the ratio we care about is the sum of s, and 1s. I.e.: . We can estimate it using our sample with: . As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: . This will be the estimand for specific values of y and w, but the statistical properties comes when including the indicator variable . This is called Ratio estimator and it is approximately unbiased for R. In this case, the variability of the ratio depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily Taylor series first-order linearization, asymptotics, and bootstrap/jackknife. The Taylor linearization method could lead to under-estimation of the variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be relatively accurate even for medium sample sizes. For when the sampling has a random sample size (as in Poisson sampling), it is as