In survey methodology, probability-proportional-to-size (pps) sampling is a sampling process where each element of the population (of size N) has some (independent) chance
to be selected to the sample when performing one draw.
[1]: 97 [2] One of the cases this occurs in, as developed by Hanson and Hurwitz in 1943,[3] is when we have several clusters of units, each with a different (known upfront) number of units, then each cluster can be selected with a probability that is proportional to the number of units inside it.
The pps sampling results in a fixed sample size n (as opposed to Poisson sampling which is similar but results in a random sample size with expectancy of n).
When selecting items with replacement the selection procedure is to just draw one item at a time (like getting n draws from a multinomial distribution with N elements, each with their own
If doing a without-replacement sampling, the schema can become more complex.
[1]: 93 This statistics-related article is a stub.