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This is my understanding:

Probability sampling refers to the situation where you know the probability of each item potentially picked.

Random sampling means each item has the SAME likelihood of being picked.

Here's where I get confused: What is SIMPLE random sampling? Using random numbers? So that means random sampling and simple random sampling are simply synonyms? If so, then does that mean systematic sampling (picking every nth) can never be considered random sampling and is thus biased?

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    $\begingroup$ I don't think these are commonly understood terms, different people will understand different things here. In particular "SIMPLE" random sampling is not clear, it could mean many things depending on context. $\endgroup$
    – Gijs
    Apr 3, 2020 at 7:37

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Expanding a bit on Gijs's comment:

Per the Oxford Dictionary of Statistics, 3rd edition (ed by Upton and Cook) on p. 378:

If each member of the sample is selected by the equivalent of drawing lots then the sample is a simple random sample or, commonly, a random sample.

But according to the Cambridge Dictionary of Statistics, 4th edition (ed. by Everitt and Skrondal), simple random sampling is

A form of sampling design in which n distinct units are selected from N units in a population in such a way that every possible combination of n units is equally likely to be the sample selected.

and they note that other methods may give each unit equal likelihood of being drawn, but only SRS gives equal likelihood to every combination. But their definition of random sampling agrees with their definition of SRS.

On the other hand, what I was taught in grad school is that simple random sampling is two forms of random sampling (with or without replacement) but that random sampling includes a lot of other methods - indeed, any method that involved random selection at all. This largely agrees with this page from Yale University (I did not go to Yale).

On the third hand, Wikipedia distinguishes between a simple random sample and a systematic random sample, with the difference being that between the two dictionaries cited above (simple random sampling is every combination, systematic random sampling is every observation). Unusually for this type of article on Wikipedia, they don't cite a source for this.

So .... TLDR ..... There's no agreement.

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    $\begingroup$ Peter, there is agreement. None of the serious sampling textbooks would give the first definition, and it is a borderline disgrace that a book titled Dictionary of Statistics would give it. The formal definition of the simple random sample is that every set of $n$ units has the same probability of selection. And you can indeed have them with without replacement (SRSWOR) or with replacement (SRSWR) depending on whether the units must be distinct or can be used more than once. SRSWOR usually makes more sense both from the precision and the logistics perspectives. $\endgroup$
    – StasK
    Apr 6, 2020 at 14:05
  • $\begingroup$ @StasK Do you have a good recommendation for an English language textbook on sampling? Best. $\endgroup$
    – Alexis
    Mar 13, 2023 at 2:54
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    $\begingroup$ @Alexis the best textbook is Lohr but it is quite pricey. $\endgroup$
    – StasK
    Apr 14, 2023 at 14:59

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