# How biased is a statistical study in which sampling was purposely made without repeats?

It is understood in mathematical statistics that a sample (as in sampling distribution) may very well contain repeatedly the same item/subject.

In practice though, it would never occur to someone conducting an experiment (e.g. in human sciences studies) to take into account the same subject twice. Samples are made

• by taking the first people who agree to take the tests and look random enough (e.g. using quota sampling).
• by taking the only $$12$$ people living in your country who fit the requirements of the study (e.g. when you're conducting research on a rare disease) (and yes, $$12$$ is ridiculously small but that's still what is being done when no more subjects are available).

In what situations can this habit turn out to be really bad - induce a huge bias on the results?

What seems clear to me is that if the total population is large enough, then it shouldn't matter too much, since samples with repeats are quite unlikely to occur and don't weigh much in the sampling distribution. When the total population is small though, it feels like we're more likely to miss something.

• This is called "sampling without replacement", and is commonly discussed in survey sampling and other contexts. The samples are not strictly independent, but they are exchangeable, which implies many of the desirable properties of iid samples (depending on what you're doing with them). Jan 8, 2020 at 17:59
• @Dougal The reason I'm asking is because all hypothesis tests I know (that are used in human sciences) assume replacement. The very first time you see this at work is in the proof that the sample variance is an unbiased estimator: it works with replacement only. Jan 8, 2020 at 22:27