I wrote in a document something similar to this:

Based on the collected samples, we hope to infer ...

I was asked to provide a citation for my claim. I guess I can claim that I don't need a citation because sample based inference is well-known, but it made me question myself.

Why can we say that we can characterize a populations based on some samples? Who came up with this and where is their research?

I was thinking of the central limit theorem, but my understanding is that it only applies to populations that have distributions that can be approximated with a normal distribution. So the question is whether the claim is even true that we can use samples to make statistical inferences even when the population distribution is unknown. If yes, what is the theorem that says this.

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    $\begingroup$ Central limit theorems refer to what results look like, not the data. But the history of sampling is as long as you like. Early people tested small amounts to make decisions on whether to eat something, bathe somewhere, choose a partner,,,, There was no sharp beginning, $\endgroup$ – Nick Cox Jul 19 at 12:27
  • $\begingroup$ it is intuitive that if you eat a fruit from a tree and you don't die, it is safe to eat more fruits, but nature doesn't have to follow human intuition. Regardless, is there anyone who formalized this? $\endgroup$ – Andrei Jul 19 at 12:40
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    $\begingroup$ Every area of research has 'knowledge' that is so commonly used and so uncontroversial that you don't need to use any reference. Publishing guides by most journals also state something along those lines. It would be like including a proof to the Pythagorean theorem in a paper that includes the calculation of the hypotenuse somewhere, or adding a reference for gravity on Earth being around $9.81$ m/s$^2$. Unless the essence of your paper is sample based inference (e.g. its history), I think this is completely unnecessary. $\endgroup$ – Frans Rodenburg Jul 19 at 12:55
  • $\begingroup$ @FransRodenburg I agree it's unnecessary, but it made me think, and now I want to know :) $\endgroup$ – Andrei Jul 19 at 12:57
  • $\begingroup$ This is a cheap reply but everything still hinges on what counts as "formalized". $\endgroup$ – Nick Cox Jul 19 at 12:58

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