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This question is solely about terminology.

Consider the following definition regarding random sample from All of Statistics by Larry Wasserman

If $X_1,\cdots ,X_n$ are independent and each has the same marginal distribution with cdf $F$, we say that $X_1,\cdots ,X_n$ are iid (independent and identically distributed) and we write $X_1,\cdots ,X_n \sim F$. If $F$ has density $f$ we also write $X_1,\cdots ,X_n \sim f$.

We also call $X_1,\cdots ,X_n$ a random sample of size $n$ from $F$.

Consider another sort of definition I came across many sources as follows

A random sample is a sample that is chosen randomly. It could be more accurately called a randomly chosen sample. Random samples are used to avoid bias and other unwanted effects. Of course, it isn’t quite as simple as it seems: choosing a random sample isn’t as simple as just picking 100 people from 10,000 people. You have to be sure that your random sample is truly random!

I am just asking about usage of term random sample in mathematics.

Is it a pure technical term as mentioned in first definition? If yes, then why there is a wrong usage across literature?

If no, then do we need to interpret explicitly based on the context to know whether it is a collection of random variables or set of samples(instances)?

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  • $\begingroup$ Your first quotation is missing things, but there is an important distinction between sampling from a distribution and sampling without replacement from a population $\endgroup$
    – Henry
    Commented Aug 16, 2019 at 11:16
  • $\begingroup$ @Henry Do you mean that first one needs more context or used the word random sample losely> $\endgroup$
    – hanugm
    Commented Aug 16, 2019 at 11:21
  • $\begingroup$ I mean that the two times it says "we write $X_1,\cdots ,X_n$" it looks as if we should write some more in these particular cases as we have already said that in general $\endgroup$
    – Henry
    Commented Aug 16, 2019 at 11:23
  • $\begingroup$ @Henry Yeah, sorry for that, rectified..... $\endgroup$
    – hanugm
    Commented Aug 16, 2019 at 11:26

1 Answer 1

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The term sample is used both to describe the individuals that provided the data (second example) and the values that were obtained (first example). I see this is a bit confusing. The use of the term in the second example stems from applied research, while the definition in the first example is purely mathematical. They are related in the following way.

Consider the sample to be the individuals that provided the values and not the values themselves. However if you randomly obtain individuals from a population, where the variable of interest has a certain distribution the values you get to work with are iid, which satisfies the requirements of the first definition. I.e. the values are a random sample from the distribution.

What matters for applied work is whether you can safely assume that your sample is representative, i.e. whether the individuals you obtain dont for some reason differ from the general population wrt the relationship or variable that you are studying. If the sample is representative you can go ahead and use the mathematical tools developed under the assumption of random sampling. Thus the second definition is somewhat misleading. If the first 100 individuals are a representative sample (in applied sense), the values you obtain by sampling them are a random sample (in theoretical sense).

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