I'm reading an Econometrics textbook and it talks about "Simple Random Samples" and "IID draws" (Independent, Identicaly Distributed variables).
While it defines the two seperately, it also seems to imply that they are the same thing.

From what I understand, a simple random sample is where n objects are selected from a population at random. While an IID draw is where n objects are randomly selected from a population of independent and identically distributed random variables.
(by randomly selected I mean that each random variable is equally likely to be selected)

So: Does is a 'simple random sample' the same thing as an 'iid draw' or are they not necessarily the same thing?


The phrase "simple random sample" typically refers to selecting $n$ items from a population of $N$ items in such a way that all $n$-element subsets are equally likely to be chosen. This is often referred to as "sampling without replacement" since if the sampling is accomplished by successive draws then the second item chosen can't be the first item chosen. This introduces a clear dependence of successive draws, so they can't be modeled as independent, identically distributed choices. IID is, in contrast, sampling with replacement. It would correspond to picking an item at random, recording it, then putting it back in the population before drawing the next item, so that it could be chosen again.

The mathematics of sampling with replacement is easier than the mathematics of sampling without replacement. If the sample size, $n$ is small in comparison with the population size, $N$, then it is common to cheat a little, and treat simple random samples as samples with replacement. This is a reasonable approximation since once the population is large enough, the chance of choosing the same item twice is quite small, so the overwhelming majority of samples with replacement in fact contain no duplicates.


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