Simple random sample

In finite-population sampling, a simple random sample of fixed size $n$ without replacement is a subset $S$ of $N$ labelled units drawn so that every size-$n$ subset has probabilityThe sample is the selected subset; simple random sampling is the procedure that generates it. This definition does not require the observed values to be distinct, only the selected population labels. The without-replacement convention is important because an independent repeated draw from a population is a different design.^[ref]

One construction randomly orders all $N$ labels and takes the first $n$. Each particular subset can occupy those positions in $n!(N-n)!$ permutations, out of $N!$, proving the probability above. Equivalently, draw successively and uniformly from the remaining labels, then discard their order.

For $N=5,n=2$, there are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ possible samples. A particular unit belongs to four of them, so its inclusion probability is $2/5$; a particular pair is one complete sample, so its joint inclusion probability is $1/10$. These are properties of the procedure before observing which subset was obtained.

A simple random sample need not resemble the whole population in every realization. For fixed values $(0,0,0,0,100)$, six of the ten size-two samples contain only zeros. Thus the sample mean is zero with probability $0.6$, although its expected value is the finite population mean $20$. The remaining four samples each have mean $50$, giving $0.6\times 0+0.4\times 50=20$. Unbiasedness concerns repeated selection, rather than a guarantee about any one sample. Coverage of the population labels and faithful implementation of the selection rule remain essential.