= Sampling without replacement
<Sampling without replacement> selects distinct labelled units: after a unit is selected it is unavailable for the remaining draws. Distinct labels can carry equal numerical values, so repeated values do not necessarily imply a repeated unit. The rule alone does not specify selection <probabilities>; a procedure favoring particular units is still <sampling without replacement> but need not be <simple random sampling>.
In uniform sequential <sampling without replacement>, the next label is chosen uniformly among those remaining. For a population of $N$ labels, any ordered sequence of $n$ distinct labels has <probability> $1/[N(N-1)\cdots(N-n+1)]$. Each size-$n$ <subset> has $n!$ possible orders, so its <probability> is $\binom Nn^{-1}$. This gives the fixed-size <simple random sampling> design. Equivalently, take the first $n$ labels of a uniformly random <permutation>. Restricting a uniformly random <permutation> to a specified label set preserves uniformity; biased modulo mappings should not replace uniform selection.
The draw positions have identical marginal <probability distributions>, but exhibit <statistical dependence>. If the first label is $i$, the <conditional probability> that the second label is $i$ is zero, whereas its marginal <probability> is $1/N$. Thus these draws are not <independent and identically distributed> when $N>1$ and $n>1$. For a marked subset of labels, counting size-$n$ <subsets> with <binomial coefficients> gives the <hypergeometric distribution>. The resulting negative <covariance> explains the <finite population correction> for both counts and <sample means>.
Back to article page
