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Identify the selection unit on each draw
With replacement, a chosen unit returns to the pool before another independent uniform draw. The same labelled unit can appear twice. Without replacement, it is removed from the eligible pool and cannot appear again in that sample.
Compare two draws from four units
| Feature | With replacement | Without replacement |
|---|---|---|
| Ordered outcomes | 16 possible ordered pairs | 12 possible ordered pairs |
| Repeated unit | Allowed | Excluded |
| First draw A and second draw A | Probability 1/16 | Probability 0 |
Carry the distinction into the mean variance
Repeated independent uniform draws use the variance of the finite value distribution divided by n. A simple random sample without replacement has a finite-population reduction under its stated variance convention. The formulas use different population-variance divisors, so retain their definitions.
Do not confuse duplicate values with repeated units
Two distinct units can both have value 10. Selecting both without replacement is still possible. The rule concerns unit identities, not whether a numerical value appears more than once in the recorded sample.