When sampling is done without replacement, it means that once an item is selected from the population, it is not put back, so it cannot be chosen again in the same sampling process.

Quick Scoop

Plain-language meaning

Think of a bowl with 10 distinct marbles.
If you draw one marble, keep it out, then draw another, you are sampling without replacement.

Each marble can appear at most once in your sample, and the total number of marbles left to pick from shrinks after each draw.

Key points at a glance

  • Once chosen, an item is removed from the pool and cannot be selected again.
  • The population size decreases after each selection (10 items, then 9, then 8, etc.).
  • Probabilities change from one draw to the next, because the composition of whatā€™s left has changed (events are dependent).
  • Common in ā€œreal lifeā€ surveys, lotteries, and samples where you donā€™t want duplicates, like picking unique households for a survey.

Tiny example

Suppose you have cards numbered 1, 2, 3, 4, 5.
You draw two cards without replacement :

  • First draw: probability of drawing a 1 is 1/51/51/5.
  • If you did draw a 1, then for the second draw there are only 4 cards left (2, 3, 4, 5), so the probability of drawing, say, a 2 is now 1/41/41/4.

The second probability depends on what happened first, which is exactly what ā€œwithout replacementā€ captures.

Quick contrast (with vs. without)

[5][1] [9][1][3] [1] [7][1] [3] [5][3] [1] [5][1][3]
Aspect With replacement Without replacement
Can you pick the same item twice? Yes, items go back into the pool.No, each item can appear only once.
Population size while sampling Stays the same.Decreases after each draw.
Relationship between draws Usually treated as independent.Dependent: each draw changes the next probabilities.
Typical uses Bootstrapping, some simulations.Surveys, lotteries, choosing unique people/items.

In short, ā€œsampling without replacementā€ = pick unique items, donā€™t put them back, and remember that each pick changes the odds for the next one.

Information gathered from public forums or data available on the internet and portrayed here.