Variance in statistics

As stated in the mission - “Measure of variability” in statistics: “The sample variance is an unbiased estimator for the population variance only when we sample with replacement.”
What does it mean “with replacement”?

Sampling is done with replacement when for each sample, every member of the population can possibly be selected. It is without replacement when an element of the population can only be sampled once.

Imagine that you have a box with 4 blue marbles and 6 red marbles.

Sampling with replacement would mean that you take a marble, check its color and then put it back in the box. This means that for each sample, the probability of taking a blue marble is 4/10 and the probability of taking a red one is 6/10.

Sampling without replacement would mean that you take a marble, check its color at leave that marble out of the box. This marble can never be selected again. In this case, the probabilities change as you perform samples. On the first sample, you have 4/10 of finding a blue and 6/10 of finding a red. However, on the second same, since we did not put the marble back in, the probabilities change.

In general, sampling with replacement maintains probabilities at each sample and sampling without replacement changes the probabilities as the data is sampled.

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thanks a million :blush: :blush: