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What is the difference between sample variance and sampling variance? They seem same. Aren't they?

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Sample variance refers to variation of observations (the data points) in a single sample. Sampling variance refers to variation of a particular statistic (e.g. the mean) calculated in sample, if to repeat the study (sample-creation/data-collection/statistic-calculation) many times. Due to central limit theorem, though, for some statistics you don't have to repeat the study many times in reality, but can deduce sampling variance from a single sample if the sample is representative (this is asymptotic approach). Or you could simulate repetition of the study by a single sample (this is bootstrapping approach).

An additional note on "sample variance". Two may be mixed in one term:

  • Estimate of population variance based on this sample. This is what we usually use, it has denominator (degrees of freedom) n-1.

  • Variance of this sample. It has denominator n.

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The sample variance, $s^2$, is the variance of the sample, an estimate of the variance of the population from which the sample was drawn.

"Sampling variance" I would interpret as "the variance that is due to sampling", for example of an estimator (like the mean). And so I would consider these two terms to be quite different.

But "sampling variance" is a bit vague, and I would need to see some context to be sure. And I'd prefer to say "sampling variation" for the general idea.

[Many people (particularly in quantitative genetics) use the term "variance" in place of "variation", whereas I would reserve "variance" solely for the particular measure of variation.]

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