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Finite population correction factor FPC($H_n^N$)
Unbiased estimation of standard deviation is a famous topic in college Statistics. Local sample variance should be divided by $$n-1$$ to become unbiased sample variance before it can be used. Many high school students cram this up without understanding. However, different sampling methods coincide with different ways of correction.

Finite Population Correction FPC($C_n^N$) is alreadly well-known. While, FPC($H_n^N$) is missing from many texts. But both proofs are similar.

The variance of Hypergeometric distribution shows a high symmetry with Finite population correction(FPC) factor. The variance of Negative hypergeometric distribution may be a strong side evidence of FPC($H_n^N$).