Why Finite Population Correction Factor is (N-n)/(N-1)


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When I read Stephen's Schaum's Outline of Theory and Problems of Elements of Statistics, in 13.10, it said :

I just wanna know how to prove it

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Start by writing the standard definition of a basic mean and variance. Then write the definition of a sample mean. Then use same mean definition as your random variable in the variance definition.

The trick is to use the definitions in expectation form.