I understand that when sampling from a finite population and our sample size is more than 5% of the population, we need to make a correction on the sample's mean and standard error using this formula:

$\hspace{10mm} FPC=\sqrt{\frac{N-n}{N-1}}$

Where $N$ is the population size and $n$ is the sample size.

I have 3 questions about this formula:

  1. Why is the threshold set at 5%?
  2. How was the formula derived?
  3. Are there other online resources that comprehensively explain this formula besides this paper?
  • 11
    $\begingroup$ You don't correct the mean! $\endgroup$ – whuber Dec 5 '10 at 21:42
  • 3
    $\begingroup$ You only correct the variance. $\endgroup$ – SmallChess Apr 10 '16 at 15:48

The threshold is chosen such that it ensures convergence of the hypergeometric distribution ($\sqrt{\frac{N-n}{N-1}}$ is its SD), instead of a binomial distribution (for sampling with replacement), to a normal distribution (this is the Central Limit Theorem, see e.g., The Normal Curve, the Central Limit Theorem, and Markov's and Chebychev's Inequalities for Random Variables). In other words, when $n/N\leq 0.05$ (i.e., $n$ is not 'too large' compared to $N$), the FPC can safely be ignored; it is easy to see how the correction factor evolves with varying $n$ for a fixed $N$: with $N=10,000$, we have $\text{FPC}=.9995$ when $n=10$ while $\text{FPC}=.3162$ when $n=9,000$. When $N\to\infty$, the FPC approaches 1 and we are close to the situation of sampling with replacement (i.e., like with an infinite population).

To understand this results, a good starting point is to read some online tutorials on sampling theory where sampling is done without replacement (simple random sampling). This online tutorial on Nonparametric statistics has an illustration on computing the expectation and variance for a total.

You will notice that some authors use $N$ instead of $N-1$ in the denominator of the FPC; in fact, it depends on whether you work with the sample or population statistic: for the variance, it will be $N$ instead of $N-1$ if you are interested in $S^2$ rather than $\sigma^2$.

As for online references, I can suggest you

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  • $\begingroup$ This formula is used for finite population, but with replacement or without replacement? $\endgroup$ – skan Jul 11 '15 at 16:10
  • 4
    $\begingroup$ @skan without replacement. $\endgroup$ – Black Milk Mar 1 '16 at 20:42

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