Resource assessment exercises: finite population correction
- This article is part of the Resource assessment exercises. See the category page for a (chronological) table of contents.
Finite population correction
If we observe all values \(y_{i\in U}\) we talk about a census. The mean is calculated, not estimated, i.e.,
SRSwoR <- sample(trees$dbh, size=N) mean(SRSwoR) ## [1] 21.05 mean(trees$dbh) ## [1] 21.05
As noted before, SRSwoR stands for simple random sampling without replacement (woR). However, if we take a sample with replacement (SRS; set replace = TRUE
) we get a slightly different value. This would be an estimate.
SS <- sample(trees$dbh, size = N, replace = TRUE) mean(SRS) ## [1] 20.96
If we are interested in a population parameter and take an SRSwoR of size \(n=N\), we get the true population value. There is no doubt about its value (assuming that measurement errors are absent). However, if we estimate the standard error for the \(n=N\) sample we get a positive value instead of a zero \(s_{\bar{y}}\).
sd(SRSwoR)/sqrt(N) ## [1] 0.07416
This cannot be! The estimator of the standard error holds for sampling with replacement. For sampling without replacement we have to correct for the fact that we took a relatively large sample. The finite population correction (fpc) for a relatively large sample is defined as,
$\text{fpc}=1-\frac{n}{N}.$ | 1 |
Obviously, if \(n=N\) the fpc becomes zero. Suppose we take a sample of size \(n=25,000\) from trees
, then
S25k <- sample(trees$dbh, size = 25000) sd(S25k)/sqrt(25000) # without fpc ## [1] 0.08099 fpc <- 1 - 15000/30000 sqrt(var(S25k)/25000 * fpc) ## [1] 0.03307 sd(S25k)/sqrt(25000) * sqrt(fpc) ## [1] 0.03307
For the parametric standard error the fpc becomes,
$\text{fpc}=\frac{N-n}{N-1}.$ | 2 |
As a rule of thumb, we apply the fpc when the sampling fraction
$f=\frac{n}{N}$ | 3 |
exceeds 0.05, i.e., 5 percent.
Related articles
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