Why isn't the standard error equal to zero?

Question

According to this page:
http://mathworld.wolfram.com/StandardError.html

It says: "It therefore estimates the standard deviation of the sample mean based on the population mean."

So for example, if I have an entire population of 5 people, their age is {8,6,7,1,5}, then its stddev is about 2.702 and mean is 5.4

If I take random sample of size 3, maybe I will come up with {8,1,5}, then its stddev is about 3.512, mean is 4.667

As I increase the size of the sample and finally come to 5, that is, the sample is exactly equal to the entire poplution in this example. Then the standard error of the mean between [the sample] and [the entire population] should theoretically be zero.

But when I use this formula SE = σ/sqrt(n),
Since I already know the stddev of the entire population, which is 2.702 and I also know the size of the sample is now 5.
Substituting these values into the formula:
SE=2.702/sqrt(5)=1.208

The SE is not equal to zero. Why?

Thanks a lot.

Answer 1

Because the standard error is based on the assumption that you are sampling from a population that is very large in comparison to the sample being taken. This is frequently true -- most polls involve trying to infer the attributes of millions of people from a sample of no more than a few thousand. However, if your entire population consists of only five people, then the assumptions behind the standard error are false, and so the standard error becomes inapplicable.

Answer 2

I think the formula is only for use with infinite populations (or at least, 20 or 30 times larger than the sample size). Otherwise, you should make a correction:

SE = [σ/sqrt(n) ] [sqrt( (N - n) / (N - 1) ]
n = sample size
N = population size

Answer 3

i really dont know why