UP504

Sampling Fraction 
Sampling Fraction  or 
why do we usually just care about the sample size, not the population size?
sample (N)
vs. population (M)
sampling fraction =
N/M
the actual formula for the standard error (standard deviation of the sampling distribution) is:
where f = sampling fraction = N / M
but since typically M >> N
then f > 0
so 1f becomes 1, and so the formula for the standard error becomes:
Comparison of Corrected and Uncorrected Standard Error
Calculations of a Hypothetical Population of 38,000 (and standard deviation
of 20,000).
sample
size (n) 
Population
size (m) 
sampling
fraction (f) 
standard
deviation of the sample 
std
error (corrected) 
std
error (uncorrected) 
Percent
difference between corrected and uncorrected standard error 
1 
38,000 
0.00003 
20,000 
19999.7 
20000.0 
0.00% 
100 
38,000 
0.00263 
20,000 
1997.4 
2000.0 
0.13% 
200 
38,000 
0.00526 
20,000 
1410.5 
1414.2 
0.26% 
400 
38,000 
0.01053 
20,000 
994.7 
1000.0 
0.53% 
800 
38,000 
0.02105 
20,000 
699.6 
707.1 
1.06% 
1,600 
38,000 
0.04211 
20,000 
489.4 
500.0 
2.13% 
3,200 
38,000 
0.08421 
20,000 
338.3 
353.6 
4.30% 
6,400 
38,000 
0.16842 
20,000 
228.0 
250.0 
8.81% 
12,800 
38,000 
0.33684 
20,000 
144.0 
176.8 
18.57% 
25,600 
38,000 
0.67368 
20,000 
71.4 
125.0 
42.88% 
37,999 
38,000 
0.99997 
20,000 
0.5 
102.6 
99.49% 
38,000 
38,000 
1.00000 
20,000 
0.0 
102.6 
100.00% 
Note that there is very little difference in using the corrected vs. uncorrected standard error until the sampling fraction gets large. For example, even with a sample of 800 (out of a total population of 38,000), the difference is only 1 percent. The two estimates of standard error only begin to deviate significantly when the sample size is more than several thousand (that is, when the sampling fraction approaches about 10% or more).
Moral of the story: it is fine  and more conservative  to use the uncorrected estimate, which is easier to calculate anyway.