Editing Margin of error (section)

== Effect of finite population size ==
The formulae above for the margin of error assume that there is an infinitely large population and thus do not depend on the size of population <math>N</math>, but only on the sample size <math>n</math>. According to [[Sampling (statistics)|sampling theory]], this assumption is reasonable when the [[sampling fraction]] is small. The margin of error for a particular sampling method is essentially the same regardless of whether the population of interest is the size of a school, city, state, or country, as long as the sampling ''fraction'' is small.

In cases where the sampling fraction is larger (in practice, greater than 5%), analysts might adjust the margin of error using a [[finite population correction]] to account for the added precision gained by sampling a much larger percentage of the population. FPC can be calculated using the formula<ref>{{cite journal|last=Isserlis|first=L.|year=1918|title=On the value of a mean as calculated from a sample|url=https://zenodo.org/record/1449486|journal=Journal of the Royal Statistical Society|publisher=Blackwell Publishing|volume=81|issue=1|pages=75–81|doi=10.2307/2340569|jstor=2340569}} (Equation 1)</ref>

:<math>\operatorname{FPC} = \sqrt{\frac{N-n}{N-1}}</math>

...and so, if poll <math>P</math> were conducted over 24% of, say, an electorate of 300,000 voters,
:<math>MOE_{95}(0.5) = z_{0.95}\sigma_\overline{p} \approx \frac{0.98}{\sqrt{72,000}}=\plusmn0.4%</math>
:<math>MOE_{95_{FPC}}(0.5) = z_{0.95}\sigma_\overline{p}\sqrt{\frac{N-n}{N-1}}\approx \frac{0.98}{\sqrt{72,000}}\sqrt{\frac{300,000-72,000}{300,000-1}}=\plusmn0.3%</math>

Intuitively, for appropriately large <math>N</math>,

:<math>\lim_{n \to 0} \sqrt{\frac{N-n}{N-1}}\approx 1</math>
:<math>\lim_{n \to N} \sqrt{\frac{N-n}{N-1}} = 0</math>

In the former case, <math>n</math> is so small as to require no correction. In the latter case, the poll effectively becomes a census and sampling error becomes moot.