# Difference between superpopulation and infinite population

What is the difference between Superpopulation and Infinite population? Please explain this with examples.

In survey sampling you have a finite population. One modeling method envisions the finite population as coming from a theoretical infinite population. This imaginary population is called a superpopulation model. On the other hand when selecting a random sample (not from a finite population) is viewed as sampling at random from an infinite population. So the term infinite population is for ordinary sampling and superpopulation specifically refers to the situation when the sample is taken from a finite population.

In the field of ecological statistics (e.g. mark-recapture) we often have long time series of data, where any individuals may only be exposed to sampling for only a portion of the total time series. In this context we can consider every individual that was exposed to sampling during the course of the experiment, a measure we call the superpopulation. This is different from the population at any given time point, which is the total number of individuals exposed to sampling at that time point. Both of these definitions of a population are finite measures.

Aside - I used the term "exposed to sampling" as population heterogeneity is often the rule rather than the exception in ecology. Subpopulations may exist that behave differently and may completely avoid detection by our survey techniques. These individuals thus are not part of our statistical population definition.

A population which is uncountable (or at least, not countable on fingertips) is called an "infinite population" — such as the number of red cells in blood, or the number of infective bacteria in the body of a patient.

An imaginary or theoretical population is called a superpopulation, or hypothetical population.

– Andy
May 22, 2014 at 12:29
• @Andy The original question did ask for specific examples, which are in short supply in the other answers, so I actually thought this answer did provide something worthwhile. The last textbook definition I saw of "infinite population" actually gave "grains of sand on a beach" as its example, which technically speaking is countable from a mathematical point of view but which is nevertheless described as an "infinite population" - so I do rather like this answer's idiomatic phrase "not countable on fingertips". A good copy-edit would help though. Apr 9, 2015 at 11:28

A superpopulation is a theoretical infinite population consisting of all events that "could have happened," not just the real population.

To give an example, imagine you have a company that has 100 employees of various races, and you want to know whether race is correlated with pay. You don't want to check the payrolls for the entire company, so you subsample a group of employees. There's two possible metrics here.

1. If you want to know "does the company actually, in the real world, pay one race or another more," you should compute population statistics.
2. If instead you want to know "is this company discriminating based on race," the important question isn't "is some race getting paid more than another." What we really want to know is whether the overall process by which we're setting pay is discriminatory. In this case, we want to calculate standard errors and confidence intervals by assuming a superpopulation.

To make the difference clearer, imagine that we sampled the entire population, i.e. all 100 employees, and we found a 0.5% difference in incomes across races. Is this caused by chance? If we test the hypothesis using all employees at the company as our population, the standard error is 0 (we know everyone's exact income!) so we'd say "There's definitely a difference." But... that's not really what we care about. What we care about is a question more like "In the long run, will this company engage in pay discrimination?" The number of cases here is infinite, so we consider the superpopulation instead of the true population, and a difference of 0.5% would probably be easy to explain by chance.

In general, superpopulations are the more interesting question in the sciences. Sometimes, though, the actual population is what we're actually interested in, e.g. in some surveys or opinion polls.

(An infinite population just happens to be a specific kind of "actual population" with infinitely many possible outcomes.)