## Finite or infinite population

*July 6, 2019*

Many authors are confused about whether the purpose of writing a research report is to describe or to generalise. Some authors also seem to believe that p-values and confidence intervals are descriptive measure that must be used to describe the importance of what has been observed in a studied group of subjects. This is not the case; p-values and confidence intervals describe generalisation uncertainty. However, the question is more complicated than this. Generalising with the help of p-values and confidence intervals must sometimes be performed differently depending on the type of population studied.

Most randomised trials, cohort studies, and case-control studies are not performed for the participating subjects themselves but for the benefit of future patients, an infinite population. A survey, on the other hand, is usually performed to learn about a finite population defined in time and space. While an infinite population only can be studied using samples, a finite population can be studied both with samples and censuses. Analysing a sample from a finite population may require different calculations than a sample from an infinite population.

A sample drawn from an infinite population is usually considered to be a simple random sample. Surveys usually have a more complicated sampling design, and this needs to be accounted for in the analysis. A finite population correction (FPC) may also be necessary. Ignoring the sampling design and analysing survey data as if they had been collected as a simple random sample, is likely to yield too small standard errors, too narrow confidence intervals, and too low p-values.

Another, more philosophical, difference is that the analysis of a sample from an infinite population prioritises the internal validity (i.e. an unbiased description of cause-effect relationships between variables). For survey data, the aim is instead to achieve as high external validity (i.e. an unbiased describtion of the populations' properties) as possible.