Does it make sense to compute confidence intervals and to test hypotheses when data from the whole population is available? In my opinion, the answer is no, since we can accurately compute the true values of the parameters. But then, what is the maximal proportion of data from the original population that allows us to use the aforementioned techniques?
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3$\begingroup$ If you use correct finite-sample methods, the variance goes to zero just as your sample hits the population size. That is, there's no maximum size; the proper formulas work as they should, right up to $n=N$. $\endgroup$– Glen_bCommented Sep 1, 2013 at 9:33
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$\begingroup$ I think you should state it more clearly if the question is about "sample=population" case or "sample from finite population" case. $\endgroup$– ttnphnsCommented Sep 1, 2013 at 10:05
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$\begingroup$ First part of question is about sample=population and second about sample from population (when sample size < population size). $\endgroup$– sitemsCommented Sep 1, 2013 at 10:19
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2$\begingroup$ Closely related questions: Test for significance with data representing the whole population? and Resources for when population data is available $\endgroup$– SilverfishCommented Sep 18, 2016 at 11:16
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$\begingroup$ A design-based inference perspective: arxiv.org/abs/1706.01778 $\endgroup$– Christoph HanckCommented May 6 at 13:03
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2 Answers
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The first question is one that has no generally agreed upon answer. My own view is like yours, but others have argued that a population can be viewed as a sample from a "super-population" where the exact nature of a super-population varies depending on context: E.g. a census of all the people living in a building could be viewed as a sample from all the people living in similar buildings; a census of the population of the USA (not that one could ever be truly complete) could be viewed as a sample from a super-population of Americans who might one day exist (or something like that). I think this is often an excuse to get to use p-values; many scientists in substantive fields aren't comfortable if they haven't got a p-value. (But that is my view).
The second question seems a bit odd to answer in a general way. When do you get a sample that is (say) even more than half of the population?
A bigger problem will be bias. Going back to the US Census, the problem isn't simply that it misses people, but that the people it misses are not a random sample of the total population; so, even if the census gets answers from (to pick a number) 95% of all people, if those 5% remaining are quite different, then results will be biased.
answered Sep 1, 2013 at 11:54
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3$\begingroup$ I think whether or not you calculate confidence intervals for a population statistic depends on if you want to make inferences on the actual population or for the hypothetical "super population". In a past job with a state health dept we reported annual statistics like very low birth weight percentages and suicide rates that bounced around from year to year. Yes, we were reporting on the whole population, yet it would be silly to hinge the State's health progress (and funding!) on every increase or decrease in these and other health indicators as a complete shift in the population's health. $\endgroup$– RobertFCommented Aug 1, 2019 at 23:22
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Suppose only 2 out of 12 committee members are women.
The proportion $\frac{1}{6}$ can be taken as a statistic descriptive of the whole population (the committee). Perhaps something ought to be done to correct the imbalance, regardless of how it arose.
Or it can be taken as an estimate of the probability of a woman's being selected for the committee—a property of the selection process. You can put confidence intervals around it, test if it's significantly different from one-half (or another relevant null hypothesis), & so on. Perhaps the process needs to be changed to make it fair.
The two views, descriptive & inferential, are not contradictory, but quite distinct.
The answer to the second question is that it makes sense to calculate confidence intervals for & test hypotheses about a population parameter even if just a single individual is unsampled. Just note that CIs & tests have to take account of a considerable proportion of the population's being sampled: see finite population correction.
answered Sep 2, 2013 at 12:32