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I have always been confused about the use of the term “population” in statistics. In my first statistics course I was taught that we need a sample, because surveying the whole population is too costly. So there is the whole population and there is small sample from it which we study.

The problem is that this intuition is just wrong outside of a few toy examples, when population is literally the whole population of the US (or world). Actually even in those few examples it is probably wrong, since world population is just one of hypothetical repeated random samples from DGP. So when in the following statistics courses we started estimating multivariate models, I was struggling to understand what the population is now and how it differs from the sample.

So I am really confused by the way statistics is taught. I feel like people use the term “population” partly because of historical reasons, partly because it makes it easier to explain the concept of the sample in Stat 101. The problem is that it teaches wrong intuition, which students have to unlearn later and creates a hole in understanding of the most fundamental statistical concepts. On the other hand, the concept of DGP is harder to introduce in elementary statistics course, but after students understand it, they will have solid conceptual foundation in statistics.

I have two questions:

  1. I would guess that there is ongoing discussion among statisticians on this issue, so can anybody give me references on this?

  2. And more importantly, do you know any examples of introductory-level statistics textbooks, which forego “population” and introduce statistics, based on concepts of DGP and sample? Ideally, such textbook will devote large space to explaining conceptual foundations of statistics and statistical inference.

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    $\begingroup$ Population, can be used in statistics in either a literal or an abstract sense. This is also true of many other words in the sciences. I think it is a horrible idea to invent new terminology (especially something as awkward-sounding 'data generating process' with yet another unnecessary acronym DGP) when there is no clear need. Seems a lot of that going around latedly. Maybe I will start referring to such as NRT (needlessly redundant terminology). $\endgroup$
    – BruceET
    May 8, 2021 at 19:10
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    $\begingroup$ @BruceET The problem is that population in abstract sense has nothing to do with population in literal sense. This problem is made worse by attempt of most introductory statistics textbooks to equate these two meanings when introducing idea of population and sample. $\endgroup$ May 8, 2021 at 19:33
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    $\begingroup$ That some textbooks may have weaknesses is not in dispute. This issue is whether making up new terminologies will help cure that. Euclid's abstract lines of zero width have survived nicely for a few centuries--some badly written geometry textbooks notwithstanding. $\endgroup$
    – BruceET
    May 8, 2021 at 19:50
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    $\begingroup$ You have two statements that are extreme, and you provide no evidence or citations in support. These are (1) "The problem is that it teaches wrong intuition...and creates a hole in understanding of the most fundamental statistical concepts", and (2) "the concept of DGP is harder to introduce in elementary statistics course, but after students understand it, they will have solid conceptual foundation in statistics." Please provide some type of justification/evidence for these claims. $\endgroup$
    – Gregg H
    May 9, 2021 at 23:22
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    $\begingroup$ @ Moysey Abramowitz When I teach survey research or intro to research methods I want students to literally understand that if they conduct a survey using a random(ish) sample there is a possibility that the answer they get from the survey will be different from what they got if they surveyed the whole population, and that there are ways to quantify this possibility. I want them to understand why political polls have margins of error around them and why a psych experiment with N=20 might find one group did better due to dumb luck. Sample/Population seems to helpfully describe these situations. $\endgroup$ May 10, 2021 at 23:34

2 Answers 2

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There are certainly already many contexts where statisticians do refer to a process rather than a population when discussing statistical analysis (e.g., when discussing a time-series process, stochastic process, etc.). Formally, a stochastic process is a set of random variables with a common domain, indexed over some set of values. This includes time-series, sequences of random variables, etc. The concept is general enough to encompass most situations where we have a set of random variables that are of interest in a statistical problem, and so statistics already has a sufficiently well-developed language to refer to hypothesised stochastic "processes", and also refer to actual "populations" of things.

Whilst statisticians do refer to and model "processes", these are abstractions that are formed by considering infinite sequences (or continuums) of random variables, and so they involve hypothesising quantities that are not all observable. The term "data-generating process" is itself problematic (and not as helpful as the existing terminology of a "stochastic process"), and I see no reason that its wide deployment would add greater understanding to statistics. Specifically, by referring to the generation of "data" this terminology pre-empts the question of what quantities are actually observed or observable. (Imagine a situation in which you want to refer to a "DGP" but then stipulate that some aspect of that process is not directly observable. Is it still appropriate to call the values in that process "data" if they are not observable?) In any case, setting aside the terminology, I see deeper problems in your approach, which go back to base issues in philosophy and the formulation of research questions.


Existents vs processes in empirical research: I see a number of premises in your view that strike me as problematic, and appear to me to misunderstand the goal of most empirical research that uses statistics. When we undertake empirical research, we often want to know about relationships between things that exist in reality, not hypothesised "processes" that exist only in our models (i.e., as mathematical abstractions from reality). Indeed, in sampling problems it is usually the case that we merely wish to estimate some aspect of the distribution of some quantity pertaining to a finite population. In this context, when we refer to a "population" of interest, we are merely designating a set of things that are of interest to us in a particular research problem. Consequently, if we are presently interested in all the people currently living in the USA, we would call this group the "population" (or the "population of interest"). However, if we are interested only in the people currently living in Maine, then we would call this smaller group the "population". In each case, it does not matter whether the population can be considered as only part of a larger group --- if it is the group of interest in the present problem then we will designate it as the "population".

(I note that statistical texts often engage in a slight equivocation between the population of objects of interest, and the measurements of interest pertaining to those objects. For example, an analysis on the height of people might at various times refer to the set of people as "the population" but then refer to the corresponding set of height measurements as "the population". This is a shorthand that allows statisticians to get directly to describing a set of numbers of interest.)

Your philosophical approach here is at odds with this objective. You seem to be adopting a kind of Platonic view of the world, in which real-world entities are considered to be less real than some hypothesised "data-generating process" that (presumptively) generated the world. For example, in regard to the idea of referring to all the people on Earth as a "population", you claim that "...it is probably wrong, since world population is just one of hypothetical repeated random samples from DGP". This bears a substantial similarity to Plato's theory of forms, where Plato regarded observation of the world to be a mere imperfect observation of eternal Forms. In my view, a much better approach is the Aristotelian view that the things in reality exist, and we abstract from them to form our concepts. (This is a simplification of Aristotle, but you get the basic idea.)$^\dagger$

Plato and Aristotle

If you would like to get into literature on this issue, I think you will find that it goes deeper into the territory of philosophy (specifically metaphysics and epistemology), rather than the field of statistics. Essentially, your views here are about the broader issue of whether the things existing in reality are the proper objects of relevance to human knowledge, or whether (contrarily) they are merely an epiphenomenon of some broader hypothesised "process" that is the proper object of human inference. This is a philosophical question that has been a major part of the history of Western philosophy going back to Plato and Aristotle, so there is an enormous literature that could potentially shed light on this.

I hope that this answer will set you off on the interesting journey into the field of epistemology. For present purposes, you might wish to take a practical view that also considers the objectives that researchers set for themselves in their research. Ask yourself: would researchers generally prefer to know about properties of the people living on Earth, or would they prefer to try to find out about your (hypothesised) "hypothetical repeated random samples" of people who might have lived on Earth instead of us?


$^\dagger$ To avoid any possible confusion among those lacking historical knowledge, please note that these are not real quotes from Plato and Aristotle --- I have merely taken poetic license to liken their philosophical positions to the present issue.

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    $\begingroup$ Not sure I follow the argument in your 2nd paragraph. Data-generating process isn't intended to oust stochastic process, but just refers to the particular stochastic process used to model the data, the observations. So you might have one stochastic process that generates times of death, another that generates times of censoring; you put the two together to get a process that generates times of death/censoring for each patient: that's your data-generating process. $\endgroup$ May 13, 2021 at 10:25
  • $\begingroup$ That is fine, but even then, notice that you had to be careful not to refer to the intermediate processes as DGPs because they do not produce observable "data". That is my point in the second paragraph. In any case, the OP seems to consider the DGP much more broadly than this. $\endgroup$
    – Ben
    May 25, 2021 at 22:53
  • $\begingroup$ statistics needs more memes! ;o) $\endgroup$ Apr 17 at 12:00
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    $\begingroup$ @Dikran Marsupial Instead of DGP I sometimes use GOD, generates our data. $\endgroup$
    – BenP
    Apr 17 at 18:00
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This is an old question, but I have been thinking about similar issues and I think this philosophical discussion in the answer and comments may be hiding some simple but concrete points about how population and data-generating process (DGP) relate to each other in empirical research.

As I see it, the concept of population itself does not necessarily specify what cause the data to occur they way the do. To fully specify this we have to indicate how the population is sampled (e.g. was it truly a random probability sample or something else), and what kind of measurement was made on each sampled individual. Considered jointly, the population and the sampling approach provide a full DGP.

Sure, one could twist the concept of population to indicate the 'population of measurements that can be made with the specified sampling approach'. However, similarly to the OP I have the impression that having such fluidity in the concept of 'statistical' population can hamper rather than help intuition when moving from introductory statistics to more advanced applications.

Personally, I find it helpful to think of the population as the set of similar elements or units that are the subject of a study and on which we would like to make some inferences, and find it helpful to include under the DGP label everything else we need to describe how our data was generated.

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