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Suppose the human population consisted of $N = 3$ people, each with a specific height. Let $X^N$ be the random variable representing the heights of this population of $N$ people. Since $X^N$ can only take $N = 3$ distinct values it is a discrete random variable with a probability mass function.

For example, we could have $N = 3$ people with heights $150$ cm, $160$ cm, and $170$ cm, and thus the probability of any particular height occuring is $1/3$.

Now, consider the case of $X^N$ when $N = 6$ billion, i.e. the heights of the real-world human population. We now have $6$ billion distinct values for $X^N$. Although there is now a very large range of values $X^N$ can take, it is still a discrete random variable as those $6$ billion discrete values are the only values of $X^N$ that can occur.

Therefore, heights of humans is actually a discrete random variable and not a continuous random variable? Everywhere I look it says that human heights is a continuous random variable with a pdf, but it seems from the above that it is actually a discrete random variable with a pmf?

Bertus101
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    This question is essentially the same as https://stats.stackexchange.com/questions/478142/do-we-need-hypothesis-testing-when-we-have-all-the-population/478540. A "population" from the point of view of statistics is infinite in size even though only a finite number of individuals actually exist. – Michael Reid Oct 15 '20 at 11:44
  • @MichaelReid It is not the same, that question that is concerned with hypothesis testing is clearly different than mine which poses a very specific question about when we can consider a random variable to be discrete vs continuous, along with an example to make what I am asking clearer. You could say it is related to my question, but then any question on this site will be naturally related to numerous other questions. – Bertus101 Oct 15 '20 at 13:57

3 Answers3

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.. each with a specific height that can be measured with infinite accuracy..

Based on this, we could say that height of a single individual, say $X$, is continuous RV, it can be any real number within a plausible range. This makes vector of heights, say $X^N$, a continuous random vector as well.

For example, we could have =3 people with heights 150 cm, 160 cm, and 170 cm, and thus the probability of any particular height occuring is 1/3.

This is only one realisation of the random vector described above. The probability $1/3$ describes something like $$P(X_1=150|\text{Three people have heights 150,160,170})=1/3$$

which is not equal to $P(X_1=150)=0$. (because a continuous RV being equal to a specific value is $0$)

If $X$ was measured with finite precision, it'd be discrete RV in nature, and even a very large $N$ like 6 billion wouldn't change the nature of $X^N$, which would be discrete.

gunes
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  • Forget about the 'measured with infinite accuracy part', that is confusing matters. I've edited that out of my post. Say we have $N=3$ people with heights exactly equal to $150$cm, $160$cm, $170$ cm. These people define our human population. This gives us a discrete random variable $X^3$. Now, in the real world, the actual human population is $X^{6 \text{billion}}$; these $6$ billion people have $6$ billion distinct heights that thus define a discrete random variable. – Bertus101 Oct 15 '20 at 10:19
  • Do you disagree? – Bertus101 Oct 15 '20 at 10:23
  • The vector [150,160,170] is only a *realization* not the random vector itself, so we can't discuss about its discreteness. But, if you're saying that these heights are obtained via sampling from a discrete set of values (like 110,120,...,200), then yes, the random vector $X^3$ would be a discrete variable. In addition, the portion you deleted in OP was making it much more unambiguous than current version. – gunes Oct 15 '20 at 10:25
  • If $[150,160,170]$ are the heights of the $3$ people in our entire population, we can't just say $X_1 = 150$cm in this instance but it could alternatively be some other value because it is random. It is not random, the person corresponding to $X_1$ has a fixed height $150$cm, they are not an instance drawn from a 'higher-level' population, $[150,160,170]$ itself is the 'top-level 'population that we can draw samples from. – Bertus101 Oct 15 '20 at 10:36
  • Similarly, the $6$ billion people in the world define $6$ billion distinct possible heights we can get if we sample from the human population, but we cannot get anything outside of those $6$ billion heights, as they are the top-level population. So human heights is a discrete random variable as far as I can see if we want to be truly accurate about it. – Bertus101 Oct 15 '20 at 10:37
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    Then, yes, it is discrete if you fix the population and sample from it. You have $M$ distinct values in $N$ people, like $M$ distinct colours in $N$ balls. – gunes Oct 15 '20 at 10:40
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    As @gunes correctly points out, if you want to view current population as the relevant population, then, yes, that results in a discrete distribution. At least to me, it is more natural to think of these (actually we are closer to 8 billion at the moment) as a sample from the (at least conceptually) infinite population of humans that have been, will be and could have been born with some height that, physically, is a continuous variable. – Christoph Hanck Oct 15 '20 at 11:49
  • @ChristophHanck That is a nice interpretation. – Bertus101 Oct 15 '20 at 13:58
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Height is continuous in principle, but reported as one of various discrete measurements in practice.

What's more, there can be many minutely different conventions both within and between datasets.

For example, in some countries original measurements might be variously in inches or cm (mm) and standardized to one or the other. That can lead to a distribution that is oddly spiky in detail, although it can take a very large sample to make that obvious.

Even if a country or a group of researchers uses just one of those units of measurement, the detail can still be complicated. In practice, observers can use different personal or shared rounding rules and there can be digit preferences (e.g. a tendency to report heights ending in 0 or 5 mm rather than any nearby values).

Although there is plenty of scope to make this rigorous by introducing notation and talk of different sample or outcome spaces, for many readers all that may be needed is a careful contrast between principle and practice.

To the fair comment that height is just an example here: sure, but there is a similar story about many other variables.

Nick Cox
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  • My question is not related to measurement, I purposely edited my original post to remove the issue of measurement because it was confusing matters. – Bertus101 Oct 15 '20 at 14:00
  • No doubt, but in this case and others my unoriginal suggestion is that discreteness is a matter of measurement conventions. I do not consider that height is a discrete variable as a matter of principle. Your making discreteness a matter of the sample size (I doubt I can do justice to your formulation) is to me a very puzzling take on the question, which I don't address directly. I am writing for people interested in the question implied by the title and not addressing your idiosyncratic formulation. – Nick Cox Oct 15 '20 at 14:15
  • In CV terms, you own the thread in the sense that you can control which if any answer is accepted, but others can chime in with different interpretations of,the question as well as varying answers. – Nick Cox Oct 15 '20 at 14:17
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A random variable $X$ is defined in terms of possible outcomes $\Omega$, not observed outcomes from trials. If $\Omega$ is defined to be countable, then $X$ is discrete, and if $\Omega$ is defined to be uncountable, then $X$ is continuous. Any finite set of draws of $X$ will not only be countable but finite, whether $\Omega$ is uncountably infinite, countably infinite, or finite.

In other words, we choose $\Omega$ to represent the possible outcomes as a modeling decision about what the possible values are. It may be useful to decide to model height as a continuous random variable because this decision allows us to use well-understood probability distributions, to have well-grounded notions of distance and ordering, and to estimate models that fit the data well. Even if this decision is wrong for fundamental physical reasons related to limits on what distances can be measured physically (N.B. I am not a physicist), it may be more useful than a discrete model with billions of ordinal outcomes.

jkpate
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