# Does it make sense to model population distributions as independent from individual distributions? (social sciences)

Let's start with an example. Consider that each individual $$i$$ in a group of size $$n$$ throws a dice, where each dice is different (random variable $$X_i$$, notice subindex $$i$$).

In social sciences, we tend to be interested on the distribution of the group's result. However, we tend to think of the latter as a stable, clearly identified distribution $$X$$, which is NOT derived as a function of individual's distributions.

Consider concrete "variables" (in the sense of objects) studied in social sciences, like annual income of workers, number of employees in firms, size of households, proportion of black students in schools, price of stocks. In each of these examples, most of the times the random variable of interest is the population's distribution (parameters/moments/etc), estimated from samples/data.

But to me, it is clear each single element of those variables is also a random variable. The size of my household, my annual income, the % of black students in the school next door, the price of my company's stock, they are all random variables, in the sense that they are processes that can take different values in different states of the world. That is, they can take different values given certain probabilities. There is a certain probability I divorce, or I have another child. There is a certain probability I am fired, or promoted. Etc. Moreover, all these probabilities are different across individuals/units. My colleagues' income is not the same as mine, and they move in different trajectories (although certainly with some degree of correlation). Same is households' size and schools' segregation levels.

So, in this context, does it make sense to think of populations having separable paremeters/moments/distributions from individual ones, and even not related to them?

Plus: in this regard, is social sciences really different from other sciences?

• Are you familiar with multilevel models? With the use of partial pooling, you could infer the population level, sub-group level and individual levels, simultaneously Commented Sep 26, 2023 at 0:21

### Population distributions are different to distributions of individual values, but they can be closely related (e.g., in the IID model)

I will give you a run-down of how this is usually treated in statistics, which I think is a good basis for the situation you are dealing with. Social science is no different to other sciences in the regard that it should use statistics in the same way as other sciences. The treatment I will show you here is the standard way that statistics treats population distributions, and it is applicable to any branch of science, including both the physical and social sciences.

Suppose we have a population of $$N$$ people and you measure some characteristic of those people (e.g., their income in the present year). We can encapsulate this information using the random variables $$X_1,X_2,...,X_N$$. In a statistical inference problem, we would typically take a random sample of people from this population and use that sample to make inferences about the population. (In the alternative event that we observe the whole population we then have a full census and so no inference problem arises.)

Strictly speaking, the "population distribution" is the multivariate probability distribution of the entire random vector $$\mathbf{X}_N \equiv (X_1,X_2,...,X_N)$$, which fully specifies the joint probabilistic behaviour of the $$N$$ random variables at issue. We can state this distribution for the random vector as follows:

$$\mathbf{X}_N \sim \text{PopDist}_N.$$

This distribution is very general, but we can often simplify things if there is some more "structure" to the problem. Additional structure to the problem can allow us to simplify this joint distribution, and perhaps even write it as a function of some simple univariate distribution. In particular, one common case where there is a substantial simplification is the IID model which we describe below.

#### Exchangeable random variables and the IID model

One particular case occurs very often in applications of probability, which is the IID model. In many sampling situations it is reasonable to assume that the order of the random variables in the population does not affect their probability distribution (i.e., the probability distribution would remain the same if you swapped the order of any of the random variables), and that this order-invariance condition would hold regardless of how many people are in the population (a condition called "exchangeability" of a sequence). This condition occurs in simple random sampling. When this condition holds, it turns out that the observable random variables are IID (independent and identically distributed) conditional on some underlying univariate distribution:

$$X_1,X_2,...,X_N \sim \text{IID } \text{Dist}.$$

In this case, the population distribution can now be written as the "product distribution" of $$N$$ lots of the univariate distribution:$$^\dagger$$

$$\text{PopDist}_N = \underbrace{\text{Dist} \times \text{Dist} \times \cdots \times \text{Dist}}_{N \text{ times}}.$$

You will notice that this means that in the IID model the population distribution is now a function of the univariate probability distribution for a single random variable. Because of this, we sometimes use $$\text{Dist}$$ as a proxy for the population distribution and we might even refer to it as the population distribution (though this involves a slight abuse of terminology). The take-home insight from this is that the distribution $$\text{Dist}$$ is both the individual distribution for a single random variable, but it is also an object that fully determines the population distribution and can therefore act as a proxy for this distribution.

It is not surprising that the individual distribuion $$\text{Dist}$$ acts as a proxy for the population distribution in this case. After all, if the underlying random variables in the population are identically distributed then there is a single univariate distribution that describes how they each marginally behave. If they are also independent then their marginal behaviour is sufficient to imply their joint behaviour and so there is a single univariate distribution that describes their joint behaviour.

Incidentally, the "exchangeability" condition that leads to the IID model is the same idea that you are talking about when you refer to a distribution as being "stable" over the population. If you believe that there is sufficient stability that the exchangeability holds, this will lead you to the IID model, which then allows you to treat the individual distribution of a random variable as a proxy for the population distribution. If you don't have this kind of stability, and exchangeability doesn't hold, then you are going to need a more complicated model, and this might lead you to a situation where there is no longer a one-to-one correspondence between the full population distribution and a univariate "proxy".

I hope the above explanation gives you a starting point to understand the relationship between the population distribution and the distribution of individual values in the population. This issue is closely related to the theory of exchangeability and the IID statistical model. You might also be interested in other questions/answers on this site that bear on this topic (see e.g., here, here, here, here, here, here, here and here).

$$^\dagger$$ Note that the $$\times$$ used here is not a regular multiplication sign --- it refers to taking the product distribution on the Cartesian product of the underlying vector spaces for the ranges of the random variables.

does it make sense to think of populations having separable paremeters/moments/distributions from individual ones,

The distribution of a population describes the variations that occur in the individuals.

Example. Say I give a large group of people randomly a six sided dice or a twenty sided dice, I let them roll three times and record the sum, then a histogram could look like:

### r-code to produce histogram

set.seed(1)
n = 10^4
rolls = function() {
dice = sample(c(6,20),1)
roll1 = sample(1:dice,1)
roll2 = sample(1:dice,1)
roll3 = sample(1:dice,1)
return(roll1+roll2+roll3)
}
hist(replicate(n,rolls(), freq = FALSE), breaks = seq(-0.5,60.5,1))

### theoretic distribution

k = 1:60
f1 = matrix(c(1,rep(0,60+61*3)),4)
f2 = matrix(c(1,rep(0,60+61*3)),4)
for (r in 1:3) {
for (ki in k) {
for (mi in 1:6) {
if ((1+ki-mi) > 0) {
f1[r+1,1+ki] = f1[r+1,1+ki] + f1[r,1+ki-mi]/6
}
}
for (mi in 1:20) {
if ((1+ki-mi) > 0) {
f2[r+1,1+ki] = f2[r+1,1+ki] + f2[r,1+ki-mi]/20
}
}
}
}

theoretic = (f1[4,-1] + f2[4,-1])/2

lines(k,theoretic)
points(k,theoretic, pch = 20, cex = 0.7)


This histogram, displaying the population distribution, includes the randomness that may happen within the individuals (the process where each individual rolls those three dices).

At the same time the histogram shows that the population might be coarse and multiple hierarchical processes of randomness might occur which make a distribution that does not look like a pretty Gaussian bell curve, and typical parameters like the mean or median might not sufficiently describe the distribution.

Also the 'population distribution' is different from some theoretical (stable) distribution. I have added points and a line for some hypothetical distribution when the sample size is infinite. This relates a bit to the question: Does it make sense to compute confidence intervals and to test hypotheses when data from whole population is available? (and I believe there are other versions around) that considers a census where the sample is equal to the population. In that case 'the population' is sometimes itselve considered to be a sample from a hypothetical population of populations. In social sciences the question might be either about the one type or the other. For example, the question “are people from country A taller than people from country B?”, will consider the set of people as the distributions. On the other hand, a question like “are people who eat olives 'healthier' than people who do not eat olives?”, might be after the mechanism and the hypothetical population, and not just after the actual population.

• Thanks. Cannot replicate diagram (default bins different and seed not set either). And, distribution DOES look quite stable, when run over different seeds, with clear bumps around the expected averages of the two dice types (3.5*3=10.5; and 10.5*3=31.5). You seem to show an example were population distribution IS quite stable and well defined (maybe even with closed form pdf/cdf) (obviously not the same as the individual ones, but that is not the question). Commented Sep 25, 2023 at 23:06
• @lunchonacho it is not stable. Running with different seeds gives different results. But, yes the variations in the values are small in this example. When you decrease the population size then these variations increase. Commented Sep 26, 2023 at 5:10