# Explain the descriptive statistics notion of population (distribution) to a measure theorist

My understanding of probability theory is a rigorous measure-theoretic one and I know almost nothing about descriptive statistics. However, I need to understand some of the commonly used notions in this field.

Say we are interested in the average weight of all people in the world and we are measuring the sizes $$x_1,\ldots,x_n$$ of $$n\in\mathbb N$$ "randomly chosen" people.

In this simple example, what do the terms "population", "population distribution", "population mean" and "sample mean" refer to?

My understanding is the following: Let $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space and $$\mu$$ be the probability distribution on $$(\mathbb R,\mathcal B(\mathbb R))$$ describing the distribution of the weights of all people in the world.

Then the experiment described above should simply mean that we are taking independent random variables $$X_1,\ldots,X_n:\Omega\to\mathbb R$$ with $$\operatorname P\circ X_i^{-1}=\mu$$ for all $$i\in\{1,\ldots,n\}$$. And the particular people we have selected correspond to an event $$\omega\in\Omega$$. So, $$x_i=X_i(\omega)$$. Now I guess the "population mean" is $$\operatorname E[X_1]=\int x\:\mu({\rm d}x)$$ and the "sample mean" is simply the unbiased estimator $$\frac1n\sum_{i=1}^nX_i$$ evaluated at $$\omega$$.

Did I get these things right? I'm not sure how "population" and "population distribution" are different. It seems like both would refer to the distribution of the random variables, which is $$\mu$$ here.

• Neither the concept nor any element of the practice of descriptive statistics requires (or even benefits) from a measure-theoretic framework, suggesting you might intend us to understand "descriptive statistics" in an unusual sense. Could you therefore clarify what you mean by this phrase? – whuber Jul 2 '20 at 14:11
• I didn't say that there is a benefit or that that it's required. I just want to know how these notions correspond to the measure-theoretic point of view. – 0xbadf00d Jul 2 '20 at 14:15
• You might be someone for whom springer.com/gp/book/9780387947174 was written. Although the author is undoubted extremely able he wrote one of the most bizarre books in the history of statistics. Volume 1 appeared in 1996, and volume 2 was promised then. – Nick Cox Jul 2 '20 at 14:51
• The "population" is always ALL the individuals (people or things) in which you're interested, and a sample is always a subset of the population. The population mean is simply an average (of whatever variable you're interested in) over the entire population (usually not known for practical reasons), and the same mean is the average taken over the sample. Most of statistics is involved in understanding how well the sample mean approximates the population mean. – Adrian Keister Jul 2 '20 at 14:54
• This might interest you: stats.stackexchange.com/questions/199280/… – kjetil b halvorsen Jul 2 '20 at 15:10