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We have population parameters ($\theta$) that are NOT random variables*: $\mu$, $\sigma$, $\rho$, etc.

We have sample statistics that ARE random variables: $\bar{X}$, $S$, $\hat{P}$, etc. Since they are random variables, it is conventional to capitalize them.

It is conventional to refer to population size as $N$ and sample size as $n$.

My question is the following: Why does this stray from our classical logic of using capital letters to denote random variables? Is it because $n$ refers to a specific number and thus is not a random variable? I am looking for the reasoning/explanation for why this notation is used; notation is naturally subjective, but I hope someone can shed light on why this specific convention is followed.

*Note that I'm only talking about frequentist statistics instead of Bayesian.

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  • $\begingroup$ There are various conventions for various things, & they aren't always consistent. The convention you're referring to is more common in the survey statistics world. In other contexts, $N$ is the total number of statistical units in an experiment, & $n$ is the number per condition. In still other cases, various subscripts are used to indicate different applications. In general, while $N$ is not a parameter, I wouldn't call it either a random variable or a sample statistic. $\endgroup$ Jun 23 at 19:27
  • $\begingroup$ There are limited numbers of small and capital latin letters. Choose them wisely! Good taste and judgment are required for naming mathematical objects clearly and efficiently, but as in any case where good taste is needed, some people will combine polka dots and stripes. Chacun à son goût-- but this has nothing to do with frequentism, Bayesianism, sample size, inference, or statistics. $\endgroup$
    – whuber
    Jun 23 at 20:19

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