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The definitions of a parameter and statistic pretty much agree: parameters and statistics are numerical characteristics or numerical summaries of a population and sample, respectively, for a given study. I don't think this is common usage, but...

Could the population size $N$ be considered a parameter? Could the sample size $n$ be considered a statistic?

After all, the size of the population or sample is a numerical summary or characteristic of the population or sample.

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I agree with the answer given by @Dave (+1) in most aspects. I also agree with the "philosophical" sentiment that population size should usually be considered $\infty$ and therefore not a parameter.

On the other hand, there are some practical applications were it is reasonable to consider the finite population (rather than the generating process) and thus the population size becomes a parameter. A classic example of this is the German Tank Problem.

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A “statistic” has the rather trivial definition of being a function of the data, so counting how many points there are is a function of the data. Sure, sample size is a statistic.

A “parameter” is a knob you turn to get some distribution to behave a certain way. If you want a normal distribution centered at 7, turn $\mu$ up to 7. If you want it spread out a lot, turn $\sigma^2$ up to 81.

“Population size” is a strange idea, and you can find differing opinions about if it can exist. You may think that if you observed every person, then you’ve observed the population. Say you found that people now are taller than people 200 years ago, having measured everyone in 1819 and 2019. If you then do a hypothesis test of their heights, you’re saying that what you’re interested in is the process that generates human heights, and the humans you observed are the ones who happened to be born.

I say that a parameter is some characteristic of a distribution (in the mathematical sense of being a CDF). Therefore, population size is $\infty$ and not a parameter.

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