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Ah, I think I'm with you now. My stats is rusty, but I think what you're asking about is one of the fundamentals of variance.

I think this[1] Stats StackExchange answer frames it well.

That is helpful, thanks. Especially this part:

> I believe that the Law of Large Numbers explains why the variance (standard error) goes down when the sample size increases. https://en.wikipedia.org/wiki/Law_of_large_numbers

So what I was looking at is the inverse of the law of large numbers.