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I'm trying to figure out if my usage of the term "statistic" has drifted over time. I was thinking that, for example, any expectation or function of an expectation was a "statistic" in a physics sense but now I see that Wikipedia defines only the actual calculation (function of samples) as the statistic:

https://en.wikipedia.org/wiki/Statistic#:~:text=A%20statistic%20is%20an%20observable,measurement%20and%20a%20population%20average.

Is anyone aware of more physics leaning definitions where an expectation operator was called a statistic and does this not transfer into probability and statistics in some contexts?

Stephan Kolassa
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mathtick
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    Very much related: [What is the difference between an estimator and a statistic?](https://stats.stackexchange.com/q/47728/1352) I personally believe that the Wikipedia definition of "statistic" is mainstream in statistics. The expectation or any other quantity derived from the *distribution* itself (without recourse to a sample) would be called a "functional". If the usage in physics is different, then you may get more specific answers at [Physics.SE](https://physics.stackexchange.com/search?q=statistic) (if such questions are on-topic there). – Stephan Kolassa Jun 09 '21 at 15:10
  • @StephanKolassa Thanks, this is very helpful. I also reach for the word "functional" but as I go cross-discipline I wonder if it is confusing. Additionally, if any moderators or high-lever users see this it might be useful to add "semantics" as a tag across all of these sites for questions like this (I can not as I do not have sufficient rank). They are in a sense non-technical, but also sit at the highest level of the field as it is about knowing what *everyone* is doing across the field. – mathtick Jun 09 '21 at 15:21
  • Hm. As you see, I added the [tag:terminology] tag. We don't have a "semantics" tag. Would you be able to give a definition that clearly separates "semantics" from "terminology"? I would assume the two would soon end up as synonyms. – Stephan Kolassa Jun 09 '21 at 15:27
  • Also, I would find it more confusing to use established terms like "statistic" with different meanings in different fields. Taking terms like "functional" that are well established in one field A and using them *with the same meaning* in a different field B looks much more reasonable to me. You can always explain the definition in field B, and people can come to field A for more information - and not be confused because A has a different meaning than B. – Stephan Kolassa Jun 09 '21 at 15:30

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Expectation is a probabilistic procedure that maps a random variable to a constant. It integrates over the known distribution of the random variable. A statistic is a functional that maps a random sample to a point. The sample average is a statistic that, by the SLLN, converges to the expectation in the long run.

For example, if $X_1, X_2, \ldots, X_n \sim \text{iid Uniform} (0, \theta)$. The maximum is a statistic, the expectation of the maximum is $(n-1)/n \theta $. Obviously if you collected such a random sample IRL, you wouldn't know $\theta$, but you can find an unbiased estimator of it.

AdamO
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  • Ok, so this appears to be the common "statistics" definition of the word I guess. If any alternative uses come up maybe we can add them to this answer since this is less about "correct" and more about "complete" answers. A funny use case for stackexchanges in general but a good one! – mathtick Jun 09 '21 at 15:23
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    @mathtick I think you are hung up on the possible existence of a so-called "statistics" definition and a "non-statistics" definition. This definition can also be found in McDonald Weiss' "A Course in Real Analysis". Up to what level is your physics training? I would think anyone in quantum or higher would have had a rigorous (measure theoretic) training in the topic. Alternately, if you're doing labs, you're probably dealing with stats at a second year level which isn't rigorous enough to connect probability and statistics. – AdamO Jun 09 '21 at 15:36
  • good point re: labs vs grad level/quantum. I am more thinking grad level and beyond. As in what one would write in a paper. – mathtick Jun 20 '21 at 13:59