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Before the realizations of the random variables $X_1,\dots,X_n$ become known, their sample mean can be regarded as a random variable.

This statement confuses me. Can you give me an example?

Tim
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John Richard
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    What is the average of three rolls of a die? It's a function of the individual rolls which are themselves random, and so the average is also random. – dsaxton Sep 14 '16 at 16:45
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    You don't want to pass on a [beautifully detailed explanation on this topic](http://stats.stackexchange.com/a/227006/67822) by @fcop – Antoni Parellada Sep 14 '16 at 17:17
  • @dsaxton oh, i see. so this means that even if we use realizations of the rvs, sample mean is itself a random variable, right? i'm not too familiar with mathematical notation, thats why im asking. – John Richard Sep 14 '16 at 17:26
  • Once you roll the die then you know what value it takes on and it's no longer random. But when we talk about the average beforehand, it is just as random as the individual rolls. – dsaxton Sep 15 '16 at 02:44
  • @JohnRichard John, if you feel like I answered your question, please consider "accepting" my answer by clicking on the tick mark on the left. – Antoni Parellada Sep 17 '16 at 06:10

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Without mathematical notation, and to elaborate on @dsaxton didactic comment...

In a probability "problem" (read, space) there is a random experiment. This is something that happens out there in the physical world: the tossing of a coin, the rolling of a die, or the temperature in Philadelphia in the morning. For God (sorry, just a construct to move the argument along) the experiment is not random; and it wouldn't be random for us if we could account for every single minute factor that comes into play (direction of the wind, force exerted on the die, exact material composite of the die, height reached upon tossing, acceleration,... you get the gist). But all we can do is observe.

What we observe is the outcome. And we assign (read, "map") this outcome to a numeric value in the real line. This function from the observation of something happening to the assignment of a numerical value to it is the random variable. And the map or function would be defined like this: "we look at the die, and we notice which side is facing up." Yes, kind of silly, but we could have decided to note the velocity of the die upon impacting the table - that would be a different random variable (a continuous one) on the same random experiment: rolling the die.

Now we have a the random variable defined. The actual realizations, say $1$ or $6$, are not the random variable - they are individual outcomes.

A different random variable that we can choose to define is on a sample - not a single experiment, but a group of experiments: multiple die rolls, say. We can define it as: "Note the number facing up in $5$ die rolls; add them; and divide by $5$". The sample mean.

This is another random variable - a statistic on the sample.

They are related, and when we have a realization of the group of experiments ($5$ die rolls), say, $\{5, 5, 6, 1,2\}$ the mean will be determined - no longer a random variable. I believe that is what (loose terminology aside) the sentence: "Before the realizations of the random variables $X_1,\dots,X_n$ become known, their sample mean can be regarded as a random variable" implies.

The sample mean is just a statistic tying this answer to the more rigorous post by @fcop. To quote a key passage of his/her answer:

So, to answer your question, a test statistic is a special case of a random variable. The test statistic is thus a random variable, related to the random variable from which the sample (used to compute the test statistic) was drawn.

Community
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Antoni Parellada
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  • This makes sense. Thank you so much, thank you everyone for their answers. – John Richard Sep 14 '16 at 18:17
  • @John Richard I'm happy it made sense - it is a difficult and fascinating topic as you can see on the post I link. I did my best to explain it stripped of baffling set theory notation. – Antoni Parellada Sep 14 '16 at 18:19