There are a few ways I think of averages of a (discrete) random variable:

  1. Average is an OLS estimate of running a regression of the random variable on a constant term. In that sense, it is a value that 'best' represents the data (minimizes the Euclidean distance)

  2. A central tendency of the data (i.e. my best guess of the random variable, without having any data on it)

Now, in many cases, the average is not included in the support of the random variable. For instance, the expected value of the outcome of rolling a die is 3.5. However, this value is not included in the support. How would one interpret the average in this case?


I think this is one of those cases in which "interpretation" (another common one is "intuition") is just not the problem. The mean is

$\frac{1}{n}\sum x_i$

You get that answer, because that's the definition. Here it may clash with intuition, because you can't roll 3.5 on a die, but that's more a problem of the intuition itself - which is not something that's always useful. If it were, we wouldn't need math.

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    $\begingroup$ Note also that awkward-looking averages can be usually made sensible by translating to totals. If gender is coded 0 or 1 then mean 0.5 may look awkward to interpret at first sight, but everyone should realise that it is just the equivalent of 50 males and 50 females, or whatever the frequencies are. There are too many feeble jokes about statistical reports of average families with 1.2 children, or whatever, but the problem is the idea that a mean must correspond to a concrete instance. $\endgroup$ – Nick Cox Feb 8 '16 at 16:59

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