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I know that the average of a discrete random variable X can be estimated as:$$\widehat{E[X]}=\frac{1}{N}\sum_{i}x_{i}$$ where $i$ are the units we are averaging over. In practice, we just call this value an estimate of the average of $X.$ For example, when $X$ is income and we average over individuals, we get average income. My question is what units is this average in? I suspect it should be units of $X$ per i. So in my example, it is income per person.

This question may sound very basic, but if true, then I just want to make sure that I am more clear in the future when referring to the average. Thanks!

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You are correct.

The units of the average, or an estimate of the average, are the same as the units of the individual $x_{i}$

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