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We usually tend to say the "Average" is whether "Mean", "Median" or "Mode" and in colloquial usage "Average" is always equivalent to "Mean".

But my question is: Is there any precise rigorous definition of "Average of a statistical population" in statistics (regardless of our knowledge about mean, median or mode)?

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  • $\begingroup$ Do you have a definition for "statistical population"? $\endgroup$
    – user9464
    Nov 17, 2016 at 18:37
  • $\begingroup$ @Jack I may define it: "Any finite countable set of numbers" which we use to apply our statistical study on it. $\endgroup$ Nov 17, 2016 at 19:29
  • $\begingroup$ Then you have a definition for the average of a finite set of (real) numbers, don't you? $\endgroup$
    – user9464
    Nov 17, 2016 at 21:28
  • $\begingroup$ @Jack By reading your answer what I get is you mean the "Average" is nothing but "Mean". But the thing is I thought it was more than that. I mean I expected that "Average" had a separate standalone meaning and it could be equal to "Mean" in just some special cases. $\endgroup$ Nov 18, 2016 at 7:04

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From a book (I forget the title) I read, the 'average' is basically a number which (we believe) can represent the whole population. Since Statistics usually deals with a huge number of data, we need a kind of 'summary' of the whole data, which is the 'average' itself. Thus, the choice of the type of average (mean, median or mode) depends on the discretion of the statistician based on the problem.

Moreover, 'standard deviation' is also used, alongside with the average, to provide the summary of the whole data.

I hope this helps!

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For continuously distributed $x$, $\langle x\rangle=\int xp(x)dx$ (appropriate limits), where $\langle x\rangle$ is the "average" or "most expected" or "expectation value" of $x$, and $p(x)$ is the distribution function. For a discrete population, $\langle x\rangle=\sum_i x_ip(x_i)$.

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To quote Wikipedia:

In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean. In statistics, mean, median, and mode are all known as measures of central tendency.

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