# Rigorous definition of "Average"

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)?

• Do you have a definition for "statistical population"?
– user9464
Nov 17, 2016 at 18:37
• @Jack I may define it: "Any finite countable set of numbers" which we use to apply our statistical study on it. Nov 17, 2016 at 19:29
• Then you have a definition for the average of a finite set of (real) numbers, don't you?
– user9464
Nov 17, 2016 at 21:28
• @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. Nov 18, 2016 at 7:04

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, $\langle x\rangle=\sum_i x_ip(x_i)$.