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