What are the different types of “averages” and when to use each one?

So, the concept of an average truly is somewhat abstract. Most statisticians define it as a "measure of central tendency." Others say it is the "center of gravity" for a set of numbers.

I personally prefer a slightly more concrete explanation: A statistic that describes the "typical", or better yet, "representative" value of a data set. We humans get to decide what "representative" actually means. Is it the most common number? The number that falls "in the middle" of a set of numbers? Etc.

I am going to list a few types of averages and am wondering if someone could provide a data set where that particular average is the best choice for describing a "typical" or "representative" value for that data set.

Types of averages:

• Arithmetic Mean: The number that you could use in place of each of the values of a data set, and still have the same sum. Formula: $$\overline{X}=\frac1N\sum\limits_{i=1}^{N} X_i$$

• Mode: The most common value in a data set. No formula that I know of.

• Median: The literal middle number of a data set where the values are listed in ascending order. No formula that I know of.

• Root Mean Square: Don't really know how to describe this average in a physical sense. Maybe the average that gives larger numbers in the data set more "weight" or "significance?" Formula: $$X_{rms}=\sqrt{\frac1N\sum\limits_{i=1}^{N} X_i^2}$$

• Mean Root Square: I just made this one up, but it seems to work well when I applied it to some random data sets. It seems to do the opposite of the RMS and gives smaller numbers in the data set more "weight" and "significance." Formula: $$X_{mrs}=\frac1N\sqrt{\sum\limits_{i=1}^{N} X_i^2}$$ EDIT: Turns out this should not be considered an average because it fails to describe a data set that consists of only one number, unlike all the other averages.

• Geometric mean: The number that you could use in place of each of the values of a data set, and still have the same product. Formula: $$GM=\sqrt[N]{\prod\limits_{i=1}^{N} X_i}$$

Feel free to add in any other popular or useful types of averages and when to use them!

• The "mean root square" of the dataset 1,1,1,1 is 1/2, so it is not an average in any sense. You should try the "square mean root" instead. – Rahul Apr 13 '17 at 18:53
• @Rahul Hmm, but I could make a similar argument for other averages as well. In this data set, 1, 1, 20, 21, 22, 23, 24, 25, the mode is 1, but certainly fails to describe the "representative" value and thus also not an average in any sense. – Nova Apr 13 '17 at 19:17
• @Rahul I don't think there is any one average that works universally. Maybe my MRS idea fails to work for some data sets, but is effective in many others. – Nova Apr 13 '17 at 19:19
• @Rahul Unless there is some mathematical rule about averages that I don't know about. For example, for any data set, an average must always be a number that is greater than or equal to the lower boundary and lesser than or equal to the highest boundary. If this is so, then yes, MRS would fail to be an average. – Nova Apr 13 '17 at 19:19
• There is no complete axiomatization that covers all the existing averaging methods as far as I know, but certainly all averages I know of have the property that if all values in the dataset are identical then the average is the same as that value. Anyway, your mean root square is just the root mean square divided by $\sqrt N$. – Rahul Apr 13 '17 at 19:43