We all know what is the median and mean of a finite set of data .

What is not so well known is the *median* is the constant that minimizes , while the *mean* is the constant that minimizes .

This raises a sort of obvious, if not at all intuitive, generalization of the median and mean of a finite data set.

Let’s define to be the constant that minimizes .

The median of is just and the mean is .

So what about and ?

As I’m writing this I have little to no intuitive feel for what these “odd-beast” measures of central tendency” are actually indicating.

Some things are sort of obvious without much thought. For example, if for all then for all .

How does vary with increasing k?

To illustrate, let’s take a data set of 100 uniformly random numbers between 0 and 100.

Then a plot of for looks as follows:

We see that after the mean, the values of increase quite a bit but then begin to level out around .

For a set of 100 numbers randomly chosen form a standard normal distribution we get the following typical plot of with k:

By randomly choosing set of 100 numbers between 0 and 100 from a U-shaped distribution we get quite a different phenomenon:

I have no idea (yet!) how varies theoretically with k for different distributions, nor any ideas (yet!) whether these odd-beasts deserve the name of measures of central tendency.

But, whatever, I’m fascinated by them.

If you’ve seen anything like this before please let me know.

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