>> there isn't really anything like an
>> average between November and January, I guess... 
> 
> Then you guessed wrong. The problem was specifically stated as cyclic,
> for instance trying to find a yearly peak in something (births, deaths,
> bird migration, etc). 

I'm not sure why you would average months to get a peak of something over
time. Average doesn't help with peaks. That's more like counting how many
instances occurred in a given period, and find the period(s) (day of year,
week, month, whatever) that have periods with fewer instances on both
sides.

> When seen it this context (which was the context
> of the original question) there *is* an average of November + Januaty,
> and it is December. 

I don't quite agree. If we say Nov 03 and Jan 04, the mid-point is Dec 03.
We are not on a cyclic scale, but on a linear one. If you go cyclic, there
are two mid-points between any given set of two different points, and the
only difference is that one is closer than the other (usually). So without
adding additional information -- which is specific to the problem that you
want to solve --, there's no general way to say which one of the two is the
"right" one.

You say that it's obvious that Dec is the mid-point between Nov and Jan.
But there might be a good reason to say that June (2004) is the mid-point
between Feb (2004) and Oct (2004), and not December...  Whether it makes
sense to calculate December (and not June) as a mid-point between two
points Feb 04 and Oct 04 depends very much on the very specific question
you want to have answered.

I'm not that fit in statistics, but I'm not sure it makes at all sense to
talk about "average" in a cyclic domain. Probably an average requires at
least an unambiguous mid-point operation for any two points, which is not
given in a generic cyclic domain. There need to be additional definitions,
and these depend on the specific problem.

Gerhard
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