From: tr.17687@z991.linuxsc.com   
      
   Ben Bacarisse  writes:   
      
   > Ben Bacarisse  writes:   
   >   
   >> Tim Rentsch  writes:   
   >>   
   >>> Ben Bacarisse  writes:   
   >>   
   >>    
   >>   
   >>>> The "other" average, minimises the sum of the of the angular   
   >>>> distances.   
   >>>   
   >>> I think you mean it minimizes the sum of the squares of the   
   >>> angular distances.   
   >>   
   >> Yes thanks.  I originally thought I'd write an abbreviation but then I   
   >> convinced myself I would not forget any of the "squares of"s.  Proof   
   >> reading might have been useful too.   
   >>   
   >>>>> and the center of mass is never on the unit   
   >>>>> circle (not counting the case when all the time values are the   
   >>>>> same).  Even so it's an interesting way to view the distinction.   
   >>>>   
   >>>> It's more interesting because it really is about chords!   
   >>>   
   >>> I don't buy the "really" here.  The center of mass is fundamental   
   >>> and always well-defined.  Furthermore it works for collections of   
   >>> points that are not all on the same circle or sphere.  Looking at   
   >>> the problem in terms of chords seems somewhat artificial, or at   
   >>> least less fundamental.   
   >>   
   >> There are other ways for something to be interesting (at least to me).   
   >> When you plot the two averages, and look at the chords vs the arcs, it   
   >> becomes clear why one average "looks" more average than the other.   
   >   
   > I can see a possible misunderstanding here.  You might have taken my "it   
   > really is about chords" to mean "it is fundamentally about chords", but   
   > I said "it really is about chords" because you thought I'd used the word   
   > chord incorrectly to refer to the lines between the centre of mass and   
   > the data points.  I was saying "I really did mean chords".   
      
   Now that I see what you meant it all makes sense.   
      
   Also I agree that comparing the chords view and the arcs view   
   gives a useful perspective.  I am starting to appreciate the   
   value of that perspective more as time goes on.   
      
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