From: ram@zedat.fu-berlin.de   
      
   ram@zedat.fu-berlin.de (Stefan Ram) writes:   
   >I waited a few days before answering to allow   
   >sufficient time to think about the problem.   
      
     (A few lines in this post contain more than 72 characters.)   
      
     I sometimes look at records of events like when someone got   
     up or went to bed on different days and wonder about the   
     "average time" he gets up or goes to bed. Some years ago,   
     I asked in another newsgroup how to average times, and from   
     answers there I kept this expression for a spreadsheet and   
     two times in cells A1 and B1:   
      
   REST(1.5+1/6.28318530717959*ARCTAN2((COS((A1-0.5)*6.283185307179   
   9)+COS((B1-0.5)*6.28318530717959))/2,(SIN((A1-0.5)*6.28318530717   
   59)+SIN((B1-0.5)*6.28318530717959))/2),1)   
      
     . I added a bit, like "REST", but the trigonometric part   
     came from that newsgroup. (Times t of a day in spreadsheets   
     often are 0 <= t < 1 because they often use the day as the   
     unit of time). At that time, I did not understand what was   
     going on! In hindsight, today, I see that this seems to   
     calculate the average via the average of the position of   
     the two points on a 24-hour clock in two dimensions.   
      
     The corresponding Wikipedia page is called, "Mean of circular   
     quantities".   
      
     I also have a copy of a page "Circular Values Math and   
     Statistics with C++11" saved. Other pages I found were   
     "Averages-Mean angle - Rosetta Code", "Averaging Angles -   
     The Math Doctors", or "algorithm - How do you calculate   
     the average of a set of angles - Stack Overflow". Maybe   
     some of these pages can still be found in the Web today.   
      
     There was also another web page that used lines some of   
     which have more than 72 characters. It contained,   
      
   |A good way to estimate an average angle, A, from a set of angle measurements   
   |a[i] 0<=i