Finding Average Times

Date: 02/15/2002 at 13:07:24
From: Wanda Rayborn
Subject: How to figure an average time

My son and I have started a physical fitness program, and we are 
keeping track of our time for the mile. At the end of the week we 
would like to figure what our average time for the week is and put it 
on a line graph. We plan to do this for 16 weeks and chart our 
progress. I have tried converting the times to total seconds and 
then dividing by the number of days. Then converting back to minutes 
and seconds but the answer I get does not seem logical. 

Date: 02/15/2002 at 14:32:39
From: Doctor Ian
Subject: Re: How to figure an average time

Hi Wanda,

Suppose that your times were

  15 minutes, 10 seconds
  15 minutes, 20 seconds
  15 minutes, 30 seconds

You could average the times; or you could average the parts of the 
times that were in excess of 15 minutes:

  10 + 20 + 30
  ------------ = 60/3 = 20 seconds
       3

So the average time would be 15 minutes, 20 seconds.  Does this make 
sense?  It it still seems a little murky, take a look at the Dr. Math 
archives for a more detailed explanation of why it works: 

   Normalization
   http://mathforum.org/dr.math/problems/michael.08.01.01.html   

In your case, I would probably set everything relative to 15 minutes.  
Then the times would be

  15 minutes + 37 seconds
  15 minutes + 35 seconds
  15 minutes - 40 seconds
  15 minutes - 70 seconds
             
37 and 35 add up to 72; minus 70 is 2; minus 40 is -42. One fourth 
of that is about 10 seconds, so the average time should be about 
10 seconds less than 15 minutes, or about 14 minutes, 50 seconds. 

But if you're really going to do this for 16 weeks, you're probably 
going to want to use a spreadsheet; or keep some kind of running total 
of the time, i.e., 

    day     time       total
    ---     -----      ----- 
     1      15:37      15:37
     2      15:35      31:12    <- 30:72
     3      14:20      45:32
     4      13:50      59:22    <- 58:82

Note that I just add the minutes and the seconds separately; and when 
the seconds are greater than 60, I subtract 60 and 'carry' an extra 
minute over to the minutes column. 

Now, when I want an average, I can do something like this:

  59:22   56:00 + 3:22   
  ----- = ------------ 
    4           4      

          56:00   3:22
        = ----- + ----
            4       4


          4*14:00   3:22
        = ------- + ----          <- This is why I chose 56:00
             4        4

                  202 sec
        = 14:00 + -------
                     4


        = 14:00 + 50 seconds or so

        = 14:50

Now, this is a lot of addition, and fooling around with minutes-
versus-seconds (to say nothing of hours), and frankly it would give me 
a headache to do it for more than a few days.  

Fortunately, you can use the normalization idea here, too. Suppose you 
pick a 'target time', like 12 minutes, that you think you'll 
eventually get close to. Then you can record just the excess over 
12 minutes (which is easy enough to do in seconds):

  target = 12:00

   day      time     excess      total excess
  ----    -------    ----------  ------------
     1      15:37    3:37 = 217           217
     2      15:35    3:35 = 215           432 
     3      14:20    2:20 = 140           572
     4      13:50    1:50 = 110           682

Now I can quickly get the average excess over 12 minutes, by dividing 
682 seconds into 4 equal parts:

  682/4 = 170.5 seconds

        = 2 minutes, 50.5 seconds

so my average time is 12 minutes plus the average excess, or 14:50.5. 

If you end up running faster than 12 minutes, all that will happen is 
that you'll start subtracting 'excess' times from the total, instead 
of adding them, which will make the total go down, which will make the 
numbers you're using even smaller, which is a good thing.  

In the long run, you might end up with a calculation that looks like

  -2415/54 = -44.7 seconds

which, subtracted from 12 minutes would give you an average time of 11 
minutes, 15.3 seconds.  

I hope this helps.  Write back if you'd like to talk more about this 
or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/