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### 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,

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.

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Division
Middle School Statistics

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