Finding Average TimesDate: 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/ |
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