Calculating Average Starting Time

Date: 06/23/2003 at 19:28:43
From: Patti
Subject: Calculating the average starting time

How do you average a series of starting times to get the average 
starting time?  Let's say the times were 9:30 a.m., 10:15 a.m., 11:45 
a.m., 1:30 p.m. and 11:30 a.m.

I've tried averaging the times as written; converting times after 
12:00 so they don't appear less than, say, 11:00; converting them to 
a decimal form; combining "easy" times (9:00 + 10:00) to get a partial 
average and then attempting to average these with the remaining times. 
Depending on how I combine the times, I get different answers. Do you 
count the earliest time as a "base" time and calculate the elapsed 
time of the other starting times from there? Do you include the 
earliest time when you average the rest?  

Thank you for your assistance.

Date: 06/23/2003 at 21:36:30
From: Doctor Rick
Subject: Re: Calculating the average starting time

Hi, Patti.

There are many ways you could do this, and if done correctly, they 
will all give the same answer. You must be doing something wrong, 
perhaps misunderstanding how to convert minutes and seconds to decimal 
form, or how to combine partial averages (which may require "weighted 
averaging").

Here is one way to average the times you have listed:

9:30 a.m.  =  9 hr 30 min =  9 + 30/60 =  9.5 hr past midnight
10:15 a.m. = 10 hr 15 min = 10 + 15/60 = 10.25
11:45 a.m. = 11 hr 45 min = 11 + 45/60 = 11.75
1:30 p.m.  = 13 hr 30 min = 13 + 30/60 = 13.5
11:30 a.m. = 11 hr 30 min = 11 + 30/60 = 11.5
                                         -----
                                         56.5 hr

56.5 divided by 5 = 11.3 hr
                  = 11 hr + 0.3*60 min
                  = 11 hr 18 min

Here is another way:

9 + 10 + 11 + 13 + 11 = 54 hours; 
divide by 5 ==> 10.8 hour = 10 hr + 0.8 * 60 min
                          = 10 hr 48 min

30 + 15 + 45 + 30 + 30 = 150 min;
divide by 5 ==> 30 min

10 hr 48 min + 30 min = 10 hr 78 min
                      = 10 hr + 1 hr + 18 min
                      = 11 hr 18 min

See, I got the same answer both ways. The first time I expressed 
everything in hours since midnight, averaged, and converted back to 
hours and minutes. The second time I averaged the hours since midnight 
and the minutes separately, then added the two averages.

Do these two methods show you what you were doing wrong? I'll be glad 
to discuss it further if you are still unsure of what you're doing.

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

Date: 06/29/2003 at 15:44:35
From: Patti
Subject: Thank you (Calculating the average starting time)

Dear Doctor Rick,

Thank you very much for your assistance.  I'm still not sure what I 
was doing wrong before, but when I applied your two methods to my 
other problems I got the same answer both times. I also tried it using 
fractions of an hour and that worked, too.  Thanks again!

Patti