Algebra: Average of an Average

Date: 10/20/97 at 18:38:24
From: Ron Cook
Subject: Algebra: average of an average

Hi!  This one is a real-life need.  I work for Andersen Consulting and 
am having a hard time remembering my basic Algebra!

If the "Wireless" line of business hires 50 people in one month, and 
the "Multimedia" line of business hires 80 people in one month, what 
is the average number of people per month we are hiring?

At first, one thinks (50 + 80)/2 = 65 is the correct answer, but it's 
not because we're taking the average of an average, right?

Don't you have to let x = something and then do 1/50 + 1/80 and 
solve... or something?

Please help!

Thanks,
R. S. Cook

Date: 10/20/97 at 20:25:21
From: Doctor Tom
Subject: Re: Algebra: average of an average

Hi Ron,

65 is the correct answer if the question is: "What is the average
number of people hired per line of work per month?"

Assuming that the entire company is made up of only those two lines
of business, and you're asking "What is the average number of
people hired by the whole company per month?", the answer is
obviously 50+80 = 130.

The operations above are perfectly reasonable. Let me show you
the kind of situation that I think you were worried about, and
you'll see why it's different from the situations above:

I take 4 exams and have an average score of 80. Then I take 2
more exams and on those 2, my average is 100. What's my overall
average for the course?  

The wrong way to do it is (80+100)/2 = 90. This is wrong because the 
averages were of different-sized groups. To get the correct answer, 
I know that on the first 4 exams I got 320 total points because 
when I divide 320 by 4, I get 80.

Similarly, for the last two exams, I must have gotten 200 points
total. So for the six exams, I got 200+320 = 520 points and
520/6 = 86.66666 = my real grade average.

For your problem, the averages you are averaging are for the same
period, so it works out. To convince you that it's true, let's just 
look at a situation where the averages came from 10 months of data.

Then in the first line of business, 500 people must have been hired, 
since 500/10 = 50. Similarly, 800 were hired in the other, since 
800/10 = 80. Altogether, 1300 people were hired in the ten months, 
or 1300/10 = 130 per month, company-wide.  Or if you're trying to get 
the average per line of work, 65 is right, since if each group had 
hired 65 people each month for 10 months, there would be 65*20 = 1300 
total hires, so it works out.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/21/97 at 08:57:12
From: ronald.s.cook
Subject: Re: Algebra: average of an average

Thanks!  You know me better than myself.

Ron