Averaging Percentages

Date: 12/11/98 at 00:34:39
From: Dominic Dionisio
Subject: Average of percentages

Hi,

I am having difficulties in explaining to several friends that you 
cannot take percentages by totalling them up and then averaging the 
total of the percentages. It does not equal the percentage of the 
total of the numbers.

Is there a rule or theory that can explain this better?

Date: 12/11/98 at 12:11:37
From: Doctor Peterson
Subject: Re: Average of percentages

Hi, Dominic. I'm not entirely sure what kind of problem you are 
referring to. Certainly there are at least some situations where you 
can average percentages. For example, if there are 50 questions on an 
exam, and three students got 20%, 30%, and 40% of them right, then the 
average number of questions they got right is 30%, or 15 questions.

I suspect what you are thinking of is cases where the percentages are 
taken from different totals, in which case weighted averaging is 
needed. For example, if I survey 20% of 50 people, and 80% of 500 other 
people, then I have not surveyed (20+80)/2 = 50% of the total 
population, but:

   .20 * 50 + .80 * 500   10 + 400   410
   -------------------- = -------- = --- = 74.5%
         50 + 500           550      550

The problem is simply that the percentages in such a problem do not 
represent fractions of the same total, so they can't be added.

For more information on weighted averages, please see:

   http://mathforum.org/dr.math/problems/smith11.2.98.html   

For another explanation of weighted averages, if you want another 
perspective, see:

   http://mathforum.org/dr.math/problems/riggins11.16.95.html   

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