Averaging Averages

Date: 11/09/1999 at 14:44:10
From: Stacy Francis
Subject: Averages

I was reading your response on averaging averages. This has come up in 
a meeting I go to regarding a report that I am responsible for every 
month. After reading your response, I want to make sure that I am 
doing this right.  

I have 5 different departments that send me an average rating to 3 
different questions from a feedback form (rated 1-5). For example, 
this is what they send me:

Question 1 - average 4 (this is the average of question 1 from all 
feedback forms this department received during a certain time period)

Question 2 - average 3 (same comment as above)
Question 3 - average 5 (same comment as above)

Total average- 4

After getting this information from all 5 different departments, I 
combine them in a total company report by taking everyone's average to 
question 1 and then average that to get an average for the whole 
program. (For this program the company average to question 1 is...) 
and so on for questions 2 and 3 and the total average.

If I am reading your response correctly to the question on averaging 
averages, this is okay to do if all information is contained within 
the same time period (which it is).

Please advise. Thank you,
Stacy Francis

Date: 11/15/1999 at 19:45:48
From: Doctor Douglas
Subject: Re: Averages

Hi Stacy,

It sounds as though you are trying to average a set of averages. As 
long as the data reflect the same measurement (or question) for each 
of the individual groups, then it is okay to proceed. That is, if 
question 1 is the same among all departments ("Please rate the chef at 
the last company BBQ on a scale of one to ten.") then it is a 
meaningful question to ask: what did the employees think of our last 
BBQ chef?

Now, when you take averages of averages, there is often a preferred 
way to do this operation. It is called "weighted averages" and 
reflects the fact that the number of observations may vary among the 
different groups. For example, let's say that the Marketing Department 
has 50 employees, and the Research Department has only 6. If the 
Marketing Department average was 8.5 ("BBQ was great!") and Research 
Department average was 3.1 ("heartburn!"), what is the correct average 
for these 2 departments? We might simply take the average of 8.5 and 
3.1, i.e. 5.8.  But it seems unfair to let the Research Dept sway the 
whole vote, having only 6 employees. The weighted average accounts for 
the differing number of employees:

     weighted avg. = (no. of M)(AVG-of-M) + (no. of R)(AVG-of-R)
                     -------------------------------------------
                         (total no. of M and R together)

                   =  (50)*(8.5) + (6)*(3.1)
                      ----------------------
                             (50 + 6)

                   =  7.92

Do you see how the final average is much closer now to the Marketing 
Department's evaluation? Another way I sometimes explain it is that 
simple direct averaging is like the way states are represented in the 
U.S. Senate (every state gets two votes), while weighted averaging is 
like the way states are represented in the U.S. House of 
Representatives (every state is represented in proportion to its 
population).

Now, it's impossible for me to say which type of averaging is correct 
for your situation, but I think it's probably better to use weighted 
averaging when you have information about the number of observations 
in each of the group. For a bit more on weighted averages, you may 
wish to check out the following in our archives:

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


I hope that helps. Please write back if anything is still unclear.

Good luck!

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