Explaining Weighted Averages

Date: 09/24/2002 at 14:22:10
From: Michelle Decker
Subject: Explaining Weighted Averages to My Assistant

Hello, 

I believe that I am a good example for students who say, "I'll never 
use this in real life." I use math and algebra every day in my job, 
something I myself never thought I would say. 

We use weighted averages to calculate broadcast rates and 
efficiencies, and I'm having trouble explaining the methodology behind 
this process to my 21-year-old assistant. I can do the calculations 
myself but I can't seem to teach her the reasons why I'm doing what 
I'm doing from a mathematical perspective.  

Can you please give me a layperson's explanation of why and how to use 
a weighted average?

Thank you,
Michelle Decker

Date: 09/24/2002 at 14:59:19
From: Doctor Jerry
Subject: Re: Explaining Weighted Averages to My Assistant

Hi Michelle,

I'll assume that by a weighted average of some numbers a1,a2,...,an,
you mean the sum

w1*a1+w2*a2+...+wn*an,

where w1,w2,...,wn are non-negative real numbers whose sum is 1, that
is, w1,w2,...,wn >= 0 and

(*)    w1 + w2 + ... + wn = 1.

I'll assume that these n numbers are (or have been converted to)
fractions like w1=p1/q1, w2=p2/q2,...,wn=pn/qn.  If I were explaining 
this part to someone with a limited background, I'd choose n to be 
fairly small and choose specific weights, maybe

w1=1/12, w2=3/12, w3=5/12, w4=2/12, w5=1/12.

Now, taking n=5 and using the weights just above, multiply equation (*)
by 12 (the common denominator of the weights).  This gives

(**)  1 + 3 + 5 + 2 + 1 = 12.

This helps us think of the weighted average

w1*a1 + w2*a2 + w3*a3 + w4*a4 + w5*a5

as

(1*a1+3*a2+5*a3+2*a4+1*a5)/12,

which is just an ordinary average of 1 copy of a1, 3 copies of a2, 5 
of a3, 2 of a4, and 1 of a5.  With three copies of a2 in the sample,
we may multiply a2 by 3 instead of including a2 three times. So, the
numerator of a weight gives the number of times a certain thing enters
into the average, that is, its "weight" or importance.

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