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/
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