Examples of Weighted Averages

Date: 03/26/2001 at 17:27:10
From: Robert
Subject: Statistics

Can you tell me what is meant by "weighted average," and give me some 
examples of when and how weighted averages are used? When should mean 
average be used, versus weighted average?

Date: 03/27/2001 at 12:55:56
From: Doctor TWE
Subject: Re: Statistics

Hi Robert - thanks for writing to Dr. Math.

A weighted average is one in which different data in the data set are 
given different "weights." Here are a few examples:

Slugging average in baseball: A batter's slugging average (also
called slugging percentage) is computed by:

   SLG = (1*SI + 2*DO + 3*TR + 4*HR) / AB

where: SLG = slugging percentage
        SI = number of singles
        DO = number of doubles
        TR = number of triples
        HR = number of home runs
        AB = total number of at-bats

Here, each single has a "weight" of 1, each double has a "weight" of 
2, etc.  The average counts home runs four times as important as 
singles, and so on.  An at-bat without a hit has a "weight" of zero!

Slugging average is sometimes referred to as slugging percentage.  
The term is a misnomer, for it is actually a weighted average, not a
percentage.  As such, it's possible for a batter's slugging average
to exceed 1.000 or 100%.

Course grades: Many teachers will use a "weighted average" when 
calculating a student's grade in a course. For example, a teacher 
might say the test average is 60% of the grade, quiz average is 30% of 
the grade, and a project is 10% of the grade. Suppose Mary got 90 and 
78 on the tests; 100, 100 and 85 on the quizzes; and an 81 on the 
project. Her course grade would be:

     Test average = (90 + 78)/ 2             = 84

     Quiz average = (100 + 100 + 85)/3       = 95

     Course grade = .60*84 + .30*95 + .10*81 = 87

Here, the tests carry a "weight" of .60 (or .30 each), the quizzes 
carry a "weight" of .30 (or .10 each), and the project carries a 
weight of .10. Note that the test average and the quiz average are not 
weighted averages, but the course grade is.

Grade point average (GPA): Most colleges assign "weights" to the 
individual course grades in the form of credits. A grade in a 4-credit 
course affects your GPA more by 33% than a grade in a 3-credit course. 
For example, suppose Joe took the following courses:

     COURSE        CR   GR
     Calculus       4    C
     Discr. Math    3    A
     English Lit.   3    A
     Chemistry      4    D
     Comp. Sci.     3    B

Most colleges use the scale: A = 4, B = 3, C = 2, D = 1, F = 0. To 
compute Joe's GPA, we multiply each course grade (converted to the 
number equivalent) by the course credits, then divide the sum by the 
total number of credits:

     COURSE        CR   GR
     Calculus       4    C     4*2 =  8
     Discr. Math    3    A     3*4 = 12
     English Lit.   3    A     3*4 = 12
     Chemistry      4    D     4*1 =  4
     Comp. Sci.     3    B     3*3 =  9
                  ----              ----
                   17                45

     GPA = 45 / 17 = 2.65

If the grades had been unweighted, the GPA would have been:

     (2 + 4 + 4 + 1 + 3) / 5 = 2.80

Why is Joe's GPA lower? Because he did less well  in the "more 
important" courses, i.e. those worth more credits.

Here are a few other explanations from our Ask Dr. Math archives that 
you can check out as well:

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

   Calculating a Weighted Average Grade
   http://mathforum.org/dr.math/problems/sam.5.5.00.html   

   Averaging Averages
   http://mathforum.org/dr.math/problems/francis.11.9.99.html   

The term "mean" means average in the conventional sense, in other 
words, an unweighted average.

I hope this helps. If you have any more questions, write back.

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