Averages and Weighted Averages
The average of a set of numbers is simply the sum of the values divided by the number of
values in the set.
Average value =
sum of all values___
# of values being summed
For example, the average of 5, 10, and 12 is equal to 9.
5 + 10 + 12
3
= 9
Sometimes, however, the values in a set of numbers do not contribute equally to the set.
For example, in determining your grade point average (GPA) in school, if you earned an A
in one three credit course, and a C in another three credit course, your average will be
exactly a B. However, if you earned an A in a two credit course, and a C in a four credit
course, your average will be lower than a B. This is because the A will count half as much
toward your GPA as the C, because the course in which you earned the A is worth half as
many credit hours.
To determine the average in this case, we must use the fact that the two values contribute
differently to the average, and we say that we calculate the weighted average.
Weighted average = (factor for value 1)(value 1) + (factor for value 2)(value 2) + …
Sum of all values
Averages and Weighted Averages Example 1:
Determine the average of two course grades when the two courses are equal in credit
hours. The quality points assigned to each grade are as follows:
A = 4.0 points
B = 3.0 points
C = 2.0 points
D = 1.0 point
The grade in the first class was an A and the grade in the second class was a C.
Solution:
Average value =
sum of all values___
# of values being summed
1
Average of one A and one C = 4.0 + 2.0 = 6.0 = 3.0
2
2
This result is exactly what we expect. An A averaged with a C will yield a B
average.
Averages and Weighted Averages Example 2:
Determine the average of two course grades when the two courses are NOT equal in
credit hours. The quality points assigned to each grade are as follows:
A = 4.0 points
B = 3.0 points
C = 2.0 points
D = 1.0 point
The grade in a two credit class was an A and the grade in a three credit class was a C.
Solution:
Weighted average = (factor for value 1)(value 1) + (factor for value 2)(value 2)
Sum of the weighting factors
Weighted average = (2)(4.0) + (3)(2.0)
2+ 4
=
14
5
= 2.8
You will encounter calculating weighed averages when you calculate the average atomic
weights of elements.
Averages and Weighted Averages Example 3:
Chlorine has two naturally occurring isotopes. Isotope A has a mass of 34.97, and
represents 75.8% of naturally occurring chlorine. Isotope B has a mass of 36.97, and
represents 24.2% of naturally occurring chlorine. What is the average atomic mass of
chlorine?
Solution:
The weighting factors are the percentages, and the values are the masses.
Weighted average = (factor for value 1)(value 1) + (factor for value 2)(value 2)
Sum of the weighting factors
Average atomic mass = (75.8)(34.97) + (24.2)(36.97) = 3545 = 35.45
75.8 +
24.2
100
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Hints for Using Averages and Weighted Averages:
How can I tell if I should use a weighted average in a calculation?
If the question has values that contribute equally to the average, you can use a simple
average calculation. However, if the values contribute unevenly to the average, you must
include the weighting factor and do a weighted average calculation. It is fairly easy to tell
which calculation to use since a weighted average question will include more numbers and
information than a simple average question.
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