Weighted Mean

Some situations require a special type of mean called a weighted mean,
The weighted mean of a set of data is

where wi = weight of the ith observation.
One important situation that requires the use of a weighted mean is the calculation of grade point average (GPA).
Example: Grade Point Average (Weighted Mean)
Suppose that a student who completed 15 credit hours during his first semester of college received one A, one B, one C, and one D. Suppose that a value of 4 is used for an A, 3 for a B, 2 for a C, 1 for a D, and 0 for an F. Calculate the student’s semester GPA.
Solution: GPA calculated by the simple mean is

But this not the correct GPA. In computing simple mean we assume that each course is of equal importance or “weight,” but this assumption ignores that fact that the number of credit hours are not the same for all courses. That is, the A was earned in a three-credit-hour English course, and the B was earned is a three-credit-hour math course, but the C was earned in a four-credit-hour biology lab course, and the D grade, unfortunately, was earned in a five-credit-hour Spanish class. This information is summarized in the following Table:

Table 5: Semester Academic Record

Course	Grade	Value xi	Credit hours wi	wixi
English	A	4	3	12
Math	B	3	3	9
Biology lab	C	2	4	8
Spanish	D	1	5	5
Total			15	34

Then

Thus the correct credit hour is 2.267 and not 2.50.
Example: Population size and per capita personal income for a random sample of five states are given in the following Table:
Table 6: Population and per capita Personal income, 2002

State	Population (in 000)	Per capita personal income
California	35,002	$ 32,989
Florida	16,692	29758
Minnesota	5,025	33,322
North Dakota	634	26,852
Vermont	616	29,764
Total	57,969	$152,685.00

Calculate the mean per capita personal income for 2002.
Solution: Since the population size varies for each state, the mean per capita personal income for 2002 is calculated by a weighted mean.
Table 7: Calculation of weighted mean

State	Population (in 000) (wi)	Per capita personal income (xi)	(wixi)
California	35,002	$ 32,989	1154680978
Florida	16,692	29758	496720536
Minnesota	5,025	33,322	167443050
North Dakota	634	26,852	17024168
Vermont	616	29,764	18334624
Total	57,969	$152,685.00	1854203356

Thus, the mean per capita personal income for 2002 is $31986.12 and not $30,537, as calculated by a simple arithmetic mean,