When you make a series of measurements, you can calculate the arithmetic mean or elementary average of the measurements by summing them and dividing by the number of measurements you made. However, in certain situations, some measurements count more than others, and to get a meaningful average, you have to assign weight to the measurements. The usual way to do this is to multiply each measurement by a factor that indicates its weight, then sum the new values, and divide by the number of weight units you assigned.
TL;DR (Too Long; Didn't Read)
Calculate the weighted average (weighted mean) of a number of measurements by multiplying each measurement (m) by a weighting factor (w), summing the weighted values, and dividing by the total number of weighting factors:
∑mw ÷ ∑w
Looking at It Mathematically
When calculating an arithmetic average, you sum all the measurements (m) and divide by the number of measurements (n). In mathematical terminology, you express this type of average this way:
∑(m1...mn) ÷ n
where the symbol ∑ means "sum all the measurements from 1 to n."
To calculate a weighted mean, you multiply each measurement by a weighting factor (w). In most cases, the weighting factors add up to 1 or, if you are using percentages, to 100 percent. If they don't add up to 1, use this formula:
∑ (m1w1...mnwn) ÷ ∑(w1...wn) or simply ∑mw ÷ ∑w
Weighted Averages in the Classroom
Teachers typically use weighted averages to assign appropriate importance to classwork, homework, quizzes and exams when calculating final grades. For example, in a certain physics class, the following weights may be assigned:
- Lab work: 20 percent
- Homework: 20 percent
- Quizzes: 20 percent
- Final Exam: 40 percent
In this case, all the weights add up to 100 percent, so a student's score can be calculated as follows:
[(Lab work score) • 0.2 + (homework) • 0.2 + (quizzes) • 0.2 + (final exam) • 0.4]
If a student's grades were 75 percent for lab work, 80 percent for homework, 70 percent for quizzes and 75 percent for the final exam, her final grade would be: (75) • 0.2 + (80) • 0.2 + (70) • 0.2 + (75) • 0.4 = 15 + 16 + 14 + 30 = 75 percent.
Weighted Averages for Computing GPA
Weighted averages are also used when calculating a grade-point average because some classes count for more credits than others. In a typical school year, a teacher would weight each score by multiplying by the number of credits the class is worth, sum the weighted scores and divide by the number of credits all the classes are worth. This is equivalent to using the formula for weighted average presented above.
For example, a student majoring in mathematics takes a calculus class worth three credits, a mechanics class worth two credits, an algebra class worth three credits, a liberal arts class worth two credits and a physical education class worth two credits. The scores for each respective class are A (4.0), A- (3.7), B+ (3.3), A (4.0) and C+ (2.3).
The sum of the weighted scores is [3 • (4.0) + 2 • (3.7) + 3 • (3.3) + 2 • (4.0) + 2 • (2.3)] = (12.0 + 7.4 + 9.9 + 8.0 + 4.6) = 41.9.
The total number of credits is 12, so the weighted average (GPA) is 41.9 ÷ 12 = 3.49