Learning Outcomes
- Calculate the weighted mean of a set of numbers
- Calculate a weighted mean using data in a table
KEY words
- Weight: a number assigned to a subset of the data to give it more or less value.
- Weighted mean: an arithmetic average where some data points have more value than others.
Weighted Mean
The weighted mean is similar to an ordinary arithmetic mean, except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. If all the weights are equal, then the weighted mean is the same as the mean.
An example of a weighted mean is college grade point average (GPA). The weights in a GPA are the number of credits allocated to each course, while the data points are the grades earned in each course converted to a numeric scale.
Table 1 shows Isabella’s grades in each of her classes for fall semester:
| Course | Number of Credits | Letter Grade | Numeric Grade |
|---|---|---|---|
| Math | 4 | A | 4.00 |
| English | 4 | B | 3.00 |
| Biology | 3 | A– | 3.70 |
| Racquetball | 1 | B+ | 3.33 |
| Table 1. Isabella’s grades for fall semester. | |||
To calculate her GPA we multiply the number of credits for each course by the numeric grade, add them up, then divide by the total number of credits:
[latex]\text{GPA = }[/latex][latex]\large\frac{\text{sum of [(number of credits) · (numeric grade)]}}{\text{total credits}}[/latex]
[latex]\text{GPA = }[/latex][latex]\frac{{4}({4.00})+{4}({3.00})+{3}({3.70})+{1}({3.33})}{4+4+3+1}[/latex][latex]=3.53583…[/latex]
Since GPAs are given to 2 decimal places, we round the weighted average to 2 decimal places: [latex]\text{GPA = }[/latex][latex]3.54[/latex].
Since her math and English classes have the most credits, they count more in the GPA calculation than any other course. Similarly, Racquetball counts the least since it is worth only 1 credit.
Example
Jane is trying to calculate her final grade, which is a weighted mean. The weights of grading categories and her grades in the categories are shown in the table.
| Category | Homework | Tests | Projects | Discussions | Final Exam |
|---|---|---|---|---|---|
| Weight (%) | 20 | 30 | 20 | 10 | 20 |
| Grade (%) | 90 | 78 | 85 | 96 | 84 |
1. Calculate her final grade in the course.
[latex]\text{Course Grade = }[/latex][latex]\large\frac{\text{sum of [(category weight) · (numeric grade)]}}{\text{total weights}}[/latex]
[latex]\text{Course Grade = }[/latex][latex]\frac{20(90)+30(78)+20(85)+10(96)+20(84)}{20+30+20+10+20}\text{ = }84[/latex]
Jane earned 84% in this course.
2. Which category has the most influence on her grade?
The largest weight is 30% for tests, so test scores have the most influence on her grade.
3. Which category has the least influence on her grade?
The lowest weight is 10% for discussions, so discussions have the least influence on her grade.
Try IT
To measure the success of an NFL quarterback, the NFL use a calculation known as the adjusted net passing yards per attempt, or ‘ANY/A’. The higher a player’s ANY/A, the better.
The formula is:
[latex]\text{ANY/A = }\large\frac{\text{[passing yards + 20(number of passing touchdowns) – 45(number of interceptions thrown) – sack yards lost]}}{\text{(number of passing attempts + number of sacks)}}[/latex]
Suppose an NFL quarterback has the following numbers:
| Completed Passes | Passing Attempts | Passing Yards | Passing Touchdowns | Interceptions | Sacks | Sack Yards Lost | |
|---|---|---|---|---|---|---|---|
| player 1 | 420 | 642 | 5620 | 58 | 12 | 15 | 95 |
| player 2 | 365 | 489 | 4190 | 46 | 6 | 19 | 21 |
- Calculate player 1’s ANY/A.
- Calculate player 2’s ANY/A.
- Which player is the better quarterback, according to his ANY/A?
- What category has the greatest effect on the measure ANY/A? Is the effect positive or negative?
- If player 1 wants to increase his ANY/A value, what should he work on? Explain your reasoning.
When we calculate a mean from a frequency table, we are calculating a weighted mean. The weights are equivalent to the frequency of each category or interval.
Example
Debajin sells tea. His tea has different quality ratings of A – D, with A being the highest grade, and most expensive.
The frequency table shows the price of each grade of tea along with the number of units he sold last month.
| Grade | Price per kilogram ($) | Number of kg sold |
| A | 32 | 1200 |
| B | 24 | 3150 |
| C | 10 | 7580 |
| D | 2 | 2875 |
Calculate the weighted mean price per kilogram of tea sold.
Solution:
[latex]\text{weighted mean = }\large\frac{32(1200)+24(3150)+10(7580)+2(2875)}{1200+3150+7580+2875}\text{ = }13.208…[/latex]
The weighted mean price per kilogram of tea sold is $13.21 per kilogram.
Try IT
Natural gas traders are often interested in the volume-adjusted average price of gas in a particular region.
There are usually many gas stations within a region. Each of these varies in both price and volume of supply.
The table shows five such gas stations within a region:
| Station | Price per gallon ($) | Volume (gallons) |
|---|---|---|
| 1 | 6.51 | 2100 |
| 2 | 7.22 | 960 |
| 3 | 5.95 | 2300 |
| 4 | 8.20 | 585 |
| 5 | 7.55 | 895 |
- Which category acts as the weights in the weighted average?
- Calculate the weighted mean representing the volume-adjusted average price of gas.
