10 Weighted Average
I’m sure you know that to calculate an average or mean, you add the numbers then divide by how many numbers you added. For example, the average of 45, 67, and 82 is calculated as
Average = [latex]\frac{45+67+72}{3}=61[/latex] normal way
In this calculation, each of the numbers contributes the same amount to the average. An alternate way to calculate an average is the multiply each number by the fraction or percent it contributes to the whole. In the example above, each number contributes 1/3 or 0.3333 to the average, so the average could be calculated as
Average = (45)(0.3333) + (67)(0.3333) + (72)(0.3333) = 61 alternate way
While you will need to calculate the average often in chemistry, on occasion, you will need to calculate a weighted average, where each of the numbers contributes a different amount (or weight) to the average. When calculating a weighted average, you will use the alternate way of calculating the average shown above, except that each number will contribute a different percent. This might come up in calculating your grade in a course or in calculating the average atomic mass of an element, as the following two examples illustrate.
Example 10.1
Suppose the syllabus for a specific course states that homework counts 15% towards your grade, the quizzes 25%, and the exams 60%. If you had the following scores
Homework = 95%
Quizzes = 86%
Exams = 77%
what would be your final percent for the course?
Answer: Multiply each score by the percent it counts towards your grade (written in decimal form) and add them up.
(95%)(0.15) + (86%)(0.25) + (77%)(0.60) = 82%
Example 10.2
The atomic mass of an element written under the element on the periodic table is the weighted average of all the naturally occurring isotopes of that element. What is the atomic mass of Si, which has the following naturally occurring isotopes?
| Isotope Mass (amu) | Isotope Percent Abundance | |
| 28Si | 27.97693 | 92.23% |
| 29Si | 28.97669 | 4.67% |
| 30Si | 29.97376 | 3.10% |
Answer: Multiply each mass by its percent abundance (written in decimal form) and add them up.
(27.97693 amu)(0.9223) + (28.97669 amu)(0.0467) + (29.97376 amu)(0.0310) = 28.09 amu
It may be that you will be given the average and need to calculate one of the component scores or masses, as the following examples illustrate.
Example 10.3
Suppose your grade for a course consists of 70% from semester exams, 10% from homework, and 20% from the final exam. If you scored an average of 86% on the exams and 98% on the homework, what score would you need on the final exam to get 90% in the course?
Answer: Your weighted average needs to be 90%, so plug that in and put an “x” in for the unknown final exam score.
(86%)(0.70) + (98%)(0.10) + (x)(0.20) = 90%
Now solve for x
70% + 0.20x = 90%
0.20x = 20%
x = 100%
Example 10.4
Sulfur has three naturally occurring isotopes. One has a mass of 31.972 amu and an abundance of 95.00%. Another has a mass of 32.971 amu and an abundance of 0.76%. What is the mass of the third isotope? (Hint: You have access to a periodic table.)
Answer. Remembering that an element’s atomic mass (the weighted average of the isotopic masses) is under its symbol on the periodic table, you can look that up. It is 32.06 amu. Since all of the abundance percentages need to add up to 100%, the abundance of the isotope whose mass you are calculating must be
100% – 95.00% – 0.76% = 4.24%
Therefore, the calculation can be set up as
(31.972 amu)(0.95) + (32.971 amu)(0.0076) + (x)(0.0424) = 32.06 amu
30.624 amu + 0.0424x = 32.06 amu
0.0424x = 1.436 amu
x = 33.9 amu (rounded to three sig figs)
