Ready to skip ahead to practice problems? Click here!
Weighted averages have been on my mind, probably because this is the time of year when students are calculating their final course grades!
The “usual” average (technically called the ordinary arithmetic mean) is easy enough: Add up all the values you have, divide them by the total number of values, and voila! You have the average. That’s great, but the ordinary arithmetic mean takes it for granted that all values have equal importance, or an equal probability of occurring. What if that’s not the case?
That’s where the weighted average (technically called weighted arithmetic mean) comes into play! The weighted average is the sum of the weights-multiplied-by-the-corresponding-values, divided by the total number of weights:
In that formula, wi stands for the ith weight, xi stands for the corresponding ith value, and N stands for the total number of weights. The values can change from situation to situation, but the weights do not. This formula bears some similarity to the ordinary arithmetic mean, since there is a sum of values divided by the total number of components. Still, this is an odd-looking formula, so let’s try some examples with it!
1) Since calculating final course grades is a common situation that requires weighted averages, let’s start there. Say that you are taking a class where homework counts for 10% of your final grade, quizzes count for 20%, the midterm exam counts for 20%, the final exam counts for 30%, a research paper counts for 15%, and participation counts for 5%. By the end of the semester, you have a 95% homework grade, 84% quiz grade, 88% on your midterm, 92% on your final exam, 94% on your research paper, and 100% participation. How do you calculate your final grade?
The weights are the amounts that each component of the course contributes to the final grade (w1=10 for homework, w2=20 for quizzes, etc.) and the values are your scores on each component of the course (x1=95 for homework, x2=84 for quizzes, etc.). There are 6 components to the course, so N=6. Since all of the components of the course must add up to 100% of the course, it should not be surprising that the sum of the weights is 100:
Nicely done – you got an A in your class!
Notice that, since the components of the course add up to 100%, we could have used a shortcut in this problem: We could have converted the percentage weights into decimal form and simply multiplied them by the scores to get the same final grade:
This shortcut only works easily when all of the weights clearly add up to 100% – be wary about using it in other situations, because it will take extra work, as we’ll see in the next two examples.
2) Grade point averages are also calculated using weighted averages. In their case, the number of credit hours are the weights, and the grades are the corresponding values!
Er, wait. How can letter grades be used in a calculation?
The grades are turned into numbers! Given that a 4.0 is the highest GPA in most gases, it should not be surprising that an A is assigned a value of 4. The grades are defined as follows: A=4, B=3, C=2, D=1, F=0.
Say that you are interested in computing your semester GPA. You took 16 credits in total: 4 in calculus (yay!), 3 in chemistry, 3 in zoology, 5 in Spanish, and 1 in music. You received an A in calculus (also yay!), an A in chemistry, a B in zoology, a B in Spanish, and an A in music. Let’s calculate your GPA using a weighted average. You took five classes, so N=5:
Again, the denominator should not be a surprise, since you took 16 credits in total.
Notice that we could have used the “shortcut” if we turned the course credits into percentages. Accordingly, 4 credits is 4/16 = 0.25 = 25% of your total semester credits, 3 credits is 3/16 = 0.1875 = 18.75%, 5 credits is 5/16 = 0.3125 = 31.25%, and 1 credit is 1/16 = 0.0625 = 6.25%. Now that we know these percentages, we can compute our weighted average in an alternative way:
We get the same answer, but it really wasn’t much of a shortcut in this case, since we had to compute the percentages separately. Still, it’s an option, and I want you to be aware of it in case you see it in class!
3) Weighted averages aren’t just for grades. You could also use a weighted average to calculate the amount of calories burned during a workout. Say that jogging burns 400 calories per hour, your cardio and conditioning routine burns 600 calories per hour, and your cool-down stretch burns 150 calories per hour. Of your hour-long workout, you are going to spend ten minutes warming up, forty-five minutes working out to your routine, and 5 minutes stretching. How many calories will you burn?
There are three components to your workout, so N=3. Since all of the components of your hour-long workout must add up to 60 minutes, it should not be surprising that the sum of the weights is 60:
You burned just over 529 calories in your hour-long workout!
Again, notice that we could have used the “shortcut” if we turned the components of our workout into percentages. Accordingly, 5 minutes is 5/60 = 0.0833 = 8.33% of an hour, 10 minutes is 10/60 = 0.1667 = 16.67% of an hour, and 45 minutes is 45/60 = 0.75 = 75% of an hour. Now that we know these percentages, we can compute our weighted average in an alternative way:
We get the same answer, but it again really wasn’t much of a shortcut in this case, since we had to compute the percentages separately.
Weighted averages might seem intimidating compared to ordinary averages, but fear not! Anytime something you want to measure is doled out in unequal amounts, a weighted average is your friend!
Let’s get some practice!
Answer the following questions using weighted averages.
1. You are making a choice between two colleges to attend, and you are using math to compare them – specifically, you have constructed a 10-point scale to evaluate a college. The reputation of the computer science department is the most important to you, and so you assign that score a weight of 4. The cost is second-most important to you, and so you assign it a weight of 3. Location is third-most important to you, so you assign it a weight of 2. Finally, the variety of student organizations offered is least important to you, so you assign it a weight of 1.
You score College A as follows: 90% on reputation, 70% on cost, 65% on location, and 100% on the variety of student organizations. You score College B as follows: 80% on reputation, 80% on cost, 70% on location, and 90% on variety of student organizations.
Based on your weighting system, which college should you choose to attend?
2a. It’s the end of the semester, and you are wondering what minimum score you must achieve on the final exam in order to get an A in the class (80% overall grade).
The class is weighted as follows: Homework counts for 10%, quizzes count for 20%, the midterm exam counts for 25%, the final exam counts for 30%, a research paper counts for 10%, and participation counts for 5%. By the end of the semester, you have a 92% homework grade, 83% quiz grade, 89% on your midterm, 89% on your research paper, and 100% participation. What must you score on the final exam in order to get a A in the course?
2b. If the final exam is worth 200 points, how many points must you score in order to get an A in the course?
0.935*200 = 187