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Weighted Average – Explanation and Examples
A weighted average is a mean composed of smaller means that do not evenly contribute to the whole.
The most common example of a weighted average is a school grade where homework, in-class work, and tests are weighted differently. That is, the contribute a greater or a lesser amount to the final grade.
Weighted averages are also important in statistics when accounting for differences in surveys and populations and in other areas of mathematics.
This section covers:
- What is a Weighted Average?
- How to Calculate Weighted Average?
- Weighted Average Formula
- Weighted Average Examples
What is a Weighted Average?
A weighted average is a mean that depends on not only the terms in the data set but also the terms’ corresponding weights.
People often associate weighted averages with grades, but they are useful in many mathematical concepts. This includes any kind of statistical analysis where samples are weighted to better match population “weight.” For example, suppose a survey consists of $60$ female respondents and $40$ male respondents in a population that is 50-50 male and female. In this case, the responses would be weighted so that female responses only held $50%$ weight despite making up $60%$ of the responses.
How to Calculate Weighted Average?
When calculating a weighted average, every term has a corresponding weight.
To calculate the weighted average, multiply each term by its weight, then divide by the sum of the weights. Often, these weights are percents. Remember to convert these percents to decimals before multiplying. In these cases, calculations also tend to be easier because the weights, when they are percents, will likely add up to $100%$, which is equal to $1$.
In some cases, it may be necessary to find a mean before going to this step. For example, when consider calculating a grade where homework accounts for $40%$ of the final score. If there are $20$ homework assignments in a semester, each assignment is not multiplied by $0.4$. Indeed, the assignments themselves may not all have the same weight because some assignments may be longer or shorter.
Instead, calculate the average homework score. Then, consider this as the term and multiply that by $0.4$ when calculating the weighted average of the grade.
Weighted Average Formula
The weighted average, $\bar{x}$ of terms $(n_1 .. n_k)$ with corresponding weights $w_1 … w_k$ is:
$\bar{x} = \frac{\sum_{i=1}^k w_i n_i}{\sum_{i=1}^k w_i}$.
This is the sum of the products of terms and their corresponding weights divided by the sum of the weights.
Weighted Average Examples
The most common examples of weighted averages are grades. In many classes, tests, quizzes, homework assignments, labs, and participation all contribute differently to the final grade. Labs and tests, for example, may be worth more than homework assignments and participation.
Another common example of a weighted average is a GPA when not all classes are the same number of credits. Many colleges, for example, offer half-credit courses, one and two credit courses, three credit courses, four credit courses, and even some five credit courses. An A in a five credit course would have a more significant impact on GPA than an A in a half-credit course.
Common Examples
This section goes over common examples of problems involving discrete data and their step-by-step solutions.
Example 1
Find the weighted average for the following.
$7$ weighted at $30%$
$8$ weighted at $55%$
$20$ weighted at $15%$
Solution
Recall that the formula for a weighted average is:
$\bar{x} = \frac{\sum_{i=1}^k w_i n_i}{\sum_{i=1}^k w_i}$.
In this case, there are three values and three weights. Plugging these values into the given formula yields:
$\bar{x} = \frac{7(0.3)+8(0.55)+20(0.15)}{0.3+0.55+0.15} = \frac{2.1+4.4+3}{1} = \frac{9.5}{1} = 9.5$.
Therefore, the weighted average is $9.5$.
Example 2
In a science class, labs are worth $50%$ of the grade, quizzes are worth $10%$ of the grade, homework is worth $15%$ of the grade, and tests are worth $25%$ of the grade.
Eliza got $25$ out of $30$ points on the first lab assignment, $50$ out of $50$ points on the second lab assignment, and $40$ out of $50$ on the third lab assignment.
She also got a grade of $70%$ in quizzes and $90%$ in homework.
On the first test, she got an $85%$ and on the second test she got $95%$.
What was Eliza’s final grade?
Solution
In this case, the weights are given, but the number associated with each weight is not. In particular, the number associated with the lab and test grades must be calculated.
For labs, Eliza got a $\frac{25}{30}$, a $\frac{50}{50}$ and a $\frac{40}{50}$. She therefore, in total got $\frac{115}{130} = \frac{23}{26}$.
For her tests, Eliza got an $85%$ and a $95%$. Without any more information, assume these tests were equally weighted. Therefore, her average score was an $\frac{85%+95%}{2} = 90%$.
Now, use the weighted average formula. Don’t forget to turn the percents into decimals for calculations.
$\frac{\frac{23}{26}(0.5)+0.7(0.1)+0.9(0.15)+0.9(0.25)}{1}$.
This is approximately:
$0.872$ or $87.2%$.
Example 3
An employer pays one employee $15$ per hour. This employee works $30$ hours per week.
The employer pays another employees $18$ per hour. These employees both work $30$ hours per week.
Another employee makes $25$ per hour and works $20$ hours per week.
How much does the employer pay an employee per hour of work on average?
Solution
There are a few ways to do this problem.
One could figure out the total amount that employer pays each week and then divide that by $4$.
Finding a weighted average, however, is essentially. In this case, use hours as the weights.
Now, there are three employees who work $30$ hours per week. Two make $18$ dollars per hour and one makes $15$ dollars per hour. The final employee makes $25$ dollars and works $20$ hours. Therefore, the weighted average is:
$\frac{15(30)+18(30)+18(30)+25(20)}{110} = \frac{2030}{110}$.
This simplifies to approximately $18.45$ per hour.
Example 4
A writer sends an invoice for work done.
She wrote $5$ blog pages for $40$ dollars each. She also wrote $2$ ads for $60$ dollars each and one instruction manual for $100$ dollars.
On average, how much did the writer make per project?
Solution
Again, it is possible to list out the price per project and divide by the number of projects. It also works, however, to use the number of projects as the weight and the price as the terms.
In this case, the weighted average is:
$\frac{40(5)+60(2)+100}{8} = \frac{420}{8} = 52.5$.
Therefore, the writer makes $52.50$ per project on average.
Example 5
A truck driver went $50$ miles per hour for three hours. Then he stopped for one hour and did not go anywhere. Then, he drove four more hours at $60$ miles per hour. What was the driver’s average speed?
Solution
As with some of the other problems, there is a way to solve this problem that does not involve weighted averages. One could calculate how far the driver drove in the first leg, second leg, and third leg then divide that by the total amount of time.
This, however, is exactly what taking the weighted average does. In this case, let the time be the weight and the speed be the terms. The weighted average, then, is:
$\frac{50(3)+0(1)+60(4)}{8} = \frac{390}{8}$.
Note that the numerator represents the total distance traveled and the denominator is the total time. This fraction is approximately equal to $48.75$ miles per hour.
Practice Problems
- Find the weighted average:
90 weighted at 5
110 weighted at 6
40 weighted at 15. - A student takes a 5 credit course, a four credit course, two three credit courses, and a one credit course. He gets a C in the five credit course, a B in the four credit course and one of the three credit courses, and an A in the other three credit course and the one credit course. On a GPA scale, C=2, B=3, and A=4. What is the student’s GPA for this semester?
- An art piece consists of three different parts. The first part has a density of $70$ pounds per foot cubed and a volume of $2$ feet cubed. The second part has a density of $20$ pounds per foot cubed and a volume of $5$ feet cubed. Finally, the third part has a density of $5$ pounds per foot cubed and a volume of $3$ feet cubed. What is the average density of the art piece?
- In a self-paced online class, there are five modules. The first four modules are worth the same, but the fifth module is worth as much as the other four combined. A student is one-fourth of the way through the last module and has done one-third of three of the other four. She has completed the first module entirely. What fraction has she completed on average of each of the modules?
- A quilter sells one blanket for $140$ dollars, two pillows for $30$ each, and twelve table runners for $20$ each. How much did she make on average per sale?
Answer Key
- $\frac{855}{13}, approximately $65.76$
- About $2.867$
- $25.5$ pounds per foot cubed
- $\frac{3}{8}$
- $29.33$ dollars