# Calculating Weighted Average: Method, Formula & Example

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• 0:02 When to Use a Weighted Average
• 2:07 Examples with Percentages
• 5:22 Examples with Frequencies
• 8:46 Lesson Summary

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Lesson Transcript
Instructor: Erin Monagan

Erin has been writing and editing for several years and has a master's degree in fiction writing.

This lesson will examine the concept of a weighted average and types of situations when it should be used instead of a standard average. It will also give some real-world examples of a weighted average.

## When to Use a Weighted Average

Most people are familiar with the idea of finding the average, or arithmetic mean, of a series of items. You simply add up all the item values and divide by the total number of items to calculate the average. However, this only works when all the items are weighted equally. For example, to calculate your average monthly electricity bill for a year, it would make sense to add up the billed amounts for the previous twelve months and divide by twelve, since each bill cycle is roughly the same period of time (one month).

Now, let's say you want to find your current average in your English class. Most classes usually assign a different weight or value to papers than to homework assignments, quizzes and tests. In this case, you might need to use a weighted average, which is an average where each value has a specific weight or frequency assigned to it, to calculate your grade.

There are two main cases where you will generally use a weighted average instead of a traditional average. The first is when you want to calculate an average that is based on different percentage values for several categories. One example might be the calculation of a course grade, mentioned earlier.

The second case is when you have a group of items that each has a frequency associated with it. In these types of situations, using a weighted average can be much quicker and easier than the traditional method of adding up each individual value and dividing by the total. This is especially useful when you are dealing with large data sets that may contain hundreds or even thousands of items but only a finite number of choices.

For example, let's say you teach one section of a chemistry course and want to find the average score on the most recent exam. However, since there are a total of 800 students in the class, across four sections, the traditional method of finding an average would involve adding up 800 individual numbers. However, using a weighted average would probably only involve using 40 to 50 different numbers, along with their frequencies. Now, we'll take a look at how to calculate a weighted average.

## Examples with Percentages

Let's take a look at some examples.

Example 1: A student is enrolled in a biology course where the final grade is determined based on the following categories: tests 40%, final exam 25%, quizzes 25%, and homework 10%. The student has earned the following scores for each category: tests-83, final exam-75, quizzes-90, homework-100. We need to calculate the student's overall grade.

To calculate a weighted average with percentages, each category value must first be multiplied by its percentage. Then all of these new values must be added together. In this example, we must multiply the student's average on all tests (83) by the % that the tests are worth towards the final grade (40%). Please note that all %'s must be converted to decimals before you multiply. Similarly, the final exam score (75) will be multiplied by its % (25). The same will be true for both the quizzes (90 * 25%) and homework (100 * 10%). Thus, the overall calculation would be (83 * .40) + (75 * .25) + (90 * .25) + (100 * .10) = 33.2 + 18.75 + 22.5 + 10 = 84.45 or 84% if rounded down.

Example 2: A student has earned the following averages in his history course: tests-90, quizzes-88, papers-85, homework-95. The overall course grade is comprised of tests (30%), quizzes (20%), final exam (20%), papers (20%) and homework (10%). We need to figure out what score must he earn on the final exam in order to earn a final grade of at least 90% before rounding.

We will use the same method for calculating a weighted average that we used in the previous problem except that now we already know the overall grade and do not know one of the category values. First, we will multiply each value by its percentage to get the following: (90 * .30) + (88 * .20) + (x * .20) + (85 * .20) + (95 * .10). Note that a variable, x, is used in place of the value for the final exam score since that is what we are trying to find. Simplifying that gives 27 + 17.6 + .20x + 17 + 9.5 or 71.1 + .2x. We'll set this equal to 90 since that is the overall minimum grade the student wants to earn giving 71.1 + .2x = 90. Subtracting 71.1 from both sides gives .2x = 18.9. Dividing both sides by .2 yields x = 94.5. Thus, this student must earn a 94.5 or higher on the final exam to achieve an overall course grade of at least 90%.

## Examples with Frequencies

Now let's look at some examples with weighted averages with frequencies.

Example 3: You are thinking about signing up for a two-year contract of new satellite television service which offers the following discounts: \$30/month for months 1-12 & \$10/month for months 13-24 (promotional pricing), \$10/month for months 1-10 (member referral discount) and \$5/month for months 1-24 (paperless discount). We have to find the average monthly savings for the first two years of your service.

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