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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

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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.

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.

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%.

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.

Although this problem could be done by using a standard average calculation, it will be much quicker to calculate the average monthly savings by using a weighted average. First, you will multiply each value (the amount of savings in this example) by the frequency with which it occurs. For example, the $30/month discount occurs 12 times since it's applied each month for the first year. Thus, $30 * 12 = $360.

The $10 promotional pricing discount also occurs 12 times so it will be multiplied by 12 as well giving $10 * 12 = $120. The $10 member referral discount is only applied for 10 months so it would be multiplied by 10, or $10 * 10 = $100.

The paperless discount, $5/month, is applied for 24 months, or $5 * 24 = $120. Then, you will add up all of these values and divide by the total number of months, since we are looking for the average monthly discount over the entire two-year (24 month) period. $360 + $120 + $100 + $120 = $700 / 24 months = $29.17/month in savings on average. Note that you could have calculated this value by adding up $30 twelve times and then adding $10 twelve times, $10 ten times and $5 twenty-four times and dividing that result by 24. However, it is much faster to simply multiply each value by its frequency and add those four numbers, which is the idea of a weighted average.

**Example 4**: This chart below shows the number of runs a major league baseball team scored in a game and the number of times each value occurred. Let's calculate the average number of runs the team scored over the entire 162 game season.

Runs | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 12 | 15 |

Frequency | 8 | 27 | 30 | 21 | 23 | 23 | 17 | 7 | 3 | 2 | 1 | 1 |

To calculate the average we need to first multiply each value (# of runs scored) by its frequency and add up the values. The calculation should look like this:

(0 * 8) + (1 *27) + (2 * 30) + (3 * 21) + (4 * 23) + (5 * 23) + (6 * 17) + (7 * 7) + (8 * 3) + (9 * 2) + (12 * 1) + (15 * 1) =

0 + 27 + 60 + 63 + 92 + 110 + 102 + 49 + 24 + 18 + 12 + 15 = 577 runs

Then divide that value by the number of games, 162, to get 572/162 = 3.56 runs/game.

The **average**, or **arithmetic mean**, of a series of items means you simply add up all the item values and divide by the total number of items to calculate the average. A **weighted average** is an average where each value has a specific weight or frequency assigned to it. There are two main cases where you'll generally use a weighted average instead of a traditional average. The first is when you'll want to calculate an average that is based on different percentage values for several categories. The second case is when you have a group of items and each has a frequency associated with it.

- Weighted average is used in cases involving percentage values for different categories or groups of objects with frequencies
- To calculate the weighted average of percentages, each value is multiplied by its percentage and added together
- To calculate the weighted average of a frequency, each value is multiplied by its frequency, then added together and divided by the total number of units in the given situation

Once you've finished, you should be able to:

- Explain the difference between an average and a weighted average
- Recall when you would use a weighted average
- Calculate the weighted average of frequencies or percentages

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High School Algebra II: Homework Help Resource26 chapters | 281 lessons | 2 flashcard sets

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