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Section 7.1 Weighted Averages

You certainly know how to find the average of several numbers by adding them up and dividing by the number of values, so for example the average of the first four positive integers is

\begin{equation*} \frac{1+2+3+4}{4} = 2.5 \end{equation*}

More formally, if \(a\) is a set with \(n\) elements then the average, or mean, value is

\begin{equation} \bar{a} = \frac{1}{n} \sum_{i=1}^n a_i = \frac{a_1 + a_2 + \dots + a_n}{n} \text{.}\tag{7.1.1} \end{equation}

This average is also called the arithmetic mean. When calculating an arithmetic mean, each number is equally important when evaluating the average. The overbar symbol is often used to indicate that a quantity is a mean value.

In situations where some values are more important than others, we use a weighted average. A familiar example is your grade point average. Your GPA is calculated by weighting your grade for each class by the credits for that class, then dividing by the total credits you have taken. The credit values are called the weighting factors.

In general terms a weighted average is

\begin{equation} \bar{a} = \frac{\sum a_i w_i}{\sum {w_i}} \label{weighted-average}\tag{7.1.2} \end{equation}

Where \(a_i\) are the values we are averaging and \(w_i\) are the corresponding weighting factors. The weighting factors may be different for each item being averaged, so \(w_i\) is the weighting factor for value \(a_i\text{.}\) In this book we will not write the limits on the sums, and understand that the intent is always to sum over all the values. Notice that if the weighting factors are all identical, they can be factored out of the sums so the weighted average and the arithmetic mean will be the same.

Weighted averaging is used to find centroids, centers of gravity and centers of mass, the subject of this chapter. All three are points located at the “center” the object, but the meaning of “center” depends on the weighting factors. Area or volume are the factors used for centroids, weight for center of gravity, and mass for center of mass.

Example 7.1.1. Course Grades.

The mechanics syllabus says that there are two exams worth 25% each, homework is 10%, and the final is worth 40%. You have a 40 on the first exam, a 80 on the second exam, and your homework grade is 90.

What do you have to earn on the final exam to get a 70 in the class?

Answer.

You need a 77.5 on the final to get a 70 for the class.

Solution.

Your known grades and the weighting factors are

\begin{align*} G_i \amp = \left [40, 80, 90, FE \right]\\ w_i \amp = \left [25\%, 25\%, 10\%, 40\% \right ] \end{align*}

Find final exam score \(FE\) so that your average grade \(\bar{G}\) is 70%.

\begin{align*} \bar{G} \amp= \frac{\sum G_i w_i}{\sum w_i}\\ 70 \amp = \frac{(40 \times 0.25) + (80 \times 0.25) + (90 \times 0.1) + (FE \times 0.4)}{(0.25 + 0.25 + 0.1 + 0.4 )}\\ FE \amp = \frac{70 (1) - (10 +20 + 9)}{0.4} = 77.5 \text{.} \end{align*}