2.4: Averages

Weighted Averages

Have you considered how your grade point average (GPA) is calculated? Your business program requires the successful completion of many courses. Your grades in each course combine to determine your GPA; however, not every course necessarily has the same level of importance as measured by your course credits.

Perhaps your math course takes one hour daily while your communications course is only delivered in one-hour sessions three times per week. Consequently, the college assigns the math course five credit hours and the communications course three credit hours. If you want an average, these different credit hours mean that the two courses do not share the same level of importance, and therefore a simple average cannot be calculated.

In a weighted average, not all pieces of data share the same level of importance or they do not occur with the same frequency. The data cannot represent a percent change or a series of numbers intended to be multiplied with each other. To calculate a weighted average, you require two components:

• The data itself—you need the value for each piece of data.
• The weight of the data—you need to know how important each piece of data is to the average. This is either an assigned value or a reflection of the number of times each piece of data occurs (the frequency).

[latex]\boxed{2.4\text{b}}[/latex] Weighted Average

[latex]{\color{red}{\text{WAvg}}}{\color{black}{\text{ is Weighted Average:}}}[/latex] An average for a data set where the data points may not all have the same level of importance or they may occur at different frequencies.

[latex]\color{blue}{\sum}\color{black}{\text{ is Summation:}}[/latex] This symbol is known as the Greek capital letter sigma. In mathematics it denotes that all values written after it (to the right) are summed.

[latex]\color{green}{w}\color{black}{\text{ is Weighting Factor:}}[/latex] A number that represents the level of importance for each piece of data in a particular data set. It is either predetermined or reflective of the frequency for the data.

[latex]\color{purple}{x}\color{black}{\text{ is Any Piece of Data:}}[/latex] In mathematics this symbol is used to represent an individual piece of data.

As expressed in Formula 2.4b, calculate a weighted average by adding the products of the weights and data for the entire data set and then dividing this total by the total of the weights.

Let’s stay with the illustration of the math and communications courses and your GPA. Assume that these are the only two courses you are taking. You finish the math course with an A, translating into a grade point of [latex]4.0[/latex]. In the communications course, your C+ translates into a [latex]2.5[/latex] grade point. These courses have five and three credit hours, respectively. Since they are not equally important, you use a weighted average.

Step 1:  In the numerator, sum the products of each course’s credit hours (the weight) and your grade point (the data). This means:

[latex]\small(\text{math credit hours}\times\text{math grade point})+(\text{communications credit hours}\times\text{communications grade point})[/latex].

Numerically, this is:

[latex]\begin{align*}\sum wx=\left(5\times 4\right)+\left(3\times 2.5\right)=27.5\end{align*}[/latex]

Step 2:  In the denominator, sum the weights. These are the credit hours. You have:

[latex]\begin{align*}\sum w=5+3=8\end{align*}[/latex]

Step 3:  Apply Formula 2.4b[latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex] to calculate your GPA.

[latex]\begin{align*}\text{WAvg}&=\frac{\sum wx}{\sum w}\\[1ex]\text{WAvg}&=\frac{27.5}{8}\\[1ex]\text{WAvg}&=3.44\text{ (GPAs have two decimals)}\end{align*}[/latex]

Note that your GPA is higher than if you had just calculated a simple average:

[latex]\begin{align*}\text{SAvg}&=\frac{\sum x}{n}\\[1ex]\text{SAvg}&=\frac{4 + 2.4}{2}\\[1ex]\text{SAvg}&=2.25\end{align*}[/latex]

This happens because your math course, in which you scored a higher grade, was more important in the calculation.

Paths To Success

The formula used for calculating a simple average is a simplification of the weighted average formula. In a simple average, every piece of data is equally important. Therefore, you assign a value of 1 to the weight for each piece of data. Since any number multiplied by 1 is the same number, the simple average formula omits the weighting in the numerator as it would have produced unnecessary calculations. In the denominator, the sum of the weights of 1 is no different from counting the total number of pieces of data. In essence, you can use a weighted average formula to solve simple averages.

Geometric Averages

How do you average a percent change? If sales increase [latex]100\%[/latex] this year and decrease [latex]50\%[/latex] next year, is the average change in sales an increase of

[latex]\begin{align*}\frac{100\%-50\%}{2}=25\%\end{align*}[/latex]

per year? The answer is clearly “no.” If sales last year were [latex]\$100[/latex] and they increased by [latex]100\%[/latex], that results in a [latex]\$100[/latex] increase. The total sales are now [latex]\$200[/latex]. If sales then decreased by [latex]50\%[/latex], you have a [latex]\$100[/latex] decrease. The total sales are now [latex]\$100[/latex] once again. In other words, you started with [latex]\$100[/latex] and finished with [latex]\$100[/latex]. That is an average change of nothing, or [latex]0\%[/latex] per year! Notice that the second percent change is, in fact, multiplied by the result of the first percent change. A geometric average finds the typical value for a set of numbers that are meant to be multiplied together or are exponential in nature.

In business mathematics, you most commonly use a geometric average to average a series of percent changes. Formula 2.4c is specifically written to address this situation.

[latex]\boxed{2.4\text{c}}[/latex] Geometric Average

[latex]\color{green}{n}\color{black}{\text{ is Total Quantity:}}[/latex] The physical total count of how many percent changes are involved in the calculation.

[latex]\color{red}{\text{GAvg }}\color{black}{\text{ is Geometric Average:}}[/latex] The average of a series of percent changes expressed in percent format. Every percent change involved in the calculation requires an additional ([latex]1+\%C[/latex]) to be multiplied under the radical. The formula accommodates as many percent changes as needed.

[latex]\color{purple}{\times 100}\color{black}{\text{ is Percent Conversion:}}[/latex] Because you are averaging percent changes, convert the final result from decimal form into a percentage.

[latex]\color{blue}{\%C}\color{black}{\text{ is Percent Change:}}[/latex] The value of each percent change in the series from which the average is calculated. You need to express the percent changes in decimal format.

Let’s use the sales data presented above, according to which sales increase 100% in the first year and decrease [latex]50\%[/latex] in the second year. What is the average percent change per year?

Step 1: The changes are [latex]\%C_1=+100\%[/latex] and [latex]\%C_2=-50\%[/latex].

Step 2: Two changes are involved, or [latex]n=2[/latex].

Step 3: Apply Formula 2.4c:

[latex]\begin{align*}\text{GAvg}&=\left(\left[\left(1+\%C_1\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}-1\right)\times 100\\\text{GAvg}&=\left(\left[\left(1+100\%\right)\times\left(1-50\%\right)\right]^{\frac{1}{2}}-1\right)\times 100\\\text{GAvg}&=\left(\left[2\times 0.50\right]^{\frac{1}{2}}-1\right)\times 100\\\text{GAvg}&=\left(\left[1\right]^{\frac{1}{2}} - 1\right)\times 100\\\text{GAvg}&=0\%\end{align*}[/latex]

The average percent change per year is [latex]0\%[/latex] because an increase of [latex]100\%[/latex] and a decrease of [latex]50\%[/latex] cancel each other out.

Paths To Success

An interesting characteristic of the geometric average is that it will always produce a number that is either smaller than (closer to zero) or equal to the simple average. In the example, the simple average of [latex]+100\%[/latex] and [latex]50\%[/latex] is [latex]25\%[/latex], and the geometric average is [latex]0\%[/latex]. This characteristic can be used as an error check when you perform these types of calculations.

Section 2.4 Exercises

1 WestJet, WestJet Fact Sheet.

THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]\begin{align*}\text{SAvg}=\frac{\sum x}{n}\end{align*}[/latex][latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex][latex]\begin{align*}\text{WAvg}=\frac{\sum wx}{\sum w}\end{align*}[/latex][latex]\begin{align*}\text{GAvg}=\left(\left[\left(1 +\%C_1\right)\times\left(1+\%C_2\right)\times\text{ . . . }\times\left(1+\%C_n\right)\right]^{\frac{1}{n}}-1\right)\times 100\end{align*}[/latex]

Attribution

“3.2: Averages” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.