2.2 Averages

No matter where you go or what you do, averages are everywhere. Let’s look at some examples:

• Three-quarters of your student loan is spent. Unfortunately, only half of the first semester has passed, so you resolve to squeeze the most value out of the money that remains. But have you noticed that many grocery products are difficult to compare in terms of value because they are packaged in different sized containers with different price points? For example, one tube of toothpaste sells in a 125 mL size for $1.99 while a comparable brand sells for $1.89 for 110 ml. Which is the better deal? A fair comparison requires you to calculate the average price per milliliter.
• Your local transit system charges $2.25 for an adult fare, $1.75 for students and seniors, and $1.25 for children. Is this enough information for you to calculate the average fare paid, or do you need to know how many riders of each kind there are?
• Five years ago you invested $8,000 in Roller Coasters Inc. During the years that followed, the stock value has changed by 9%, −7%, 13%, 4%, and −2%, and you wonder what the average annual change is and whether your investment kept up with inflation.
• If you participate in any sport, you have an average of some sort: bowlers have bowling averages; hockey or soccer goalies have a goals against average (GAA); and baseball pitchers have an earned run average (ERA).

Averages generally fall into three categories. This section explores simple, weighted, and geometric averages.

Simple Averages

An average is a single number that represents the middle of a data set. It is commonly interpreted to mean the “typical value.” Calculating averages helps in understanding and comparing different data sets, particularly if there is a large amount of data.

For example, what if you want to compare year-over-year sales? One approach would involve taking company sales for each of the 52 weeks in the current year and comparing these with the sales of all 52 weeks from last year. This involves 104 weekly sales figures with 52 points of comparison. From this analysis, could you concisely and confidently determine whether sales are up or down? Probably not. An alternative approach involves comparing last year’s average weekly sales against this year’s average weekly sales. This involves the direct comparison of only two numbers, and the determination of whether sales are up or down is very clear.

In a simple average, all individual data share the same level of importance in determining the typical value. Each individual data point also has the same frequency, meaning that no one piece of data occurs more frequently than another. Also, the data does not represent a percent change.

To calculate a simple average, you require two components:

1. The data itself—you need the value for each piece of data.
2. The quantity of data—you need to know how many pieces of data are involved (the count), or the total quantity used in the calculation.

Using these pieces of information, you can calculate the simple average of [latex]x_1, x_2, \ldots, x_n[/latex] as follows:

In the abbreviated form, [latex]SAvg[/latex] represents the simple average and the symbol [latex]\sum[/latex] is the Greek capital letter for sigma, or S, representing the word sum. The symbol [latex]\sum\limits^n_{i=1}x_i[/latex] is used as shorthand for adding up many terms, and the symbols telling you that there are [latex]n[/latex] terms to add up, each numbered by [latex]i[/latex] and written as [latex]x_i.[/latex] The collection of symbols [latex]\displaystyle\sum\limits^n_{i=1}x_i[/latex] is read as “the sum from i equals 1 to n of x i”.

As expressed in the formulas above, you calculate a simple average by adding together all of the pieces of data, then taking that total and dividing it by the quantity.

For example, suppose you want to calculate an average of three pieces of data: 95, 108, and 97.

Note that we have no information about one piece of data being more important or valuable than another and each appears only once, thus having the same frequency. As a result, the average to be calculated is just a simple average.

There are three pieces of data: [latex]x_1=95, x_2=108[/latex] and [latex]x_3=97[/latex], and [latex]n=3[/latex].

Therefore,

[latex]SAvg=\frac{x_1+x_2+x_3}{3}=\frac{95+108+97}{3}=100[/latex]

Important Notes

Although mentioned earlier, it is critical to stress that a simple average is calculated only when all of the following conditions are met:

1. All of the data shares the same level of importance toward the calculation.
2. All of the data appear the same number of times.
3. The data does not represent percent changes or a series of numbers intended to be multiplied with each other.

If any of these three conditions are not met, then either a weighted or geometric average is used, depending on which of the above criteria failed. We discuss this later when each average is introduced.

Give It Some Thought

It is critical to recognize if you have potentially made any errors in calculating a simple average by considering your final answer and whether it makes sense in the context of the question you are answering. Review the following situations and, without making any calculations, determine the best answer.

1. The simple average of 15, 30, 40, and 45 is:
   a. lower than 20. b. between 20 and 40, inclusive. c. higher than 40.
2. If the simple average of three pieces of data is 20, which of the following data do not belong in the data set? Data set: 10, 20, 30, 40
   a. 10 b. 20 c. 30 d. 40

Solutions:

1. b (a simple average should fall in the middle of the data set, which appears spread out between 15 and 45, so the middle would be around 30)
2. d (if the number 40 is included in any average calculation involving the other numbers, it is impossible to get a low average of 20)

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 school 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 is not appropriate in this case. This is where the weighted average comes into play.

In a weighted average we can have pieces of data that have a different level of importance or that occur with different frequencies. Note that, however, a weighted average is not appropriate when the data pieces 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:

1. The data itself—you need the value for each piece of data.
2. 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).

Using these pieces of information, you can calculate the weighted average of [latex]x_1, x_2, \ldots, x_n[/latex] as follows:

In the abbreviated form, [latex]WAvg[/latex] represents the weighted average and [latex]w_i[/latex] the weight of the data piece [latex]x_i[/latex].

Note that the simple average is just a special case of a weighted average where the weights are all equal and of value 1. Then [latex]w_ix_i=x_i[/latex] for all [latex]i[/latex] from 1 to [latex]n[/latex], and [latex]w_1+w_2+\ldots+w_n=n[/latex].

As expressed in the formulas above, you can 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.

For example, let’s stay with the illustration of the math and communications courses and your GPA, discussed briefly above. 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 4.0. In the communications course, your C+ translates into a 2.5 grade point. These courses have five and three credit hours, respectively. Suppose that you want to calculate the average grade point.

Since you want to calculate the average grade point, the grades are your data pieces.

Since the grade points carry different importance, based on the credit hours, to calculate the average grade point you must use a weighted average.

There are two  pieces of data: [latex]x_1=4.0,[/latex] and [latex]x_2=2.5[/latex], with weights [latex]w_1=5[/latex] and [latex]w_2=3[/latex], respectively.

Therefore,

[latex]WAvg=\frac{w_1x_1+w_2x_2}{w_1+w_2}=\frac{5\cdot 4.0+3\cdot 2.5}{5+3}=3.4375[/latex]

Note that your GPA is higher than if you had just calculated a simple average of [latex]\frac{{42.5}}{2} = 3.25[/latex]. This happens because your math course, in which you scored a higher grade, was more important in the calculation.

Things To Watch Out For

The most common error in weighted averages is to confuse the data with the weight. If you have the two backwards, your numerator is still correct; however, your denominator is incorrect. To distinguish the data from the weight, notice that the data forms a part of the question. In the above example, you were looking to calculate your grade point average; therefore, grade point is the data. The other information, the credit hours, must be the weight.

Give It Some Thought Answers

1. The price of the drinks is the data, and the number of drinks is the weight.
2. The purchases per week is the data, and the number of people is the weight.

Geometric Averages

Note that we said that neither the simple average nor the weighted average are appropriate when trying to measure an average of a series of percent changes.

So how do we average a percent change?

The problem is this: if sales increase 100% this year and decrease 50% next year, is the average change in sales an increase of [latex]\frac{100\%+(-50\%) }{2} = 25\%[/latex] per year? The answer is clearly “no.” If sales last year were $100 and they increased by 100%, that results in a $100 increase. The total sales are now $200. If sales then decreased by 50%, you have $100 decrease. The total sales are now $100 once again. In other words, you started with $100 and finished with $100. That is an average change of nothing, or 0% 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.

We calculate the geometric average of a series of percentage changes [latex]\Delta_1,\Delta_2,\ldots,\Delta_n[/latex] as follows:

In the abbreviated form, [latex]GAvg[/latex] represents the geometric average and the symbol [latex]\prod[/latex] is the Greek capital letter for pi, or P, representing the word product. The latter is a symbol that is used as shorthand for multiplying many terms, with the counter [latex]i[/latex], and the symbols telling you that there are [latex]n[/latex] terms to multiply, each numbered by [latex]i[/latex] and written as [latex](1+\Delta_i).[/latex] The collection of symbols [latex]\displaystyle\prod\limits^n_{i=1}(1+\Delta_i)[/latex] is read as “the product from i equals 1 to n of one plus delta i”.

In business mathematics, you most commonly use a geometric average to average a series of percent changes. The formula above is specifically written to address this situation.

For example, let’s use the sales data presented above, according to which sales increase 100% in the first year and decrease 50% in the second year. What is the average percent change per year?

The task is to find the average of two sequential percent changes, [latex]\Delta_1=100\%[/latex] and [latex]\Delta_2=50\%[/latex].

Therefore, geometric average is the most appropriate:

[latex]\begin{align*} GAvg&=\sqrt[2]{(1+\Delta_1)(1+\Delta_2)}-1\\ &=\left((1+1)(1-0.5)\right)^{1/2}-1\\ &=0 \end{align*}[/latex]

Therefore, the average percent change per year is 0%. This makes sense because an increase of 100% and a decrease of 50% cancel each other out.

Important Notes

A critical requirement of the geometric average formula is that every [latex](1 + \Delta)[/latex] expression must result in a number that is positive. This means that the [latex]\Delta[/latex] cannot be a value less than -100%, else the geometric average of percent change cannot be used.

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 above, the simple average of +100% and -50% is 25%, and the geometric average is 0%. This characteristic can be used as an error check when you perform these types of calculations.

Give It Some Thought Answers

1. Simple; each item has equal importance and frequency.
2. Geometric; these are a series of percent changes on the price of stock.
3. Weighted; each item has a different frequency.
4. At least one of the numbers is wrong since a geometric average is always smaller than or equal to the simple average