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 millilitre.
• 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, 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. The stock value has changed by 9%, −7%, 13%, 4%, and −2% over these years, 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 facilitates easier comprehension of and comparison between 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 do 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.

The Formula

As expressed in Formula 3.3, you calculate a simple average by adding together all of the pieces of data then taking that total and dividing it by the quantity.

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. Review the following situations and, without making any calculations, determine the best answer.

1. The simple average of 15, 30, 40, and 45 is:
   1. lower than 20.
   2. between 20 and 40, inclusive.
   3. 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
   1. 10
   2. 20
   3. 30
   4. 40

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:

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).

The Formula

As expressed in Formula 3.4, 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.

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.

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.

Give It Some Thought

In each of the following, determine which information is the data and which is the weight.

1. Rafiki operates a lemonade stand during his garage sale today. He has sold 13 small drinks for $0.50, 29 medium drinks for $0.90, and 21 large drinks for $1.25. What is the average price of the lemonade sold?
2. Natalie received the results of a market research study. In the study, respondents identified how many times per week they purchased a bottle of Coca-Cola. Calculate the average number of purchases made per week.

Geometric Averages

How do you average a percent change? If sales increase 100% this year and decrease 50% next year, is the average change in sales an increase of 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 a $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.

The Formula

In business mathematics, you most commonly use a geometric average to average a series of percent changes. Formula 3.5 is specifically written to address this situation.

Important Notes

A critical requirement of the geometric average formula is that every (1 + Δ%) expression must result in a number that is positive. This means that the Δ% cannot be a value less than -100% else Formula 3.5 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, 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

For the first three questions, determine whether you should calculate a simple, weighted, or geometric average.

1. Randall bowled 213, 245, and 187 in his Thursday night bowling league and wants to know his average.
2. Cindy invested in a stock that increased in value annually by 5%, 6%, 3%, and 5%. She wants to know her average increase.
3. A retail store sold 150 bicycles at the regular price of $300 and 50 bicycles at a sale price of $200. The manager wants to know the average selling price.
4. Gonzalez has calculated a simple average of 50% and a geometric average of 60%. He believes his numbers are correct. What do you think?