4.1 Simple and Weighted 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 (weight) 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.

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

SOLUTION:

We will need to calculate a simple average for the first quarter in each of 2013 and 2014. Then convert the numbers into a percentage.

WHAT WE ALREADY KNOW:

You know the three quarters annually:

2013: [latex]x_{1}=$413,200,\:  x_{2}=$328,986,\: x_{3} =$350,003[/latex]

2014: [latex]x_{1} = $455,876, \:x_{2} = $334,582,\: x_{3} = $312,777[/latex]

STEPS:

For each year, perform steps 1 to 3:

• Step 1: Sum the data.
• Step 2: Count the total quantity of data involved.
• Step 3: Calculate the simple average using Formula 3.3.

PERFORM:

Step 1

• total of 2013 = $413,200 + $328,986 + $350,003 = $1,092,189
• total of 2014 = $455,876 + $334,582 + $312,777 = $1,103,235

Step 2

• for both years, n=3

Step 3

• For 2013, [latex]\text{Simple Average }=\frac{ $1,092,189}{3} = $364,063[/latex]
• For 2014, [latex]\text{Simple Average }=\frac{$1,103,235}{3} = $367,745[/latex]

FINAL ANSWER:

The average monthly sales in 2013 were $364,063 compared to sales in 2014 of $367,745.

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

To calculate the weighted average you add the products of the weights and data for the entire data set and then divide this total by the total of the weights.

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

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].

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 4.0. In the communications course, your C+ translates into a 2.5 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 (math credit hours × math grade point) + (communications credit hours × communications grade point). Numerically, this is = (5 × 4) + (3 × 2.5) = 27.5.

Step 2: In the denominator, sum the weights. These are the credit hours. You have 5 + 3 = 8.

Step 3: Apply the weighted average formula to calculate your GPA.

[latex]\text{Weighted Average} = \frac{27.5}{8}=3.44[/latex]  (GPAs have two decimals).

Note that your GPA is higher than if you had just calculated a simple average of 4 + 2.5 = 3.25. 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 backward, 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.

Solution:

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.

Step 1:

Find the sum of weights x grade = [latex](4 \times 3.0) + (5 \times 4.0) + (3 \times 3.5) + (4 \times 2.0) + (3 \times 4.5) + (4 \times 1.0) = 68[/latex]

Step 2:

Find the sum of the weights = [latex]4 + 5 + 3 + 4 + 3 + 4 = 23[/latex]

Step 3:

[latex]Weighted Average = \frac{68}{23} = 2.96[/latex]