Averages and Weighted Averages In mathematics, there are several different ways to describe an “average” score or data value. Mean To find the mean of a set of values, add up the values and divide by how many values there are. The symbol for the mean is ̅ . In a formula, the mean is given by: ⋯
̅
, where ,
,…,
are the values and represents how many values there are. Example: The heights, in inches, of six students are: 58, 61, 62, 65, 65, and 70. Find the mean height. Solution: ̅
63.5 Median To find the median of a set of numbers, first list the numbers in order, from smallest to highest. Then, find the number in the middle of that set. If there are two numbers in the middle, find the mean of those two numbers. Example: Find the median of each set of numbers. a. 28, 15, 33, 45, 21 Solution: There are five numbers in this data set. Ordering the numbers yields: 15, 21, 28, 33, 45. The number in the middle is 28, so the median is 28. b. 58, 61, 62, 65, 65, 70 Solution: There are six numbers in this data set. They are already ordered from smallest to largest. The two numbers in the middle are 62 and 65. The median is 63.5 Mean versus median? When do we use the mean versus the median when reporting the “average?” Let’s look at a few data sets, find the mean and the median, and decide which one might be more appropriate.  Problem (You do this!): A high school math teacher has two Algebra II classes. He gives each class the same exam. The scores are reported below. Class 1 Student 1 Student 2 Student 3 Score 60 62 65 Class 2 Student 1 Student 2 Student 3 Score 60 62 65 Student 4 Student 5 Student 6 Student 7 Student 8 Student 9 Student 10 66 68 70 71 72 72 73 Student 4 Student 5 Student 6 Student 7 Student 8 Student 9 Student 10 66 68 70 71 72 72 99 Class 2 Mean Class 2 Median Find the mean and median for each class. Class 1 Mean Class 1 Median Explain any differences you see between the two data sets of test scores. Use complete sentences. How do the differences change the mean? Use complete sentences. For class #2, which would be a better “average” to report, the median or the mean? Summary: The mean is sensitive to extreme scores, sometimes known as outliers. The median is not as affected by outliers. For this reason, the median is often used when reporting average salaries or the average sales price of a home. Mark Zuckerberg’s salary is an outlier, and the median would not be affected by it! Likewise, a few luxury homes can pull up the mean home price, but not the median home price. Weighted Averages Weighted averages, or weighted means, are used often to compute GPA’s and class averages. The word “weight” here indicates the importance we attach to a category. The weighted mean is found by multiplying each value by its corresponding weight, adding these up, and dividing by the sum of the weights. Example: Computing a class average. In Patricia’s math class, her grade will be computed as follows: attendance is worth 5% of her grade, quizzes are 10%, exams 60%, and the final exam is 25%. Patricia’s grades are as follows: Attendance: 100% Quizzes: 93% Exams: 86% Final Exam: 85% Determine Patricia’s course average. Solution: The weights are 5% for attendance, 10% for quizzes, 60% for exams, and 25% for the final. 5 ∗ 100
10 ∗ 93 60 ∗ 86
5 10 60 25
25 ∗ 85
8715
100
87.15  Problem (You do this!) : In Gabe’s math class, his grade will be computed as follows: quizzes are worth 10% of his grade, projects are worth 10% of his grade, exams are worth 60% of his grade, and the final is worth 20% of his grade. Gabe’s grades are as follows: Quizzes: 75% Projects: 95% Exams: 72% Final Exam: 75% Compute Gabe’s course average.