There are five people in a group that are 61, 61, 63, 64, 66, and 90 inches tall.

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1 Lesson #2 Outliers, Weighted and Trimmed Means Outliers: An outlier is an observation that is numerically distant from the rest of the data. It s a value that lies outside (and is much larger or smaller than) the other values in a set of data. Example 1: There are five people in a group that are 61, 61, 63, 64, 66, and 90 inches tall. a) Determine the mean, median, and mode. b) What is the outlier? Remove it from the data and recalculate the mean, median, and mode. Trimmed Mean: The trimmed mean is calculated by discarding a certain number or percentage of the lowest and highest scores, and then calculating the mean based on the remaining scores. A trimmed mean is less susceptible to the effects of extreme scores (outliers) than the arithmetic mean. Page 1 of 7

2 Example #2: Suppose a gymnast in the London Olympics received the following scores. 7.5, 8, 9.5, 6.5, 7, 7.5, 8, 7.5, 8, 7 Her final score is based on removing the highest and lowest scores and then finding the mean. Calculate the following: a. The arithmetic Mean. b. The trimmed Mean. Example #3 The trimmed mean can also be found by dropping a percentage of the highest and lowest scores. Calculate the trimmed mean for Example 2 except this time solve trimming 10% of the data. (Round final answer to nearest integer) Step 1: Count the number of observations, labeled n. Step 2: Reorder the scores from smallest to largest. Step 3: Find the proportion trimmed from the data. p = where P = % trimmed. Therefore Step 4: Calculate the total number of scores to be removed. n X p = Therefore, we will remove the highest and the lowest score to find out the mean trimmed 10%. Page 2 of 7

3 NOTE: If you were to trim 10% and there were 20 scores then, n = 20 p = = 0.1 n X p =.1 X 20 = 2 Therefore, you would remove 2 scores from the top and 2 scores from the bottom. Weighted Mean: The weighted mean is often used when giving grades in school. It is similar to the arithmetic mean except some data points may be more valued than others. Example #4: Mr. Krahn has 2 math classes. One class has 5 students while the other has 10. Each class scored the following grades: Class 1: 55, 69, 80, 84, 62 Class 2: 70, 90, 55, 84, 88, 93, 78, 69, 98, 75 The mean for class 1 is 70. The mean for class 2 is 80. If you calculate the mean for both classes you get a mean of 75. This answer does not account for the different number of students in each class. The proper mean for both classes can be found by totaling all of the grades and dividing by 15 students. Therefore: OR: You can use the weighted mean: Mean = Page 3 of 7

4 Assignment #2 1. What is the difference between the arithmetic mean and the trimmed mean? 2. Louise makes purses for extra income. The following numbers show how many hours she spent making each purse. 6, 8, 37, 3, 9, 6, 8.5, 5, 7.5, and 9 a. What is the mean number of hours Louise spent on each purse? b. What is the trimmed mean of hours spent on each purse if you remove the highest and lowest number? c. Which mean best represents the number of hours that Louise spent on the purses and why? d. What might explain the outliers in this case? Page 4 of 7

5 3. A teacher has two grade 12 math classes. The 25 students in class A scored the following test results: 65, 75, 92, 53, 87, 59, 32, 80, 76, 37, 68, 79, 67, 69, 81, 57, 66, 71, 90, 73, 90, 72, 61, 67, 53, The 20 students in Class B scored the following test results: 98, 79, 83, 58, 69, 84, 77, 86, 89, 63, 78, 76, 59, 89, 74, 55, 69, 64, 87, 98 a. What is the arithmetic mean for class A? and B? b. Calculate the weighted mean for the 2 classes combined using 2 different methods. 4. Bill wants to know what his average score in golf was over a season but is not sure which mean to use. He golfed at 2 different golf courses during the summer, here are his scores. Bridges Golf Course 75, 82, 83, 77, 68, 88, 86, 84, 79, 80, 84, 95 John Blumberg Course 77, 76, 74, 67, 75, 74, 99, 80 a. Find his overall mean. b. Find his mean for each course. Page 5 of 7

6 c. Find his weighted mean based on the 2 courses. d. Find his trimmed mean for both courses if he took his lowest and highest 2 scores away. e. Find his 20% trimmed mean for both courses. f. What does all this information say about Bill s golf game and the courses that he plays at? Page 6 of 7

7 5. Frank is in the Engineering Faculty at U of M and earned the following marks. 8 / 10 on an assignment 7 / 10 on a quiz 7.5 / 10 on a presentation 9 / 10 on the second assignment 10 / 10 on the second quiz 82% on his final exam His final grade is calculated by the following: 10% for assignments, 15% for quizzes, 25% for presentations, and 50% for final exams. What is Frank s final grade? Page 7 of 7