Summary
Highlights
The trimmed mean is introduced as a method to handle unusual numbers, or 'outliers', in a dataset. An example set of numbers (17, 18, 19, 19, 21, 21, 22, 34) is presented, with 34 identified as an outlier. The rule for trimming is explained: if there's an outlier (e.g., 34), you also remove the corresponding number from the opposite end (e.g., 17) to maintain balance, even if that number isn't an outlier.
The arithmetic mean of the original dataset (171 / 8 = 21.375) is first calculated as a baseline. To calculate the trimmed mean, the outlier (34) and the lowest number (17) are removed from the total sum (171 - 34 - 17 = 120), and the count of numbers (n) is reduced accordingly (8 - 2 = 6). The trimmed mean is then calculated as 120 / 6 = 20. The trimmed mean often provides a more dependable average when outliers are present.
The concept of a weighted mean is introduced using an example of student grades: a quiz (10%), a test (40%), and a final exam (50%). A student's scores are 65% on the quiz, 95% on the test, and 80% on the exam. It's emphasized that simply averaging these scores (65 + 95 + 80) / 3 = 80% would be incorrect because the components have different weights.
To calculate the weighted mean, each score is multiplied by its corresponding weight (expressed as a decimal). For example, Quiz: 65% * 0.1 = 6.5, Test: 95% * 0.4 = 38, Exam: 80% * 0.5 = 40. These weighted contributions are then summed: 6.5 + 38 + 40 = 84.5. Therefore, the student's true final weighted grade is 84.5%.