In his NPR radio program "A Prairie Home Companion," Garrison Keillor describes the mythical town of Lake Wobegon as a place where "all the women are strong, all the men are good-looking, and all the children are above average." The average (mean) score on a recent test given to a class of 30 students at Lake Wobegon Elementary School was 75 (out of 100).
(a) explain why it is possible for every student in the class to have scored above average on the test.
(b) what is the largest number of students who could have scored above average on the test?
The thing that comes to mind immediately is how the average is rounded. What if all students made a score of 75.01? The average, rounded, is 75, but all students were above average.
The thing that comes to mind immediately is how the average is rounded. What if all students made a score of 75.01? The average, rounded, is 75, but all students were above average.
You are correct. The phenomenon described in the question is actually a result of how averages are calculated and rounded. In this case, the average score on the test is 75. Since the average is calculated by summing up all the individual scores and dividing by the total number of students, it is theoretically possible for every student to have scored above this average.
For example, let's say there are 30 students in the class and they all scored exactly 75.01. When you add up these scores and divide by 30, you get an average of 75.01. However, if you round this average to the nearest whole number, it would be 75. So, even though all the students scored above the rounded average, technically they all scored above the true average.
To answer the second part of the question, the largest number of students who could have scored above average depends on how close to the average the individual scores can be. In this case, if we assume that students can score up to the nearest hundredth (like 75.01, 75.02, etc.), then it is mathematically possible for all 30 students to have scored above average.