I work for a school. I have two sets of data. Do I need to compare an apple with an orange?
Set 1: test scores from our school. Min score is 0. Max score is 100.
Set 2: test scores from third-party exams outside school. Min score is 100. Max score is 500.
I try to use data from Set 2 as a baseline to see the progress of students. First, I convert Set 1 and 2 to z scores and then compare them. For example, Peter scores 70 in Science and this is converted to a z score of 1.1. He also scores 350 in the external exam and this score is converted to 1.5.
The third-party exam was conducted in 2015. School test is conducted in 2017. Since 1.5 is greater than 1.1, maybe I can say his Science has improved since 2015?
However, there is a problem. Let's say Peter has done really well in external exams, his z score may become 3.0. He is also a top student in our school and he has received a score of 95 on our first school test. This 95 may be converted to 2.7. Since 2.7 - 3.0 is -0.3, his growth is negative. This is not right. Because the two populations have different means, the z scores will be different. Even if Peter has done so well in all school tests, his z scores of all of them may not reach 3.0. Therefore his growth will always be negative.
My question is: we have 2 sets of data with the same population but the nature of the data is not the same. How do I perform a meaningful comparison between them in statistics?