I work for a school. I have two set of data. 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 base line 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 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, may be 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 in 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 set of data with same population but the nature of the data are not the same. How do I perform a meaningful comparison between them in statistics? Thank you.