# Compare z scores with same population different period different exams

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?

• The example you provide of a student who does well on an initial exam who then does worse on follow-up exams is virtually identical to the textbook example of regression towards the mean. Indeed the Wikipedia page uses exam results as an example as well. You certainly need to be careful in such cases. I would suggest carefull reading up on how to avoid regression towards the mean. As mentioned by @user3903581 below, comparing quantiles rather than z-scores will work better here. Mar 31, 2017 at 3:30
• @AaronDefazio. Thank you for the link. I will do some research on "regression towards the mean". Mar 31, 2017 at 4:12