I'm doing a regression about how practice of sports affect the GPA. I'm using a panel data, and I'm going to do a fixed effects approach. The thing is, I need to standardize the GPA and the sports as well somehow so that I can give some kind of weight to the students who have a lower/upper GPA, or are going more to the gym, because they are seeing more or less credits. For example, a student with a GPA of 4/5 but who's seeing 20 credits isn't quite easily comparable to one that got a GPA of 4/5 as well but just with 6 credits.
So I first thought about multiplying credits by GPA, but I think the difference is exaggerated with this approach.
Then I thought I could standardize the GPA with a formula somehow like this: (x-u)*(std/2) > with x being the observation, u the mean and std the standard deviation. My intuition here is that x-u gives positive if the credits are higher than the mean so the effect on GPA will be positive, and when x is lower than the mean, well, it will go down a bit. I think this is somehow arbitrary and I think it may be wrong for me to mess with the data like this?
Then I thought I could do an auxiliary regression like this: GPA=credits+credits^2. The intuition is that credits^2 will let me now for a fact the form of the curve, and credits alone would give me a coefficient which I can add or subtract depending on whether the GPA in that semester is over que top or lower bottom of the curve, and is in fact affected upwards or downwards by the amount of credits. In this case, with the sports practice the regression would be similar.