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Two years ago, I took the LSAT and probably shouldn't have because I did not really study. At this point, I didn't make time for studying and I wasn't even positive law school was going to be my future. This summer, I put my all into studying for the September LSAT. A year after graduating from undergrad, I had a fresh perspective and am confident a career in law is what I want. How do I best address my score difference (from 157 to 173) in an addendum to my applications?

You don't need an addendum for this. Your reason for the improvement is the reason anyone seeing the two scores would assume given the temporal gap and nothing else. You took it cold, did poorly. Later you decided to put in the work and study, did well. That will speak for itself.

Don't bother with an addendum. Schools only care about your 173.