I'm assuming that you're applying to enter a PhD or Masters program. I'm a STEM PhD candidate at a research-intensive US university, and I'm basing my advice off of what professors here have told me, as well as successful applications I've seen of admitted students:
Keep in mind that often a committee will read your statement of purpose before any other statement. Thus, you want the statement of purpose to present yourself in a way that makes them excited about reading the rest of your application. In some places, if they're not hooked there, they might not even keep reading your longer sections. In this vein, it's okay to (briefly!!) mention things you explain at more length elsewhere in the application.
Of course, how to craft a statement that does that depends on the particular individuals reading your application, so disclaimer that this is only one set of advice and you likely want to recruit multiple viewpoints. Common themes in similar writing that has been successful in the work I have seen:
- Your motivation. Briefly: Why do you want to be in [program]? (In STEM at least, most graduate programs will care nearly-exclusively about your passion for research.)
- Why this program? Why is [program] the best fit for you?
- Why you? Why should [program] want you? What do you bring to the table that they want?
Those second and third points can be very hard to get a handle of if you've not been affiliated with or visited the school before. Scouring the website is helpful, but you can also email current graduate students in the program. At least in my program, we'd all be happy to answer questions in an email or talk on the phone with a potentially-applying student.
PS: I apologize for not being able to comment first to clarify some of my assumptions, I don't yet meet the 50 reputation requirement.
