What is the approximate composition of a statement of purpose in Mathematics? I am writing a statement of purpose right now for US graduate schools in mathematics, and am I bit bogged down on figuring out the approximate ratio of my background versus my actual research goals. I currently have three paragrahs of background, mainly involving seminars and projects, and a discussion of the non-traditionality of my situation, and one of my actual research goals, and I am trying to determine if this is what is desired? The only source of examples I could find are here, and I am not sure if these are actually good examples of what is desired.

Edit: I keep using the wrong word since statement of purpose and Personal statement means different things to different departments (for example, Northwestern seems to want only one essay, which they call a personal statement)
When I review graduate applications, the statement serves three purposes:


If you are not a native English speaker, it may tell me something about your English proficiency.  However, it cannot be relied upon too heavily since you may have gotten at least editorial help from a friend or even a professional.
More importantly, it is an opportunity to demonstrate that you know something about a research area that interests you.  This is more important for PhD applications than for MS applications, obviously.  If you have already done some research, hopefully your recommendation letters speak about that.  Your statement can sometimes be useful in showing that you keep up with some research area even though you haven't yet contributed to it.
Finally, your statement can address issues in your application, like a period of low grades related to something in your personal life, a gap in your schooling/employment, etc.


There is in my opinion no correct ratio of personal versus research components; it depends on the individual.