What is a good length for the statement of purpose in a U.S. PhD application?
Depending on the program I would keep it between 1.5 to 2 pages. I framed mine to answer the following questions:
- Why am I applying to this school.
- What are my research interests.
- Previous academic accomplishments.
- Previous professional accomplishments.
- Personal story of triumph (optional, I talk about having to drop out of high school to support my family)
- What is your goal after graduation.
Addition:
I would also target schools that are pumping out the research that matches with your interests... this will help when you are stating why you are applying like:
A major factor in helping me choose Awesome University was Professor Awesome's research on being awesome. I feel that this aligns well with my research interests, and would make for a great fit between me and the department.
Also, if you talk about any abstract ideas be sure to elaborate on them to give a level of specificity. They want to see you be able to translate abstract ideas into specific thoughts.
If you have a statement of purpose written for one school, you should be able to adapt it to other schools, too. Writing a fully customized SoP is a waste of time, unless you are applying to just three places (and then you are either overconfident, or just don't know what you are doing). If you've done a 1.5-2 page SoP, as JohnB suggested, just change a couple of paragraphs, and that should do it.
First paragraph: define yourself professionally ("I am a professional golf player who likes to deal with computers. I assembled my first PC at the age of 7, and hacked the school district network at the age of 12. Since then, I have been receiving straight As throughout the school")
Second paragraph: define where you want to be after Ph.D. ("I want to become a physicist to understand the nuances of friction between the golf ball and the air, and the ball and the grass. I also want to get involved with biology on the side, so that I could understand the ecology of the golf courses better.")
Two-three paragraphs: describe why this department is a particularly good fit. It's here where you describe the work you've done, and how it relates to the work other people in the department have been doing. ("I have found the work that Prof. Feynman has performed on the stability of beer foam as a function of ambient temperature and concentration of fine particulate matter (cigarette smoke) in pubs of Ireland to be very intriguing, and cited his papers on the topic in my own research on flotation of potato chips on beer surface, see Appl. Phys. Letters forthcoming in 2013").
Conclude with re-stating that you think this is a great program, and you think you are a great fit for it.
Just some clarifications that the other answers seem to have missed. No school needs you to tell them how awesome you think they are (I believe you are better than Harvard) or how much you love their location (I love being by the ocean / I've always dreamt living in Manhattan). Explain why they will not regret accepting you. Show them what you're good at and why you will be a good fit for their department. Describe your plans and ambitions for after you've graduated. Overall keep it short but well-written. Don't waste their time with long formal introductions and greetings. Don't be repetitive.