P2P No. 4 — State your business
A statement of purpose is about what you can give, not what you can get.
Writing a statement of purpose is not an enviable task, particularly for it is yet another thing we should somehow master without being shown how to do it. The most useful practical guideline I know of is that
when writing an application, imagine yourself in place of the evaluator.
When the goal is to convince, the proper question is, what does the other side want?
A good statement of purpose is not about why it is beneficial for you to get the job/award/money, but why it is beneficial for the other side to choose you.
It is hard to imagine being the evaluator, but practicing this skill will make your application stand out. It is tacit in the application itself that we would benefit from a positive decision, but emphasizing that would be very bad marketing: the best strategy to sell a house is not stating how much you need the money. On the other hand, showing the courtyard pool with a breathtaking view will convince many people that this is what they need.
We should do the same in our applications. Beware of a caveat: applying for a Ph.D. is not a one-time purchase, so bending reality is advised against. You might convince someone famous to hire you, but if this means deceiving them into believing you are a match, you have dug yourself into a hole. Why? Because the personal dimension is the most important—as I have elaborated previously—and now you are stuck for years with someone whom you potentially need to show your make-believe façade.
Beware of cajoling. The professor you applied to knows what they did, so awarding them metaphorically with the Nobel Prize is unwise - if this is the only way of getting to yes, then that's a clear no, since, ideally, we would like to work with someone who sees the merit of both our work and personality.
If the would-be supervisor's research is meaningful and exciting, that's a fair point to state though this can lead in the same direction as being flattering. The dichotomy is why composing a genuine statement is difficult. Stating genuine personal excitement or devotion to achieve real-world impact might sound like a cliché, though if this is your personal drive, give this puzzle piece away. Your would-be supervisor can then assess whether your interests align. The same holds for finding elegance in an approach.
If you have relevant research, then write about it. If your research is not relevant, that's also fine; most people are looking for curious applicants who want to learn and can express their thoughts clearly. Thus, showcasing any project that has taught you transferrable skills is also a good idea.
I encourage everyone to think in the broadest terms since everyone has a unique path, and someone might be looking for the exact combination of skills you have. But you do need to let them know that you have it.