It doesn't how smart you are. If you cannot market yourself, you are nothing!
I was sitting around the NASA intern lunch table, and one of the interns was talking about his application for grad school application for Bio-physics. However, he didn't know how to write his statement because it asked for all of his internship and research experience and how it's related to biology. He told us about his research regarding space dust, and then ended up saying, "I'm not gunna get into grad school."
What should one do in this situation?
Well first of all, he totally procrastinated on the application. It's due in 3 days. Don't procrastinate on applying for a program that's going to take at least 2 years of your life. It just doesn't make sense.
Talk about your experience. Everyone has stories, regardless if it's related to whatever! You'll have experience relevant to what you are going to be doing in your job/graduate lab. Sure, it wasn't bio related. But are you going to be collect samples? Yes. Are you going to work in a team? Yes. Are you going to organize your data into a report? Yes! If this isn't relevant to graduate school, I don't know what is.
Talk about your journey to today. You didn't just decide to go to grad school yesterday (well I hope you didn't). Why are you going to grad school? There has to be a reason!
What difference do you want to make in the world? People who read applications like doing so because they see themselves. It helps to remind themselves of the hope that the younger generation sees. Be like Obama: promote the CHANGE for this world.
It's really hard to brag while writing. I recently read essays for a high school scholarship. There was one applicant who said, "I did ___. And I did ___. Oh, I also did ___." Sure, the essay was poorly executed. But I didn't know anything about those activities unless they told me. If you say, "I'm awesome," that's bragging. But mostly likely you are just stating the facts. Make the reader say, "Wow, that applicant has potential" by showing off a little.
*Update: I found this website that gives you advice on admission essays, regardless of which admission. The link is found HERE