Yeah, that apostrophe rule is definitely only for contractions. Circumventing possessives (the cat of Mattabet) is circuitous, which actually makes writing worse.
Unless you're writing your personal statement en francais, in which case cirmcumventing possessives is the rule.
My personal statement actually had two contractions. Haven't and can't. I'm of the opinion that a personal statement should be semiformal. Not quite a night at the opera but hardly fish n' chips in the local pub.
My rule of thumb is as follows for semiformal writing: write your draft and revise for content and clarity. Ask someone to read it and comment. All of this should be done without real regard for formal language. Up to now you are writing to develop a thesis and defend it. You are writing for the strength and clarity of your ideas.
Once the content and argument are up to par, then you edit for flow. In formal writing, remove all contractions and passive voice. For semiformal, bold all contractions (as passive voice) and attempt to remove them; however, if the sentence becomes awkward or you want to intentionally voice a specific tone, leave them in. Often this will be the case when telling a story where you want the reader to feel as if you were talking directly to them.
Example: Which sounds better and is more likely to draw a reader in,
A) "While I cannot remember what drew me to the girl sitting in the corner of the emergency department waiting room..."
B) "While I can't remember what drew me to the girl sitting in the corner of the emergency department waiting room..."
While A is more formal, B has a more spoken flow and may be more successful in having the reader feel as though they are inside the mind of the writer.
Finally, edit for precision. Make sure that every sentence has a purpose and that it fulfills that purpose.
Lastly: Remember that the people reading your essays are going to read 100's if not 1000s of essays just like it. An essay this is too formal and too cookie cutter cannot stand out. Honestly, who wants to read a 1000 essays that never use contractions. It literally wastes time (since you're adding syllables) and decreases readability/enjoyment. At the same time, you need to present yourself as someone who is not lost on all formalities and didn't just throw together a PS in a few days.
If someone tells you in your interviews that the were "gripped" reading your essays and they "really enjoyed them," you've succeeded.
