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if i'm a first generation college student and working class (father worked for the Navy though he is now deceased and i probably don't have to list it and mother worked at a grocery store for 20 years) should i write a diversity statement? will that make my application look better?

It could help. First generation college students tend to get some sympathy. But make sure it's about overcoming adversity rather than just telling them you had adversity.

my PS is about overcoming my abusive home life and my father's suicide, but i think adding a DS to clarify my working class background might be worth it. thanks!