Where does the expression “no skin off one’s nose” come from and What does it mean?

The expression “no skin off one’s nose” means: nothing of concern to one; not one’s affair.

In today’s slang, nose is sometimes replaced by ear, elbow, or back, but the implication is equally evident:

If one doesn’t butt into, or stick one’s nose into, an affair that is none of one’s business, he is not likely to suffer abrasions upon any prominent portion of his anatomy by being thrown out upon his nose, elbow, back, shoulder, or the like.

The allusion is American, at least fifty years old.