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I've never seen a mathematician claim that statistics is not a branch of mathematics. Nor would that really make sense.

You can certainly practice statistics in a way that mathematicians would take issue with, but it's intrinsically a branch of math.

I've seen a number of mathematicians take umbrage at the idea of describing statisticians as mathematicians, or even applied mathematicians.

Thinking that we are somehow special makes us feel better. The truth is that we are not.