Prove: $\operatorname{std}(\frac{v}{\operatorname{std}(v)}))=1$

Question

Prove: $\operatorname{std}(\frac{v}{\operatorname{std}(v)}))=1$ for any vector $v$ where std is the standard deviation as $\sqrt{\frac{1}{n-1}\sum_{i=1}^n (v_i-\bar{v})^2}$

I think I have to cancel out the $(n-1)$s somewhere to get 1 in the end because of this proof for a z score but I can't figure out how to get to that point.

Any help is appreciated!

Answer 1

Let $v=(v_1,\dots, v_n)$. One has $$std(\frac{v}{std(v)}) = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (\frac{v_i}{std(v)}-\bar{\frac{v}{std(v)}}})^2 = \frac{1}{std(v)}\sqrt{\frac{1}{n-1}\sum_{i=1}^n(v_i-\bar{v})^2} = \frac{std(v)}{std(v)} = 1.$$