Prove: $\operatorname{mean}(v-\operatorname{mean}(v)) = 0$

Question

My professor assigned me to prove $\operatorname{mean}(v - \operatorname{mean}(v))=0$ where $v$ is any vector.

I know this to be true intuitively because subtracting the mean from every point centers the points around 0 so taking the mean of it again produces 0 but I'm having trouble putting that on paper.

I have to use that fact that the mean is equal to:

$\frac{1}{n}\sum_{i=1}^n x_i = \bar x$

I got as far as:

$\frac{1}{n}\sum_{i=1}^n x_i = \frac{1}{n^2}\sum_{i=1}^n (nx_i-\sum_{i=1}^nx_i)$

(but I don't know if that's true)

Thank you for any help!

Answer 1

mean$(x-\bar{x}) = \frac{1}{n} \sum(x_i - \bar{x})$

$= \frac{1}{n}\sum x_i - \frac{1}{n} \sum \bar{x}$

$= \frac{1}{n}\sum x_i - \frac{1}{n} n\times \bar{x}$

$= \bar{x} - \bar{x}$

$=0$

I get from line 2 to line 3 by seperating the summation, noting that subtracting $\bar{x}$ $n$ times is just the same as subtracting $n*\bar{x}$