This is essentially what Sasha said:
We interpret $\bar x$ as ${1\over n}(x_1+\ldots + x_n)$. Then the Jacobian $J:=\bigl[{\partial y_i\over\partial x_k}\bigr]_{i,k}$ looks as follows: $\left[\matrix{{1\over n}&{1\over n}&\ldots&{1\over n}\cr -{1\over n}&1-{1\over n}&\ldots&-{1\over n}\cr \vdots\cr -{1\over n}&-{1\over n}&\ldots&1-{1\over n}\cr}\right]\ .$ If you add the first row to all the other rows, which does not change the determinant, you are left with the matrix $\left[\matrix{{1\over n}&{1\over n}&\ldots&{1\over n}\cr 0&1&\ldots&0\cr \vdots\cr 0&0&\ldots&1\cr}\right]\ ,$ whose determinant is obviously ${1\over n}$.