Show that the centred variable has zero mean.
\[\bar{x}^{(C)} = \frac{1}{n} \sum_i^n x^{(C)}_i = \frac{1}{n} \sum_i^n \big( x_i - \frac{\sum_j^n x_j}{n} \big) = \frac{1}{n} \sum_i^n x_i - \frac{1}{n} \sum_i^n \frac{\sum_j^n x_j}{n} = \frac{1}{n} \sum_i^n x_i - \frac{1}{n} \frac{n \sum_j^n x_j}{n} = 0 \]
Show that normalized variable has unit (~ equal to one) variance.
\[\sigma_{x^{(N)}}^2 = \frac{1}{n} \sum_i^n (x^{(N)}_i - \bar{x}^{(N)}_i)^2 = \frac{1}{n} \sum_i^n (x^{(N)}_i - 0)^2 = \frac{1}{n} \sum_i^n (x^{(N)}_i)^2 = \frac{1}{n} \sum_i^n \big(\frac{x - \bar{x}}{\sigma_x} \big)^2 = \frac{1}{\sigma_x^2 \, n} \sum_i^n (x - \bar{x})^2 = \frac{1}{\sigma_x^2} \sigma_x^2 = 1 \]