Mean of centered data

Start from the definition of mean

\[\bar{x}^{(C)} = \frac{1}{n} \sum_i^n x^{(C)}_i\]

then substitue for \(x^{(C)}_i\) and simplify.

Variance of normalized data

Start from the definition of variance

\[\sigma_{x^{(N)}}^2 = \frac{1}{n} \sum_i^n (x^{(N)}_i - \bar{x}^{(N)}_i)^2\] then substitue for \(x^{(N)}_i\) and simplify. (Also remember what the mean of centered/normalized data is.)