How to "denormalize" a standard deviation?

Question

Lets say I have a vector of variables which have all been standard normalized with $(\mu_n, \sigma_n)$ by doing the operation $\frac{\mathbf{y} - \mu_n}{\sigma_n}$. I then have a model which predicts the y values with $(\mu_*, \sigma_*)$. If I want to bring the $\mu_*$ value back into de-normalized terms, I could just do $\mu_* \sigma_n + \mu_n$

How can I bring the $\sigma_*$ term back into a de-normalized terms?

Answer 1

$$\frac{y-\mu_n}{\sigma_n}\leftrightarrow(\mu_*,\sigma_*)$$

$$y-\mu_n\leftrightarrow(\sigma_n\mu_*,\sigma_n\sigma_*)$$

$$y\leftrightarrow(\sigma_n\mu_*+\mu_n,\sigma_n\sigma_*)$$