Considering underlying process (of some signal with number of samples $N$ and standard deviation $\sigma$) in DSP guide book, page 18, the definition of its typical error is following:
$$Typ.err = \frac{\sigma}{N^{1/2}}$$
my question is why so?
I didn't find in the book the explanation of this equation.
This section describes estimation of the mean of an underlying process. Consider i.i.d. samples $X_1,\ldots, X_n$ with mean $\mu$ and standard deviation $\sigma$. Then, $$ \bar{X}_n=\frac{1}{n}\sum_{i=1}^n X_i $$ is an estimator of $\mu$. Note that $$ \mathsf{E}\bar{X}_n=\frac{1}{n}\sum_{i=1}^n \mathsf{E}X_i=\mu, $$ and $$ \operatorname{Var}(\bar{X}_n)=\frac{1}{n^2}\sum_{i=1}^n\operatorname{Var}(X_i)=\frac{\sigma^2}{n}. $$ The "typical error" is just $\sqrt{\operatorname{Var}(\bar{X}_n)}=\sigma/\sqrt{n}$.