Standard deviation of the spectrum of white noise
true white noise has infinite power (because the power spectrum with height $ \frac{\eta}{2}$  has infinite area) which means infinite variance $\sigma^2$ which means infinite standard deviation.  white noise is a conceptual instrument.

to do any analysis of a system with white noise in it, you must somehow determine the (single-sided) bandwidth of the system, $B$, and then the power or variance is $\eta B$.