In the question Estimating Filtered Noise Variance i've found that the variance of filtered noise can be found by
$$\min_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2\sigma_x^2\le \sigma_y^2\le\max_{\theta\in [\theta_1,\theta_2]}\left|H(e^{j\theta})\right|^2 \sigma_x^2\tag{1}$$
The derivation is based on spectral density functions and the relationship $$S_y(e^{j\theta})=S_x(e^{j\theta})\left|H(e^{j\theta})\right|^2\tag{2}$$ with $$\sigma_x^2=\frac{1}{2\pi}\int_{-\pi}^{\pi}S_x(e^{j\theta})d\theta= \frac{1}{\pi}\int_{\theta_1}^{\theta_2}S_x(e^{j\theta})d\theta\tag{3}$$
The important point for me is that the variances $\sigma_x^2$ and $\sigma_y^2$ are related by a quadratic relationship.
I'm interested in the relation of the standard deviations $\sigma_y$ and $\sigma_x$, not the variances. So the question is: How is the standard deviation $\sigma_y$ related to $\sigma_x$?
My naive idea (I'm yet not too experience with random variables in the frequency domain) is to take the square root and find $\sigma_y= |H(\omega)|\sigma_x$. So the variances should have a linear relationship.
Is this correct or am I missing some point?