How do I get from some standard-deviation/variance of $x$ and $y$ coordinates to Standard Distance Deviation/RMSD?
Normally I'd calculate Standard Distance Deviation via $$ \hat\sigma=\text{RMSD} = \sqrt{\frac{1}{n}\sum_i\left((x_i - \bar{x})^2 + (y_i - \bar{y})^2\right)} $$
But in my dataset, the original points are lost, I'm left with the variances. From the top of my head, I was thinking this:
$$ \hat\sigma = \sqrt{\sigma_x^2 + \sigma_y^2} $$
.. but I'm not sure if this is valid and fail to come up with a derivation for it. Does anyone have an idea? Cheers!
$$ \frac{1}{n}\sum_i\left((x_i - \bar{x})^2 + (y_i - \bar{y})^2\right) = \left(\frac{1}{n}\sum_i\left((x_i - \bar{x})^2\right)\right) + \left(\frac{1}{n}\sum_i\left((y_i - \bar{y})^2\right)\right) $$