# Standard deviations and confidence intervals (weighted) running average

My question is related to this one. I am calculating averages, actually as many as I have samples because I calculate a running average, and for equal weighting I know how to calculate the $95\%$ CI, with $\mu\pm1.96*\sigma_{original}$. However there are two things that I dont know how to do. Can I get the $\sigma_{original}$ from the same method (convolution through Fourier transform)? How do I calculate the $95\%$ CI if the weighting is not equal (so if I convolute with a window that is not flat)?

The first term inside the brackets ($\sum x_i^2$) you get with the convolution $f^{2}\ast g$, the second part ($[\sum x_i]^2$) by the convolution $(f \ast g)^{2}$. Where $f$ is the signal and $g$ is the window. The squares are pointwise. Then apply the rest of the formula and youre golden. Note that $f \ast g$ is your running average which you already computed.
This straightforward implementation is only valid for periodic signals and I think it is also only valid for non-weighted averages / flat windows. For non periodic signals you need padding and for weighted averages you need to change the formula for $s$. How to change this I still dont know. If I ever figure it out Ill add it here.