I have some data and I added Gaussian noise with zero mean ($\mu=0$) and standard deviation ($SD=SD$). I was interested to see the behaviour of the noise to the data after integrations. I integrated once: after integration the mean still zero $\mu=0$ but the standard deviation it's getting bigger over time i.e. $SD$, $\sqrt{2}SD$,$\sqrt{3}SD$, $\sqrt{4}SD$ etc. When I integrated twice the mean doesn't change over time so $\mu=0$ and standard deviation becomes: $SD$,$\sqrt{3}SD$,$\sqrt{6}SD$,$\sqrt{10}SD$ etc. When I integrated my data twice I noticed a systematic error in my results.
When I integrate once the change in standard deviation is uniformly $(+SD^2)$. In the double integration the difference is not uniform.
My question:
It's possible the systematic effect in double integration caused because the difference in the standard deviation is not uniform?
How's the standard deviation affects the median?