Relationship between median and standard deviation

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:

1. It's possible the systematic effect in double integration caused because the difference in the standard deviation is not uniform?

2. How's the standard deviation affects the median?

-
you wrote "$SD = 0$" is that a typo? –  Brad S. Mar 19 at 21:22
Yes, sorry about that. I corrected it. –  user105627 Mar 20 at 9:16
What integral are you doing? Can you lay out the calculation? Its a little too abstract as written. –  Eupraxis1981 Mar 20 at 13:11
I am doing trapezoidal rule. I am integrating twice the Gaussian noise with $\mu=0, SD=SD$. I noticed that after the double integration the mean is not closed to zero. The mean and standard deviations are two different things. Why the mean affected by the standard deviation? Or if this is not the case. What causes the systematic error in the results? –  user105627 Mar 21 at 11:48