calculating the RMS noise of an accelerometer
You've missed a point. You do not, in fact, know the bandwidth. You've confused the Nyquist Limit with a bunch of other things. 

The "sampling rate is twice the bandwidth" concept is actually turning things inside out. The Nyquist sampling theorem says that, for a given signal bandwidth, you must sample at a rate at least twice that in order to accurately reconstruct the signal from the digitized data.

So, given a fixed sampling rate, you must put an analog filter before the data with a bandwidth about half the sampling rate. For accurate results, the filter should be better than a simple RC filter. The more poles, the better (and the harder to make).

Once you've done that, you can take data and do your calculations.

And as for calculating the standard deviation, have you even tried to look it up on Wikipedia? http://en.wikipedia.org/wiki/Standard_deviation