Is noise density over a bandwidth a measure of the noise's standard deviation?
The RMS signal level for Gaussian white noise is measured in units per square root of bandwidth. And RMS has the same formula as when calculating the statistical standard deviation of a series of points. So, for instance;

For each point, calculate the distance from the mean value (call it 0 volts), square the distance. Repeat for all other points and sum all the squared distances. Then divide by the number of samples (take the mean of the squares) then, take the square root. Sound familiar?