What does it mean when people say “mean squared”?

The average value of a noise waveform is zero. The square of the waveform mean is also equal to zero. The square of the noise signal and the mean of the square are non-zero. This is because the negative values associated with the zero-mean noise waveform are made positive by squaring, and the entire waveform is positive. Taking the root of the averaged square of the waveform yields the RMS.

The mean of the squared (“mean square”) noise waveform is the noise power with respect to a 1 Ohm resistor (units: V^{2}/Ω=W, “power” if noise signal is a voltage signal, and units I^{2}/Ω=W, also “power” if noise waveform is current).

The power spectral density is the power of the signal in a unit bandwidth.

What is a current noise power spectral density?

The correct definition of current noise spectral density is the mean of the squared current per hertz, <i^{2}>. The units are A^{2}/Hz.

The square of the mean is equal to zero, because the mean of the noise waveform is zero and squaring that number remains zero. The mean of the square is a non-zero number. Taking the square of a noise current results in a positive valued current waveform. Taking the average of the square is a non-zero number used for the spectral density.