Noise Spectral Density Standard
I have never heard of an IEEE standard regarding noise. I also think it isn't needed as a noise specification should be clear by itself.

A figure like \$12.4 nV/\sqrt{Hz}\$ doesn't mean anything on its own.

An example of a proper specification would be:

\$V_{noise} = 12.4 nV/\sqrt{Hz}\$  at \$1 kHz\$  (spot noise)

or 

\$V_{noise} = 12.4 nV/\sqrt{Hz}\$  between \$1 kHz\$ and \$1 MHz\$ (flat noise)

or

the 1/f noise is:

\$V_{noise} = 12.4 nV/\sqrt{Hz}\$ at \$1 kHz\$

(spot noise, which is 1/f so halves at 2 kHz and doubles at 500 Hz)

Often the 1/f noise will "flatten out" at some higher frequency into a "flat" (thermal) noise. For example:



Above is all "spot noise" so the noise at a particular frequency and in a 1 Hz bandwidth.
If you want integrated noise then you have to integrate the spot noise value over the frequency, this is easiest when the noise is flat over frequency as we can then just multiply it with the square root of the bandwidth.

For example let's take the flat noise from above:

\$V_{noise} = 12.4 nV/\sqrt{Hz}\$  between \$1 kHz\$ and \$1 MHz\$ (flat noise)

and let's say we want to integrate that over 100 kHz to 1 MHz = 900 kHz that will result in:

\$V_{noise} = 12.4 nV/\sqrt{Hz} * \sqrt{900 kHz} = 11.7 \mu V\$

Note the unit, it is Volts now!

More info in this TI Application Report.