Narrow band to 1/3 Octave, Sum or Average?

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[size=12.6667px]https://www.eng-tips.com/viewthread.cfm?qid=391295
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[size=12.6667px]Lately I have came across a vibration design criteria with 1/3 Octave Band. Been doing a lots of dynamic analysis on narrowband but this 1/3 octave band is rather new to me. Some suggest we assign narrow band amplitude (fixed interval) into octave bin and sum them, some suggest to average them.

[size=12.6667px]Gut feelings tells me it has something to do with Y-axis terms. So my questions are:

[size=12.6667px]When we are using velocity rms, do we sum it or average it? and why?
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[size=12.6667px]Here's my Matlab working. And it's the kind of thing I still do in Matlab, when I need to get back to basics. Note that my time vector is just the right length so that all signals are exactly periodic within the window. The n-1 is important here, don't want the start point duplicated.

[size=12.6667px]n=1024; % Power of 2

[size=12.6667px]t=(0:n-1)'/n; % Sample length is 1s, FFT will have 1Hz resolution

[size=12.6667px]x=2*sin(2*pi*23*t);
[size=12.6667px]y=5*cos(2*pi*24*t);
[size=12.6667px]z=8*sin(2*pi*25*t+pi/5);

[size=12.6667px]s=x+y+z;

[size=12.6667px]fs=abs(fft(s)); % Lose phase info
[size=12.6667px]famp=fs(1:n/2)*2/n; % Scale for amplitude
[size=12.6667px]frms=fs(1:n/2)*2/n/sqrt(2); % Scale for RMS

[size=12.6667px]famp(24:26) % amplitudes at 23,24,25 Hz
[size=12.6667px]frms(24:26) % rms at 23,24,25 Hz

[size=12.6667px]sqrt(mean(s.^2)) % RMS of whole signal in time domain
[size=12.6667px]sqrt(sum(frms.^2)) % RMS of whole signal in frequency domain


[size=12.6667px]= = = = =

[size=12.6667px]The console output:

[size=12.6667px]>> tmp

[size=12.6667px]ans =

[size=12.6667px]2.0000
[size=12.6667px]5.0000
[size=12.6667px]8.0000


[size=12.6667px]ans =

[size=12.6667px]1.4142
[size=12.6667px]3.5355
[size=12.6667px]5.6569


[size=12.6667px]ans =

[size=12.6667px]6.8191


[size=12.6667px]ans =

[size=12.6667px]6.8191