Say I open an audio segment with Audacity. Then I perform a size 4096+ FFT with a rectangular window. The program outputs bin magnitude values in dB. If I add them all together like this:
sum = 10*log10(10(bin1/10) + 10(bin2/10) + 10(bin3/10)...).
I get a value very close to the file's calculated "total RMS amplitude". (Or in this case, 3dB off since Audacity considers a full-scale square, instead of sine, to be 0dBFS).
Is this expected, a coincidence, or just an approximation?