Consider a "cloud" of points in the complex plane, where the X axis corresponds to the Real part and the Y axis corresponds to the Imaginary part.
The "complex mean" is the same operation as the "real mean", namely you add up all the numbers and divide by N, the number-of-numbers. Since adding complex numbers means "add the real parts and add the imaginary parts", it (operationally) is the same as adding the X-Y vectors. Similarly, dividing by N, a real, just means dividing the summed real and imaginary parts. What you end up with is the centroid of the cloud of points, the "point in the middle of the cloud".
Now consider two measurements, one with a S/N of 100 that gives you a phase of 0°, and one with a S/N of 0.1 that gives a phase of 90°. What phase would you like to report as your "best estimate" of the true phase? You could average the phases and get 45°, but I'm sure you'd agree that you'd want to put more "weight" on the measurement with higher S/N. One way of weighting the estimates would be to assume a fixed noise level, call it 1, and say "the first measurement corresponds to a signal, or a gain, of 100, the second to a gain of 0.1". Now you have two vectors, in X-Y terms, of (100, 0) and (0.1, 1). Their average is (50.5, 0.5) which corresponds to a phase of 0.6°, which is consistent with "trusting" the phase of 0° returned by the high S/N measurement more, and weighting it more heavily.
Does that make sense?
Bob Schor
