"Averaging" gets used in many contexts, I was using it to refer to the RTA's Averaging setting, not to whether one should measure in more than one position. To understand why averaging is essential with random pink noise it is worth reflecting on what the frequency content of random pink noise looks like on an RTA. You might be tempted to say that's trivial, it's a flat line, right? Here is an example of the frequency content of about 1.5 seconds of pink noise (one 64k FFT's worth at 44.1 kHz sampling), first the span out to 20 kHz then a closer look at the span to 200 Hz:

Not terribly flat, is it? Any short section of random pink noise has a similar amount of variation. The overall frequency content of random pink noise converges to flat as the time period increases, but to have something that is flat to within a couple of dB typically means averaging about 30s or so:

The idea of playing pink noise and viewing the result on the RTA is that any deviation from flat is due to the overall system's response, but that is only true of you wait long enough for the input to the system (the pink noise signal) to average to something close to flat. That means to get a reasonably accurate (to within a few dB) view of the response at a single position you would need to leave the mic stationary there for about 30s, with the RTA averaging all the while. Random pink noise is pretty useless for measuring responses.

To get around the deficiencies of random pink noise we can use periodic pink noise. The pink PN signal is mathematically constructed so that any segment of the length of the RTA's FFT contains every frequency the FFT can resolve at exactly the right amplitude to have a perfectly pink profile, so it is exactly flat on the RTA. To get an accurate view of the response at a single position with pink PN the mic only needs to be there for one FFT duration, or about 1.5s for the 64k FFT at 44.1 kHz used in the sample plots above. There is no need to do any averaging as every FFT gives exactly the same result (background noise contributions aside). However, if you move the mic away before that time has elapsed you won't see the contributions of all frequencies for that location, and it's impossible to say which frequencies will have been missed. If you move the mic around for your measurement then move it slowly and go back to the same places more than once if you want a reasonable approximation to the effect of averaging measurements made at each position.