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Bill Morris you wrote:  I'm not a mathematician and have an uncertain grasp of statistics having once thought them to be a form of contact adhesive, but can say with the conviction of the possibly ill-informed that taking the median ensures that less weight is given to outliers.

I have to admit I threw this in to give the mathematicians something to chew on.  It seems to me that there are a number of different problems here, which all overlap slightly.  The first is observer decides when to make an observation versus sextant decides.  If the observer decides, you can have a mean value (the RAF MkIX), or a median value (RAF Mk8, early A10, A12). Both are subject to the semi circular canals playing tricks, because a steady, and I mean really steady, acceleration produces a steady bubble, albeit hopelessly in error.  Such outliers are less likely to affect the final result if a median value is available.  Automatic averaging probably gives an overall fairer result, because there’s less subjectivity, but you are restricted to a mean value, so any outliers are automatically fed into the final result.

You also wrtote: And, mathematicians, how does continuously integrating as in the SOLD and Hughes Marine Bubble sextant, compare with averaging sixty observations two seconds apart?

One aspect of this which interests me is why the RAF reverted to ‘clockwork averaging’.  The first Smiths/Kelvin/Hughes periscopic sextant, called the Mk1A in the RAF, used a cone and roller integrator, but the Mk2s reverted to using the same wind-up clock mechanism as the hand held MkIXAs and Bs, except it was on the back rather than the front of the sextant.  Did Smiths have a factory full of clock parts left over from WW2, which they wished to use up?  Were the old hands, now promoted  into the Operational Requirements Branch, still in love with their MKIXBMs, or was the cone and roller found to be less accurate, or more prone to unserviceability?  It would be interesting to know.  DaveP