David,
You are correct that the variance from the addition of two random numbers is the sum of their variances. However, when they also are multiplied by a constant, such as in calculating a semi-diameter, they must be multiplied by the square of that constant. Here you would multiply the sum of the variances of sun limb contacts on and off the arc by the square of one half). That reduces the variance of the index error.
Attached as a photograph are summaries of some data from when I was on top of my game. Variation is expressed as sample standard deviation (calculated using n-1 degrees of freedom). The last column is the standard deviation of the semi diameter calculated as above in the first paragraph, and the next to last column is the standard deviation of the semi-diameter itself, without accounting for fact it was obtained from two measurements. The standard deviation is about 0.07’ of arc by either method. They appear to be quite similar.
I do recall learning that the sum of two random variables is itself a random variable; why its variance would need to rest on the variance of its components is unclear to me.
I may have read that the upper/lower limb measurement was more precise in the Sextant Handbook; I’m not sure where I read it. My copy is hiding from me at the moment. It also was my impression after shooting almost 600 semi-diameters that the touching method was much more accurate. If you wish, I will summarize more of these data. Another argument agains superposition of the sun is that it’s probably not a good idea to be adjusting side error all the time.
Measuring sun semi diameter also can tell you whether you and your sextant are having a “bad hair day,” as they say. If the observed SD differs even 0.1’ of arc from that reported in the Nautical Almanac, it’s a bad hair day! They happen.
I also am far from being a statistician, but do apply statistics in my work.
Fred
Bill B.
Thanks so much for your thoughtful and stimulating analysis of the measurements I presented.
I was bothered by the difference in my measured semidiameter and the published SD and you are right it is due to my failure to properly account for the off arc readings. In theory I knew that, but I forgot to put it into practice.
Thanks also for the link to the David Burch document. I will have to study it in detail and perhaps get ideas for further measurements. I am interested verifying experimentally the common knowledge surrounding sextant useage.
You indicated your Astra had insensible backlash, perhaps I should verify same for mine.
I'm not a professional statistician either but I am aware that in computing a difference of two independent numbers each of which is uncertain (SD1 and SD2 ). The computed number has an uncertainty SDc = Square Root( SD1^2 + SD2^2) = Square Root(2) x SD for SD1=SD2=SD. This is the essential point I wanted to make concerning the overlap vs edge to edge IC measurement. Variation is same for of each measurement but overlap method gives more precission in the result for same effort, number of measurements.
