We have numerous examples that show that our custom long-term sun almanac can usually provide declination to within about ±1' and GHA to within ±2', and then we have approximations to the sun's semi-diameter (we call it 16' always) and we have abbreviated Refraction and Dip tables, so on the Latitude reckoning we have about a ± 2' uncertainty, with more like ± 3' on the Longitude reckonings.
We have a sextant (the Mark 3) that can only be read to ± 2’ and we have to effectively do this twice counting the index error measurement. Independent uncertainties add as the square root (sqrt) of the sum of the squares, so this sextant uncertainty is about sqrt (4 + 4) = 2.8'.
So for a Latitude sight we have a total uncertainty of about sqrt(2.8^2 + 2^2) or about 3.4'. We thus estimate that we can find latitude as:
Latitude ± (3 to 5) nmi
This is to be compared to best possible with a good metal sextant and full almanac and a 1000-sight experience of ± 0.5’ for either a Lat or an actual fix of both Latitude and Longitude.
For Lon we can assume the watch is right, so sextant error is not really a factor, and we are left with finding peak time of the LAN curve of sextant heights versus UTC, which we combine with our GHA uncertainty of ± 3'.
We can typically find the center of that curve to within about 1 minute if we have good sight data. Sometimes half that, but for safety let's call this a 1 minute uncertainty, which in turn leads to a Lon uncertainty of ± 15'. We combine this with the GHA uncertainty of ± 3' for a final value of sqrt(15^2 + 3^2) = 15.3. Thus for longitude sights at LAN we have:
Longitude ± (10 to 20) nmi.
We expand the 15' a bit as there can be other uncertainties if we are moving very much in the N-S direction during the time we are measuring the curve of Hs vs UTC. This distorts the Hs curve, and the center does not reflect the actual location of the sun crossing our meridian. For typical yachts this is usually a small factor.
This backup method has a large Lon uncertainty, but we must remember that this does not mean the Lon is wrong by that much. It most likely is not, but it is just uncertain by that much and we have to keep that in mind during decision making. Indeed if we do several LAN fixes over a few days and our DR is agreeing with these fixes, then we have learned that we are likely doing better than this large uncertainty.