NavList:

Aside from all those math exercises, least square fits & etc. We have the simple practicality of two cases.

One:  you are in the middle of the ocean.  There isn't a reef, sea mount or danger point (other than container ships) for hundreds of miles.  Which point you pick relative to the cocked hat is fundamentally irrelevant.  Inside, outside, closest to the short side, symmedian point, danger point; its all noise.  Plop your point on the chart and get on with other tasks.  You are about "here" and you need to get to about "there".  Fussing over mathematical purity is a waste of time.  It buys you nothing.

Two: there is a nearby reef.  You do not want to run your vessel up onto that reef, because that means you will have a really bad day.  After obtaining your cocked hat, you pick the point in the cocked hat closest to the reef so as to chose a course well away from said reef.  Logical: Avoid reef.  Practical: save boat, stay alive.  You steer for safety. 

Would anyone really steer very close to that reef because someone said the best point was by the short side?  Because someone said the symmedian point was the most probable?  That's foolhardy.  

It appears from this seat in the peanut gallery that the only reason this topic comes up is to somehow improve CN, such that the position obtained matches up to GPS.  All of this conversation about probability curves, symmedian points, ellipses, & etc;  looses directly when it comes to the simple, logical, practical advice offered by Admiralty, when advising real mariners faced with real life navigation.  

Follow Admiralty advice.  Stay alive.  Save the boat.  Of course, that advice won't matter if you never leave port and only engage in philosophical transactions.  

Brad