# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Still on LOP's**

**From:**Brian Whatcott

**Date:**2002 Apr 23, 06:44 -0500

At 01:06 PM 4/22/02, Geoffrey Kolbe wrote: >... >If you now fire three or four more shots, you will have a distribution of >holes in the target - what we call a group. You can now crank the >statistical handle to find where the centre of the group is. The more shots >you fire, the greater the level of certainty of your calculated centre of >group. This level of certainly will increase (and your confidence circles >decrease) as the square root of the number of shots in the group. > >Similarly, if you only have one 'hat, then placing the MPP in the centre of >it is the best you can do. But if you perform multiple observations >resulting in a number of 'hats, then the MPP can be placed at the centre of >the distribution of 'hats. You can now also start doing statistics on the >distance of the 'hats from the MPP and so establish circles (or ellipses) >confidence. > >When thinking of MPP's and circles of confidence, I think the trick is to >consider the centre of the 'hat as one datum point sitting somewhere on a >probability distribution around the MPP. Until you do some more >observations and establish some more 'hats, you will not know where the >centre of that first 'hat is on the probability distribution of 'hats. When >thinking of the relationship of 'hats to MPP's in this way, you can see >that the actual size of the 'hat does not matter, only the position of its >centre. > >However, if you do perform multiple observations on each of the three >landmarks, then producing a distribution of 'hats is not an efficient use >of the data when determining the levels of confidence of your MPP. It is >much more efficient to find the mean bearing and standard deviation from >the mean for the measurements on each landmark. Plot the mean bearings and >there will be a 'hat at the intersection. The centre of _this_ 'hat should >now be a good MPP and from the standard deviations around your three mean >bearings, you can establish ellipses of confidence. > >Once again though, this 'hat formed from the mean of multiple observations >on each of three landmarks only has once chance in four of enclosing the >actual position...! > >You will see why when you also plot bearings one standard deviation away >from the mean bearing for each landmark. This will enable you to draw an >ellipse which is in effect a 50% confidence ellipse The actual position has >a 50% chance of being inside this ellipse Your 'hat should lay comfortably >inside this ellipse with perhaps just the vertices outside it. If it >evidently bigger than half the 50% confidence ellipse then there is some >systematic error which is opening up your 'hat. Your compass needs >calibrating or, when considering LOP's, you have an index error on your >sextant. > >Geoffrey Kolbe. Nice post. I would have welcomed a nod towards the Central Tendency of any distribution.... Brian Whatcott Altus OK Eureka!