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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!

   

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