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    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)
    >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
    >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
    >Geoffrey Kolbe.
    Nice post. I would have welcomed a nod towards the Central Tendency of any
    Brian Whatcott
       Altus OK                      Eureka!

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