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

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Re: Cocked hats, again.
From: Dave Walden
Date: 2007 Mar 16, 06:07 -0700

```An attemp to summarize and establish consensus.

Assertions, for equally spaced observations with a common bias and
normally distributed uncertainties with the same standard deviation in
altitude:

1. The most likely location is the center of the cocked hat.

2. The probability of being within the cocked hat is 25%.

3. The magnitude of the common bias has no effect on the location of
the most likely location.

4. Small cocked hats exist that do not contain the true location.

5. Large cocked hats exist that do not contain the true location.

Notes:

1. Doesn't say it's there, doesn't say how likely it is to be there
(although given the right data, it can be calculated), just says there
is no more likely location.  (The size of any single cocked hat, a
random result, does not provide enough information to establish the
standard deviation of the uncertainty in altitude.)

2.  I believe this has been conclusively demonstrated in this group
probabilistically, statistically, heuristically, by experience and by
Monte Carlo.  (The somewhat confusing case of the eight is the one
where the point lies on the outside of all three lines.  It turns out,
after some thought, that the most likely location is the center of the
cocked hat.)

3. The great advantage of an equally spaced round of sights!  A
changing common bias opens or closes the hat, but doesn't move its
center.

4. and 5.  Some examples from a Monte Carlo simulation were given in
an earlier post.  As was pointed out in a recent post, it is easier
for many to conceptualize if you consider all possible cocked hats,
rather than try to generalize from one or a few special cases.  Or,
trying to say it another way, we may be better off to start by not
thinking  about where the true location is for a given cocked hat, but
where the cocked hats are for a given true location.  Then, after some
thought, we can say some useful things about the first case.

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