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    Resume of "Averaging"
    From: Alexandre Eremenko
    Date: 2004 Nov 4, 22:35 -0500

     I afraid that some people on this list had difficulties
    following the long and convoluted discussion
    on the "Averaging". (I had difficulties following it myself).
    Because I think the subject is important,
    here is my summary of this discussion.
    1. We are talking of "position line navigation",
    Sumner, St. Hilaire methods etc., and Lunar distances.
    (Excluded is thus determinatrion of the latitude  by LAN
    observation, which requires somehwat different techniques).
    2. You can reduce the random error in your observations
    by a factor of 2 to 3, (in many cases much better)
    by taking several altitudes (or distances) of the same body
    in a short interval of time. The recommended time interval is 5
    minutes, or less. You cannot reduce any systematic error
    this way.
    The method also helps to reject blunders, thus making your
    position line more reliable.
    3. The way to do it is to take several measurements of
    the same quantity (altitude or distance) in a short period,
    recording the time of each measurement.
    By inspection of the resulting table, you can reject
    blunders. Use your common sense to decide which observation
    (if any)
    was a blunder (and thus has to be rejected). The points
    (TIME, ALT) have to lie approximately on a straight line.
    If one point (of five, say) is far away from the straight line
    approximately fitting the other points, this one point is
    likely to be a blunder, so reject it.
    4. Do NOT reduce each observation to obtain several
    position lines. The position lines will be almost parallel,
    and their intersection will tell you nothing useful.
    5. Instead, take the average (altitude or distance) and average
    time, and reduce the result
    as you would do with one single observation.
    As a result, you obtain ONE position line. It will
    be more reliable than a position line obtained from one
    single observation.
    6. There are rare circumstances when the method is not recommended.
    This is when you take altitude of a body which is close to meridian
    AND on high altitude. For example, it is not recommended for LAN
    observations. For these observations, there are other averaging
    methods that help you reduce the random error.
    7. The averaging method is recommended, no matter how you reduce your
    sights (using tables, or manual computations, or a calculator
    or computer,
    or abacus).
    8. There was a long discussion on this list, addressing various
    issues involved. One issue was about the "rules to reject blunders".
    The consensus was that there are no formal rules: just
    use your common sense. (Or don't reject anything if you are really
    confident in each of your measurements).
    Another issue was whether the procedure is "mathematically sound".
    It is.
    The potential objection was "non-linearity of altitude as a function
    of time". This was resolved by massive computation: the result
    was that this non-linearity is small enough to be neglected
    for practical purposes (except the above mentioned case
    of meridian passage of a body at high altitude).
    9. The procedure is recommended by many books,
    some other books don't mention it, but nobody objects.
    Except one person: Herbert Prinz. I do not see why.
    You can read his objections under "Averaging" discussion,
    together with my replies.
    10. My credentials: I teach statistics at Purdue University,
    on undergraduate and graduate (MS, PhD) level.
    That is to the future engineers,
    scinetists and to the future statistics teachers.
    Alex Eremenko.

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