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    Re: Lunar distance accuracy
    From: Frank Reed
    Date: 2007 Nov 04, 21:31 -0500

    I wrote earlier:
    "So you're saying these observations exhibit excess kurtosis "
    
    And you replied:
    "Let us stick to simple language:-)"
    
    Well, you said that the distribution of the observations has more weight in
    the tails than a common normal distribution. Apart from the very colloquial
    "fat tailed distribution", the only name for this is kurtosis. I certainly
    agree with you that it is something that you should worry about when you're
    dealing with distributions of real-world data. Almost all data which is said
    to be approximately normally distributed does, in fact, show some kurtosis
    if you study it long enough. This is responsible for all sorts of crazy
    misunderstandings in divers fields. For example, if you were to study price
    changes in currency exchange rates, you could easily calculate a standard
    deviation, and based on that you might predict that the odds of a three
    standard deviation move are something like 300-to-1 against. But if you then
    look at the data again and COUNT the number of moves of that size
    historically, you discover that the odds of a large move are considerably
    larger. In other words, outliers are much more common than a normal
    distribution would suggest. A currency trader who ignores this fact can get
    into some mighty serious trouble (have a look at a brief article called
    "Kurtosis Risk" on Wikipedia for another take on this). For the navigator,
    the same issue can arise. If results tend to cluster near a mean, you still
    should worry that there may be some aspect of your observations, or your
    sextant's behavior, or the nature of the observation itself, that leads to a
    greater than expected number of outlying points. BUT, and this is a BIG BUT,
    it is fairly easy to imagine that there are more outliers than normal when
    you worry about them too much. The only RELIABLE measure is a direct
    calculation of the sample kurtosis, and fortunately, most modern
    spreadsheets can do this easily. If it's not significantly above zero
    ('significantly' has to be defined), then you cannot claim that the
    distribution has "fat tails". White's observations do NOT show significant
    kurtosis.
    
    I wrote:
    "Enter those values into an Excel spreadsheet."
    
    And you replied:
    "I did! And wrote about this to the list 4 (four) times. Are my messages get
    lost?? You can see the results on
    www.math.purdue.edu/~eremenko/accuracy.html"
    
    No, Alex, your messages are not getting "lost". Let's not go through that
    again. You can easily check to see if your messages are distributing by
    looking for them in the list archive at http://www.fer3.com/arc. And yes, I
    did look at your bar graphs. Thanks for making them.
    
    Also, please note that I spelled out detailed instructions because in any
    conversation on this list, even if it seems that it is a conversation
    between two people, there are often other people reading along, other people
    who may want to participate, so in general, I try to give instructions "as
    if" the other person is not 100% familiar with what we're discussing. So,
    YES, I know you had already entered the data in a spreadsheet. But I also
    note that your web page does not include any values for standard deviation
    or kurtosis
    
    And then you asked:
    "Look at these error distributions. If you do not want to look at other
    results out of principle, look at White's results plotted. And then tell us:
    on your opinion, does this plot confirm that "one can measure a distance
    with 0'2 accuracy, reliably"."
    
    White's observations show a standard deviation on individual observations
    (which may be multiple actual sights or may be single sights) of 0.25
    minutes of arc. And if ew average sets of four, the s.d. drops to about 0.1
    minutes of arc. That means that about 68% will lie within one standard
    deviation of zero error. Whether that fits your definition of the word
    "reliably" is up to you. White's observations also show no significant
    kurtosis. There are no "extra" outliers, no evidece of a non-normal sample.
    
    Of course, at some point there almost have to be highly non-normal outliers
    in sextant observations. So-called "blunders", like writing down 20 degrees
    for 30 degrees, count as errors but the odds against them would be
    millions-to-one (or worse!) if we believed all recorded sextant observations
    obeyed a true normal distribution. So as I said at the top, I agree with you
    that we should be on the lookout for outliers. This applies to all celestial
    navigation, whether it's Noon Sun latitudes or intercepts for LOPs or time
    sights. There's nothing special about lunars on this score.
    
    
    In another post, replying to Henry, you wrote:
    "This I consider a violation of the rules:-) If one chooses this way, then
    why not to use
    a 20x scope, and a theodolite, or some other instrument more accurate than a
    sextant, after all?"
    
    Well, because some people shoot lunars for other reasons, and I think this
    is worth remembering. For a long time, I thought that you personally, Alex,
    were shooting lunars because you wanted to determine the arc error of your
    instrument. If that's the case --and it is for some people who play this
    "game"-- then mounting the instrument on a stand (maybe an old telescope
    mount with a clock drive) and attaching a more powerful telescope to the
    sextant might very well count as "playing by the rules".
    
    And you wrote:
    "My ultimate interest is in the measurement at sea. Unfortunately I have no
    possibility to sail very frequently."
    
    Yeah, I think we should find a corporate sponsor! Buy us a couple of berths
    aboard a nice tall ship for five or six of us, first-class cabins would be
    nice, and sail us around the world for a year so we can get some good hard
    data on lunars... Aahhh...
    
     -FER
    
    
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