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    Index error
    From: Bill Morris
    Date: 2008 Jun 4, 22:50 -0700

    I have been following in a desultory way the current discussion
    between George Huxtable and Frank Reed about errors in lunars. My
    interest is perhaps more in the instruments of navigation than in its
    practice. I did once do a lunar about thirty years ago, under the
    tutelage of a retired surveyor who spent his holidays as �fifth mate
    for navigation� aboard a South Pacific islands steamer. We used seven
    figure log tables, so you will understand that I was not necessarily
    keen to repeat the experience.
    
    In a previous posting about sextant calibration, I gave the re-setting
    error of my ex-USSR SNO-T sextant as about 4 seconds, its micrometer
    error as about 2 seconds and its backlash error as up to 12 seconds.
    
    On the occasions when I have directed a sextant at the sky rather than
    at an autocollimator instrument, I have noticed that my determination
    of index error is seldom the same twice running. At first I put all of
    it down to my inexperience  and shaky right arm, but I have quite a
    lot of experience with making and using optical instruments and am a
    careful observer. George�s requests for �Error budgets� led me to try
    to find out what sort of spread of results I(and possibly others)
    might expect when finding the index error of their sextants.
    
    I made three sets of thirty careful observations for each method:
    using a sharply defined land horizon about 6 km away; using the sun�s
    limbs; and using a 2nd magnitude star. To make things easier and to
    avoid fatigue, I clamped the sextant atop a theodolite sextant so I
    could make each observation a leisurely one. I also glued a simple but
    effective paper vernier over the micrometer index to reduce to some
    extent a tendency to bias the results in a more or less favourable
    direction when estimating tenths. Here are my results for the standard
    deviations:
    
            Method                    S.D.      95% range
    Horizon, reflected image up to direct image     0.142        0.56
    Horizon, direct image up to reflected image     0.157        0.62
    Sun�s limbs                         0.155        0.61
    Star                                0.174        0.68
    
    For those who might think that statistics is a new form of contact
    adhesive, I should point out that the standard deviation is a measure
    of the dispersion of the results about the mean value; and 1.96
    standard deviations each way will �capture� 19 out of 20 or 95% of
    results. 1 S.D. each way will include about 64% of results. So, my
    sextant-eye-brain system will give an index error of more than 0.3
    minutes away from the best estimate, the mean, one time in twenty.
    About 45% of the time it will be more than about a minute and a half
    away from the mean.
    
    Physicists are supposed to be good at error estimation and my
    education was in the biological sciences, so I leave the rest to
    George and others...
    
    Bill Morris
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