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    Re: Averaging
    From: Alexandre Eremenko
    Date: 2004 Oct 19, 00:01 -0500

    I think the question whether the averaging of
    observations (before reduction) is justified,
    or under what conditions is it justified is important
    and worth further discussion.
    
    In his last message on the topic, Herbert Prinz challenged
    this practice referring to non-linearity of altitude change.
    I agree that under some circumstances this non-linearity
    may introduce substantial error,
    however I feel that these circumstances should be rare.
    
    I am sure that Herbert's statement that:
    "manual averaging is a thing of the past"
    is an exaggerration.
    
    Many modern manuals do recommend such averaging, and I am sure
    that people who tried it know that it usually increases
    presision, without any substantial increase in volume
    of computations (which will happen if we reduce each sight
    separately).
    
    On other aspects of averaging recently discussed.
    
    1. There is NO mathematical reasons for averaging even
    or odd number of observations. (This I can say as a mathematician
    who teaches statistics classes:-)  If there are any other
    (non-mathematical) reasons, I don't know, and it is hard
    to imagine what these other reasons could be.
    
    2. Median (middle value) is indeed frequently used in statistics
    (like median income, median housing price, or median score
    on an exam). But this has nothing to do with the problem
    we discuss: of reducing random errors in observations.
    The situations where using median is recommended are DIFFERENT.
    Median can be computed independently of whether the volume
    of the sample is even or odd. In the even case they take
    the average of the two middle values.
    But I repeat, median is not the best thing to do to reduce
    random errors in a series of measurements.
    I can give specific examples which explain why sometimes median
    is used but this will be definitely out of scope of this list.
    
    To reduce random independent errors in a measurement of
    a quantity that changes linearly (or does not change at all),
    average is the proper thing to compute.
    
    Alex.
    
    
    

       
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