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    Re: Lunars using Bennett
    From: Peter Fogg
    Date: 2008 Apr 5, 04:31 +1100

    Alexandre you say:
    
    >  "Error" is a standard term in statistics. For example,
    >  "rounding error".
    
    With the greatest of respect, the issue is whether using rounded
    values added together does, to a significant extent, cause error.  I
    say that it doesn't; that these imprecisions tend to cancel each other
    out and have, using randomly generated values that mimic the use of
    Bennett tables, shown this.
    
    >  They will do both things: cancel but still
    >  accumulate.
    
    Come on Alex, its one or the other, surely.
    
    >  More precisely, suppose
    >  that one application of some table introduces
    >  a random error x. Then n applications of this table
    >  can be expected to introduce an error of x times sqrt(n).
    
    If there are 6 roundings then:
    "one in six to the power of six, or 0.00002, or [the chances of the
    potentially maximum error being introduced amount to, on average]
    0.002%.
    
    So which is it - accumulated or, in 99.998% of cases, cancelled out to
    some extent?
    
    Alex, I'm a little disappointed that you seem to have so blithely
    rejected my argument without, apparently, having looked at the
    evidence.  In order to save you a click I have pasted it below, with
    apologies to everyone else for the wasted kilobytes. If there is a
    problem with my statistical analysis, then please let me know.
    
    Quoted passage:
    With minutes of arc of, for example, anywhere between 23.0 and 23.5,
    23 is adopted. Between 23.5 and 24.0, 24 is adopted. If a value
    rounded up is followed by a value rounded down, a cancelling out
    effect occurs. On the other hand, if a rounded up value is
    consistently followed by rounded up values (or down by down) then the
    rounded amounts can add up to error. If 15.4 is followed by 57.3 then
    followed by 19.4, all rounded down, then an error of a little over one
    minute of arc has been introduced. If 15.4 is followed by 57.8 then
    followed by 19.9, then the rounded amounts have tended to cancel each
    other out.
    
    The objection raised was that the errors introduced by this rounding
    process are potentially accumulative when a series of rounded values
    are used; as during the sight reduction process.
    
    I propose to use an example to show why this is not so, why the
    rounded values tend to cancel each other out and do not tend to
    accumulate. Let's call the values rounded up 'boys' and the values
    rounded down 'girls' � for want of better names �
    
    Couples setting out to make a family have a 50/50 chance of having a
    boy or girl (actually this is not strictly correct, for biological
    reasons which need not concern us here).  Thus a probability of 0.5 of
    either, expressed as a decimal. The chance of having a girl as a
    second child, GIVEN THAT the first was a girl, is 0.5 x 0.5 = 0.25.
    Thus the chance of NOT having two girls in a row is 75%. By extension,
    the chance of not having three girls in a row (or three boys) is
    87.5%.
    
    There are 5 or 6 values (6 for stars, 5 for other bodies) entered
    during the sight reduction process used in George Bennett's book. So
    the chance of all of them conspiring towards error (all boys or all
    girls) is about 1.6%. The chance of this NOT happening, of the amounts
    rounded tending to cancel each other out, is about 98.4%.
    
    To test this idea, I used Excel to randomly generate 100 series of 6
    values between zero and 60, the same random numbers repeated in the
    adjoining column. One was expressed to one decimal place; the other
    was rounded to the nearest whole number by Excel. Then the software
    added the values in each column.
    
    Here is a sample:
    
    55.2     55
    43.1     43
    22.9     23
    19.9     20
    34.3     34
    3.8         4
    
    179.2  179
    
    And here are the summed amounts:
    
     1)  106.4, 106   254.8, 254   164.5, 165   252.2, 252   222.5, 222   265.5, 266
     2)  184.6, 186   228.2, 228   130.9, 132   191.4, 192   154.4, 155   195.5, 197
     3)   26.4,   26   198.0, 198   168.7, 169   174.7, 174   187.1, 188
     222.0, 223
     4)  198.0, 198   207.3, 208   166.0, 167   214.5, 215    242.6, 241  184.1, 184
     5)  206.8, 207   164.8,165    151.3, 153   239.9, 240    237.9, 238
    181.3, 180
     6)  152.6, 153   207.0, 207   178.6, 179   155.9, 155   228.2, 229
    131.0, 131
     7)  191.8, 192   128.8, 129   162.7, 163   221.6, 222   239.0, 239   170.1, 170
     8)  177.2, 177   238.4, 239   206.4, 206   216.5, 217   204.7, 205   146.0, 146
     9)  247.9, 248   163.8, 164   192.3, 194   191.8, 193    180.8, 181
    272.0, 272
    10)  194.1, 194   178.8, 179   142.6, 144   136.0, 137   228.0, 229
     25.1,   25
    11)  159.3, 159   255.3, 256   145.4, 146   200.8, 200   163.1, 163   144.8, 145
    12)  206.4, 207   159.5, 160   194.8, 194   137.3, 138   140.6, 140
     57.0,   57
    13)  211.9, 212   160.6, 161   106.2, 106   211.9, 212   267.5, 268   201.5, 202
    14) 180.1, 180   168.1, 169   187.7, 187   177.6, 178   173.5, 175    98.6, 100
    15) 145.5, 145   179.7, 180   215.5, 217   132.6, 134   264.6, 265   175.8, 176
    16) 147.0, 146   178.7, 180   147.0, 146   178.7, 180   212.7, 213   129.8, 129
    17) 131.5, 132   236.4, 235   191.5, 192   204.3, 205
    
    Out of these 100 samples, in 86 cases the sum of the whole numbers is
    within one whole number of the sum of the numbers expressed to one
    decimal point, and in 14 cases it is within 2 whole numbers. I was
    expecting a few cases of larger differences, but guess that this
    non-occurrence is due to the admittedly small sample of only one
    hundred.
    
    The conclusion is that rounding to whole numbers in a series does not
    lead to a great chance of the rounded amounts adding up to significant
    error.
    
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