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    Re: Lunars using Bennett
    From: Bill Noyce
    Date: 2008 Apr 4, 15:51 -0400

    I hope we can all agree with Alex: it's a well-established fact that independent
    errors in n observations accumulate as sqrt(n).  I'm sure that examining Peter's
    data will show that.
    
    I only bothered to look at the first two lines of data presented, but
    it's clear you
    can't say the errors "tend to cancel".  The (absolute value of the)
    initial error in the
    6 inputs averages 0.25 with a max of 0.5, but in the sums the (absolute) errors
    average 0.67, and only two of them are less than 0.5.
    0.25*sqrt(6)=0.61, so these results aren't too far from what we ought to expect.
    
    The truth is that the summation neither adds all the errors together,
    nor completely
    cancels them, but somewhere in between.  The precise statement of that is the
    sqrt(n) law.  Let's not argue about well-established facts.
        -- Bill
    
    >  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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