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Re: Lunars using Bennett
From: Peter Fogg
Date: 2008 Apr 5, 04:31 +1100
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. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---