# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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. > > > > > > --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---