Most of the analysis done by the gcp project uses methods to give a numerical value of the likelihood that the data produced in a given time was influenced by something anomalous. The nature of the influence is not known at this time, and it is important to rule out any possible influences that may be inherent to the generator itself. Comparing all the data generated to theoretical values will give an indication of how well the generators conform to expectation. The random number generators (called eggs) used to create the data are electronic devices, and even when the same components are used to make them, each generator will have its own characteristics. The result of this is that although the generators will behave well, there will be small differences between them.
Each egg will make 200 decisions per second, with each decision having an equal chance of becoming a one or a zero (a digit representing one or zero is known in the binary number system as a bit). The number of one bits is summed to give a value that should have a mean of 100, and a standard deviation of 7.071. To ensure the probability will be exactly 0.5, 100 of the 200 bits are inverted. The mean value for the eggs has been confirmed by testing 18 months of data generated by the eggs of the gcp.
The theoretical SD of the groups of 200 should be the square root of 50, and to test this value, the SD of each egg was found for a six month period. To be certain that the SD is not changing with time, three groups of six month periods were tested. The SD does differ from expected in some eggs, but each egg's SD seems to remain at a stable rate over the 16 months tested.
Some of the eggs had SD's greater than the theoretical value, and some were less than the theoretical value. I have constructed a table to compare the results of using the theoretical SD vs empirical SD in calculations. Only eggs that reported data for at least 2/3 of the possible seconds in a given six month period will be used here. If the egg has met this criteria for more than one of the six month groups, an average value was used. For the table below, 36 eggs have empirical SD's for the time period used.
The standard method of analysis computes the stouffer z-score for each second. The method used to find each seconds z-score is to compute (X-100) / SD for each 200 bit trial in a second, sum the z scores for the second, and divide this by the square root of N. The seconds z-score can then be squared and added to other seconds z-scores, and the resulting sum will have a chi squared distribution.
The monthly z-score is found by adding each seconds squared z-scores
together for an entire month. The result is then computed using the
following formulae:
( (Sz2)
- DF) / (sqrt(2) * sqrt(DF)).
with DF = number of seconds used.
A second method of analysis adds the z-squares for all groups of
200 in a given period of time. This is labeled in the table as the
z-squares method. For each 200 bit group, a z-score is found by dividing
(result - 100) by the standard deviation. The z-score is then squared and
summed with other z squares, increasing the degrees of freedom by one.
The monthly z-score is then found with the formulae:
( (Sz2)
- DF) / (sqrt(2) * sqrt(DF)).
| MONTH | STANDARD METHOD
(theoretical SD) |
STANDARD METHOD (empirical SD) | Z-SQUARES METHOD (theoretical SD) | Z-SQUARES METHOD
(empirical SD) |
| Jan 2000 | 1.10 | 0.75 | 1.07 | -0.69 |
| July 2000 | 0.74 | 0.44 | 0.74 | -0.88 |
| Sept 2000 | -0.85 | -1.12 | 1.54 | 0.24 |
| Oct 2000 (1) | 1.12 | 0.80 | 1.97 | 0.31 |
| Nov 2000 | -1.18 | -1.44 | 2.48 | 1.37 |
| Dec 2000 | 2.04 | 1.73 | 1.07 | -0.38 |
| Jan 2001 | 1.70 | 1.36 | 1.76 | -0.07 |
| Feb 2001 | 0.02 | -0.30 | 0.68 | -1.61 |
| Mar 2001 (2) | -0.04 | -0.38 | 1.68 | -0.11 |
| April 2001 (2) | 1.48 | 1.14 | 2.75 | 0.74 |
| May 2001 (2) | 1.42 | 1.06 | 2.70 | 0.56 |
| June 2001 (2) | -0.43 | -0.80 | 2.72 | 0.62 |
Notes: 1. data from egg #1000 is not used in this month
2. data from egg #28 is not
used in these months
The values in the table are the values that would have resulted if the entire month had been predicted to have been influenced. The results of the standard method using both the theoretical and empirical SD values for the calculations show that the method will work with theoretical values.
The z-squares method theoretically calculated result and empirical
result are not well matched, with the theoretical method producing a biased
result. Only short duration events should be done with the z-squares
theoretical method.
Empirical Values
The following table lists the values of each eggs standard deviation
during three periods of six months.
Only periods with eggs that have reported over 10 million trials
have been entered,
with most of the eggs reporting about 15 million trials in a six
month period. The average was found by performing the following calculations:
( (SD1*N1) + (SD2*N2) + (SD3*N3) ) / ( N1 + N2 + N3)
| EGG ID# | empirical SD
Jan00 to June00 |
number of 200
bit groups (*106) Jan00 to Jun00 |
empirical SD
Jul00 to Dec00 |
number of 200
bit groups (*106) Jul00 to Dec00 |
empirical SD
Jan01 to Jun01 |
number of 200
bit groups (*106) Jan01 to Jun01 |
avg SD |
| 1 | 7.06756 | 14.7 | 7.07004 | 14.3 | 7.07015 | 15.0 | 7.06925 |
| 28 | 7.06999 | 15.3 | 7.06785 | 15.6 | 7.06891 | ||
| 33 | 7.07017 | 13.9 | 7.07227 | 12.4 | 7.07140 | ||
| 37 | 7.06718 | 15.2 | 7.07141 | 15.9 | 7.07086 | 15.4 | 7.06894 |
| 100 | 7.07187 | 15.1 | 7.07307 | 13.6 | 7.07244 | ||
| 101 | 7.07417 | 14.6 | 7.07311 | 15.3 | 7.07219 | 14.9 | 7.07315 |
| 102 | 7.07322 | 15.7 | 7.07173 | 15.9 | 7.07182 | 14.0 | 7.07227 |
| 103 | 7.07236 | 15.7 | 7.07284 | 15.8 | 7.07012 | 10.9 | 7.07196 |
| 105 | 7.07329 | 14.5 | 7.07393 | 13.7 | 7.07320 | ||
| 106 | 7.07471 | 12.6 | 7.07361 | 14.9 | 7.07637 | 15.5 | 7.07493 |
| 107 | 7.07391 | 15.7 | 7.07342 | 15.3 | 7.07250 | 15.6 | 7.07327 |
| 108 | 7.07527 | 15.7 | 7.07495 | 15.9 | 7.07544 | 15.3 | 7.07522 |
| 109 | 7.07402 | 15.4 | 7.07457 | 15.9 | 7.07425 | 13.9 | 7.07428 |
| 110 | 7.07558 | 15.3 | 7.07355 | 15.8 | 7.07280 | 15.6 | 7.07396 |
| 111 | 7.07161 | 13.9 | 7.07181 | 15.5 | 7.07196 | 15.6 | 7.07180 |
| 112 | 7.06970 | 15.1 | 7.07176 | 15.8 | 7.06898 | 15.3 | 7.07016 |
| 114 | 7.07246 | 15.7 | 7.07282 | 15.8 | 7.07269 | 14.6 | 7.07266 |
| 115 | 7.07655 | 15.4 | 7.07594 | 15.3 | 7.07585 | 14.3 | 7.07612 |
| 116 | 7.07055 | 11.8 | 7.07415 | 15.5 | 7.07709 | 12.8 | 7.07402 |
| 118 | 7.07162 | 14.8 | 7.07349 | 12.1 | 7.07354 | 13.0 | 7.07282 |
| 119 | 7.07442 | 15.8 | 7.07151 | 14.5 | 7.07303 | ||
| 134 | 7.07350 | 15.6 | 7.07350 | ||||
| 142 | 7.07254 | 14.3 | 7.07254 | ||||
| 161 | 7.07185 | 15.4 | 7.07185 | ||||
| 1000 | 7.07078 | 12.8 | 7.07078 | ||||
| 1005 | 7.07279 | 15.7 | 7.07180 | 15.3 | 7.07086 | 13.8 | 7.07185 |
| 1021 | 7.06866 | 15.7 | 7.07225 | 15.7 | 7.07107 | 15.6 | 7.07066 |
| 1022 | 7.07188 | 15.7 | 7.07011 | 14.0 | 7.07151 | 15.6 | 7.07120 |
| 1024 | 7.07107 | 13.7 | 7.06905 | 13.3 | 7.06936 | 14.9 | 7.06982 |
| 1025 | 7.06964 | 15.7 | 7.07194 | 12.5 | 7.07008 | 15.5 | 7.07045 |
| 1026 | 7.07271 | 12.3 | 7.07284 | 12.2 | 7.07278 | ||
| 1027 | 7.07205 | 14.6 | 7.06987 | 15.6 | 7.06933 | 15.1 | 7.07039 |
| 1029 | 7.06946 | 15.2 | 7.06955 | 15.6 | 7.06951 | ||
| 2000 | 7.06898 | 15.6 | 7.06898 | ||||
| 2002 | 7.07284 | 14.6 | 7.07284 | ||||
| 2173 | 7.07529 | 15.0 | 7.07529 |