Table 2. Characteristic numbers for low-energy contamination.
The first column gives the name of the data set, the second the percentage of multiples of three and the third the percentage of rare events from multiples of three (which should be close to the value in column two when low-energy contamination is not present). The absolute numbers are also given. Column four gives the absolute numbers of m 3 in the class of the strongest 20% of residuals and the corresponding percentage, column five displays the significance of the shift of the mean weighted residuals from zero, and column six the significance of the positive excess residuals according to a random walk criterion with equal probability for positive and negative steps. Column seven shows a theoretical reference value from a Gaussian distribution for the mean values in the next two columns, which are the separate mean value of the positive weighted residuals (column eight) and the absolute mean value of the negative weighted residuals (column nine). These values in column eight and nine are equal within the limits of statistical fluctuations, and equal to the reference value in column seven when no systematic errors apply. Column ten shows a theoretical reference value from a Gaussian distribution for the next two columns, which display the separate mean values of the positive squared weighted residuals and of the negative squared weighted residuals. These two values are equal within statistical fluctuations and in accordance with the reference value from column ten when no systematic errors apply. Squaring the residuals emphasizes outliers. Column 13 shows the weighted agreement factor as a percentage value, column 14 gives the goodness of fit and column 15 the alternative goodness of fit. In columns eight, nine, 11 and 12, statistical fluctuations are indicated by a 3σ error bar.
| Data set | m 3 (%) | |ζ| > 3 from m 3 (%) signif† | #m 3 in largest 20% of ζ2 | 〈ζ〉/σ(〈ζ〉) | (#ζ+ − #ζ−)/(N obs)1/2 | (2/π)1/2α | 〈ζ+〉 | 〈|ζ−|〉 | α2 |
|
|
wR(F 2) (%) | GoF | aGoF |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1_uncorr | 3.54 | 50.00 (13/26) 3.35 | 28/60 (46.67) | 3.85 | 1.00 | 0.73 | 0.81± 0.10 | 0.65± 0.06 | 0.85 | 1.57± 0.63 | 0.70± 0.04 | 12.65 | 1.12 | 1.05 |
| 1_corr | 3.54 | 0.00 (0/18) – | 15/60 (25.00) | 2.96 | 1.29 | 0.73 | 0.81± 0.08 | 0.71± 0.06 | 0.85 | 1.34± 0.40 | 0.84± 0.05 | 11.13 | 1.09 | 1.14 |
| 1_filter | 3.63 | 8.70 (2/23) 0.82 | 17/62 (27.42) | 2.74 | 1.23 | 0.73 | 0.83± 0.08 | 0.74± 0.06 | 0.85 | 1.28± 0.27 | 0.93± 0.06 | 11.05 | 1.10 | 1.23 |
| 2_uncorr | 3.69 | 41.38 (12/29) 3.15 | 24/57 (42.11) | 2.33 | 1.40 | 0.76 | 0.76± 0.09 | 0.68± 0.07 | 0.91 | 1.35± 0.58 | 0.90± 0.07 | 10.75 | 1.09 | 0.90 |
| 2_corr | 3.69 | 0.00 (0/27) – | 16/57 (28.07) | 1.44 | 1.76 | 0.76 | 0.78± 0.07 | 0.77± 0.08 | 0.91 | 1.10± 0.25 | 1.15± 0.09 | 9.63 | 1.09 | 0.98 |
| 2_filter | 3.75 | 0.00 (0/26) – | 11/58 (18.97) | 1.10 | 1.70 | 0.76 | 0.77± 0.07 | 0.78± 0.08 | 0.91 | 1.06± 0.24 | 1.19± 0.10 | 9.92 | 1.08 | 1.00 |
| 3_uncorr | 3.78 | 38.33 (23/60) 4.32 | 66/162 (40.74) | 5.47 | 2.69 | 0.74 | 0.78± 0.05 | 0.67± 0.04 | 0.86 | 1.32± 0.32 | 0.78± 0.03 | 7.04 | 1.07 | 1.11 |
| 3_corr | 3.78 | 12.24 (6/49) 1.69 | 44/162 (27.16) | 4.03 | 2.78 | 0.74 | 0.77± 0.04 | 0.71± 0.04 | 0.86 | 1.08± 0.16 | 0.90± 0.04 | 6.50 | 1.03 | 1.09 |
| 3_filter | 3.73 | 6.82 (3/44) 0.78 | 34/158 (21.52) | 2.80 | 3.06 | 0.74 | 0.76± 0.04 | 0.75± 0.04 | 0.86 | 1.02± 0.15 | 0.96± 0.04 | 6.81 | 1.03 | 0.94 |
| 4_uncorr | 3.61 | 27.54 (19/69) 3.79 | 90/314 (28.66) | 6.40 | 4.10 | 0.76 | 0.80± 0.03 | 0.72± 0.03 | 0.91 | 1.17± 0.15 | 0.84± 0.02 | 11.48 | 1.03 | 0.79 |
| 4_corr | 3.61 | 4.62 (3/65) 0.38 | 57/314 (18.15) | 5.37 | 3.80 | 0.76 | 0.80± 0.03 | 0.75± 0.03 | 0.91 | 1.08± 0.10 | 0.88± 0.02 | 10.91 | 1.02 | 0.86 |
| 4_filter | 3.67 | 2.82 (2/71) 0.43 | 66/325 (20.31) | 5.74 | 3.44 | 0.76 | 0.81± 0.03 | 0.75± 0.03 | 0.91 | 1.14± 0.10 | 0.90± 0.02 | 11.13 | 1.03 | 0.86 |
| 5_uncorr | 3.71 | 19.44 (7/36) 2.14 | 36/91 (39.56) | 2.92 | 2.18 | 0.76 | 0.83± 0.06 | 0.77± 0.06 | 0.90 | 1.24± 0.22 | 0.99± 0.05 | 6.80 | 1.09 | 1.50 |
| 5_corr | 3.71 | 3.33 (1/30) 0.11 | 24/91 (26.37) | 2.36 | 2.50 | 0.76 | 0.81± 0.06 | 0.79± 0.06 | 0.90 | 1.17± 0.20 | 1.03± 0.05 | 6.68 | 1.08 | 1.47 |
| 5_filter | 3.71 | 3.57 (1/28) 0.04 | 19/91 (20.88) | 2.58 | 1.27 | 0.76 | 0.85± 0.06 | 0.78± 0.06 | 0.90 | 1.27± 0.22 | 1.03± 0.05 | 6.74 | 1.10 | 1.46 |
The significance of the 3λ signal is calculated by Δ%/σPoisson,%, where Δ% is the difference in percentage points between the multiples of three (in percent) and the contribution of multiples from three to all rare events |ζ| > 3 (in percent), and σPoisson,% = 100[#|ζ| > 3(m 3)]1/2/(#|ζ| > 3) is the standard deviation based on Poisson statistics, expressed in percentage points. This is calculated by taking the square root of the number # of rare events |ζ| > 3 from multiples of three m 3, divided by the total number of rare events #|ζ| > 3 (which gives the fraction of rare events from multiples of three), multiplied by 100 to obtain the percentage points.