S-over-root-B 1 2 3 4 5 6 7 1 2 3 4 5 6 7 ^1.15/^0.35 /sqrt( 2.828 0.102 0.697 1.291 1.886 2.480 2.828 0.092 0.834 1.695 2.621 3.592 2.000 0.707 1.414 2.121 2.828 3.536 2.000 0.785 1.741 2.775 3.864 4.994 1.414 0.493 1.333 2.174 3.015 3.856 4.697 1.414 0.479 1.505 2.641 3.847 5.104 6.404 1.000 1.000 2.000 3.000 4.000 5.000 6.000 1.000 1.000 2.219 3.537 4.925 6.365 7.850 0.707 0.348 1.538 2.727 3.916 5.105 6.294 7.484 0.707 0.275 1.517 2.932 4.445 6.030 7.672 9.362 0.500 0.707 2.121 3.536 4.950 6.364 7.778 9.192 0.500 0.574 2.032 3.656 5.383 7.187 9.053 10.970 0.354 1.087 2.769 4.451 6.133 7.814 9.496 11.178 0.354 0.871 2.553 4.407 6.371 8.418 10.534 12.706 0.250 1.500 3.500 5.500 7.500 9.500 11.500 13.500 0.250 1.167 3.092 5.199 7.428 9.748 12.143 14.602 0.177 1.958 4.336 6.715 9.09311.472 13.850 16.228 0.177 1.466 3.659 6.050 8.57411.201 13.910 16.691 0.125 2.475 5.303 8.132 10.96013.789 16.617 19.445 0.125 1.776 4.266 6.975 9.83112.801 15.865 19.008 0.088 3.066 6.430 9.793 13.15716.521 19.884 23.248 0.088 2.102 4.925 7.989 11.21914.577 18.039 21.591 1 2 3 4 5 6 7 1 2 3 4 5 6 7 ln(1/poisson prob. for .ge. from ln(1/poisson prob. for .ge. from bkgnd) bkgnd) 2.828 0.621 1.157 1.853 2.694 3.666 2.828 0.621 1.157 1.853 2.694 3.666 2.000 1.129 1.946 2.944 4.101 5.396 2.000 1.129 1.946 2.944 4.101 5.396 1.414 0.884 1.772 2.894 4.212 5.695 7.324 1.414 0.884 1.772 2.894 4.212 5.695 7.324 1.000 1.331 2.522 3.964 5.610 7.428 9.394 1.000 1.331 2.522 3.964 5.610 7.428 9.394 0.707 0.679 1.843 3.352 5.123 7.104 9.261 11.567 0.707 0.679 1.843 3.352 5.123 7.104 9.261 11.567 0.500 0.933 2.406 4.241 6.347 8.667 11.165 13.813 0.500 0.933 2.406 4.241 6.347 8.667 11.165 13.813 0.354 1.211 3.005 5.174 7.61810.279 13.120 16.112 0.354 1.211 3.005 5.174 7.61810.279 13.120 16.112 0.250 1.509 3.631 6.137 8.92211.927 15.111 18.448 0.250 1.509 3.631 6.137 8.92211.927 15.111 18.448 0.177 1.820 4.276 7.122 10.25113.599 17.128 20.810 0.177 1.820 4.276 7.122 10.25113.599 17.128 20.810 0.125 2.141 4.935 8.124 11.59615.289 19.163 23.191 0.125 2.141 4.935 8.124 11.59615.289 19.163 23.191 0.088 2.470 5.604 9.136 12.95316.991 21.211 25.585 0.088 2.470 5.604 9.136 12.95316.991 21.211 25.585 1 2 3 4 5 6 7 1 2 3 4 5 6 7 ratio /sqrt / ln(1/poisson ratio ^1.15/^0.35 / prob.) ln(1/poisson prob.) 2.828 0.164 0.602 0.697 0.700 0.677 2.828 0.147 0.721 0.915 0.973 0.980 2.000 0.626 0.727 0.721 0.690 0.655 2.000 0.695 0.895 0.943 0.942 0.925 1.414 0.557 0.752 0.751 0.716 0.677 0.641 1.414 0.542 0.849 0.913 0.913 0.896 0.874 1.000 0.751 0.793 0.757 0.713 0.673 0.639 1.000 0.751 0.880 0.892 0.878 0.857 0.836 0.707 0.513 0.834 0.813 0.764 0.719 0.680 0.647 0.707 0.405 0.823 0.875 0.868 0.849 0.828 0.809 0.500 0.758 0.882 0.834 0.780 0.734 0.697 0.665 0.500 0.616 0.845 0.862 0.848 0.829 0.811 0.794 0.354 0.898 0.922 0.860 0.805 0.760 0.724 0.694 0.354 0.719 0.850 0.852 0.836 0.819 0.803 0.789 0.250 0.994 0.964 0.896 0.841 0.797 0.761 0.732 0.250 0.773 0.852 0.847 0.832 0.817 0.804 0.792 0.177 1.076 1.014 0.943 0.887 0.844 0.809 0.780 0.177 0.806 0.856 0.849 0.836 0.824 0.812 0.802 0.125 1.156 1.075 1.001 0.945 0.902 0.867 0.839 0.125 0.829 0.864 0.859 0.848 0.837 0.828 0.820 0.088 1.241 1.147 1.072 1.016 0.972 0.937 0.909 0.088 0.851 0.879 0.875 0.866 0.858 0.850 0.844 Conclusion: S/sqrt(B) tends systematically to favor Conclusion: S^1.15 / B^0.35 does a better job of background that is too low,relative to more representing ln(1/poisson prob) within the range of accurate poisson measure. interest. From: CSA6::STROVINK 19-APR-1997 16:23:22.87 To: FNALD0::HOBBS CC: STROVINK Subj: Variations on S/sqrt(B) John, I fooled around a bit with variations on S/sqrt(B) in light of the more exact expectations that one obtains from poisson statistics. Here's what I did: I considered a grid of expected S+B in the range 1 thru 7 events in steps of 1, and of expected B in the range 0.088 to 2.828 in steps of a factor of root(2). (A few of these points are nonphysical, and were not used). I assumed that the number of candidates that one would obtain in the experiment would (for an average experiment) be equal to expected S+B. I then calculated (X) ln of 1 / (poisson probability for observing .ge. S+B from a fluctuation of background only). (This is what we want to optimize.) (Y) S/sqrt(B) (This is what is easy to optimize, since it doesn't require an assumption about S, and because it is a simple expression). I then printed out (Y) / (X) over the grid. If (Y) is a superb approximation to (X), one should see a constant value. Instead I found that (Y)/(X) increases by about *1.4 going from large to small expected B. Evidently S/sqrt(B) overstates the value of having a small background. In hindsight this seems reasonable, because the distribution of background events is not gaussian but rather poisson with a long high tail. (Y)/(X) also decreases by a smaller amount as S+B goes from small to large values. Evidently S/sqrt(B) slightly understates the value of large expected S. I then looked for small modifications to S/sqrt(B) that would dramatically improve the uniformity of the ratio (Y)/(X), over this particular range of values of expected S+B and expected B. I found that S^1.15 / B^0.35 produces a (Y)/(X) which is well within 10% of being uniform over almost all of the grid points. So my suggestion is that this function be substituted for S/sqrt(B) in the optimization used to select the cuts. It's just as easy to compute. Optimizing it will bring us closer to an analysis with maximum significance. In the event that the systematic error on B is nonnegligible, I would suggest continuing with your practice of adding (delta B)^2 to B at the outset. I can provide more details, including spreadsheets, if you are really interested. Hope this turns out to be useful. --------------------------------------------------------------------------- Converted from Excel format on 4/28/97 by Greg Landsberg