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Statistical Analysis of Decay Data

Discrepant Data. Program LWEIGHT Edgardo Browne Decay Data Evaluation Project Workshop May 12 – 14, 2008 Bucharest, Romania. Statistical Analysis of Decay Data. Relative g -ray intensities a -particle intensities Electron capture and b intensities

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Statistical Analysis of Decay Data

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  1. Discrepant Data. Program LWEIGHT Edgardo Browne Decay Data Evaluation ProjectWorkshopMay 12 – 14, 2008Bucharest, Romania

  2. Statistical Analysis of Decay Data

  3. Relative g-ray intensities • a-particle intensities • Electron capture and b intensities • Recommended standards for energies and intensities • Statistical procedures for data analysis • Discrepant data

  4. Relative g-ray Intensities Ig = A × e(Eg) A ± DA … Spectral peak area ± De … Detector efficiency Several measurements with Ge detectors: Ig1= A1× e1, Ig2 = A2 × e2, … e1, e2, … are determined with standard calibration sources, thus they are not independent quantities. Best value of Ig is a weighted average of Igi. A realistic uncertainty Digshould not be lower than the lowest uncertainty in the input values. Same criterion applies to g-ray energies.

  5. Precise half-life values are important for g-ray calibration standards The IAEA Coordinated Research Programme (CRP) gives: dT1/2/T1/2 0.00144 T1/2 /T1, where T1 is the maximum source-in-use period for a given radionuclide (15 years or 5 half-lives), whichever is shorter. Then the contribution to the uncertainty in the radiation intensity calibration using this radionuclide will not exceed 0.1%. Example: 133Ba - T1/2 =10.57± 0.04 y - T1 = 15 y, then dT1/2/T1/2 = 0.00144 x 10.57/15 = 0.0010, Experimental value is 0.04/10.57 = 0.0039. The contribution to the uncertainty is >0.1%.

  6. A = A1(434) + A2(614) + A3(723) The areas of the individual peaks are not independent of each other. DO NOT use A1(434), A2(614), and A3(723) to determine T1/2(434), T1/2(614), and T1/2(723), respectively, and then average these values to obtain T1/2. Use “A” to determine T1/2.

  7. 2. a-particle Intensities Ia = A × e A ± DA … Spectral peak area e ………..Geometry (semiconductor detectors) e is the same for all a-particle energies. Best value of Ia is a weighted average of Iai. Uncertainty is the external (multiplied by c) uncertainty of the average value. Same criterion applies to a-particle energies, but because of the use of standards for energy calibrations, a realistic uncertainty should not be lower than the lowest uncertainty in the input values.

  8. Electron Capture and b Intensities Most electron capture and b intensities are from g-ray transition intensity balances. Ib or Ie = OUT - IN Ib,e IN OUT

  9. 4. Recommended Standards for Energies and Intensities Recommended standards for g-ray energy calibration (1999), R.G. Helmer, C. van der Leun, Nucl. Instrum. and Methods in Phys. Res. A450, 35 (2000). Update of X Ray and Gamma Ray Decay Data Standards for Detector Calibration and Other Applications, IAEA-Report, Vienna 2007. Recommended Energy and Intensity Values of Alpha Particles from Radioactive Decay, A. Rytz, Atomic Data and Nuclear Data Tables 47, 205 (1991)

  10. I strongly suggest reading the following paper Decay Data: review of measurements, evaluations and compilations, A.L. Nichols, Applied Radiations and Isotopes 55, 23 (2001).

  11. 5. Statistical Procedures for Data Analysis

  12. Averages Unweighted x(avg) = 1 / n xi sx(avg) = [ 1 / n (n – 1)(x(avg) – xi)2]1/2 Std. dev. Weighted x(avg) = Wxi /sxi2 ; W = 1 /sxi-2 c2 = (x(avg) – xi)2 /sxi2Chi sqr. cn2 = 1 / (n – 1)(x(avg) – xi)2 /sxi2 Red. Chi sqr sx(avg) = larger of W1/2 and W1/2cn. Std. dev.

  13. Discrepant Data • Simple definition: A set of data for whichcn2 > 1. • But,cn2has a Gaussian distribution, i.e. it varies with the number of degrees of freedom (n – 1). • Better definition: A set of data is discrepant ifcn2 is greater thancn2(critical). Wherecn2(critical) is suchthat there is a 99% probability that the set of data is discrepant.

  14. c2n (critical)[n=N-1] n c2n (critical) n c2n (critical) ----------------------------------- 1 6.6 11 2.2 2 4.6 12 2.2 3 3.8 13 2.1 4 3.3 14 2.1 5 3.0 15 2.0 6 2.8 16 2.0 7 2.6 17 2.0 8 2.5 18 - 21 1.9 9 2.4 22 - 26 1.8 10 2.3 27 - 30 1.7 > 30 1 + 2.33  2/n

  15. Limitation of Relative Statistical Weight Method (Program LWEIGHT) • For discrepant data (c2n > c2n(critical)) with at least three sets of input values, we apply the Limitation of Relative Statistical Weight method. The program identifies any measurement that has a relative weight >50% and increases its uncertainty to reduce the weight to 50%. Then it recalculatesc2nand produces a new average and a best value as follows:

  16. Ifc2n c2n(critical), the program chooses the weighted average and its uncertainty (the larger of the internal and external values). • Ifc2n > c2n(critical), the program chooses either the weighted or the unweighted average, depending on whether the uncertainties in the average values make them overlap with each other. If that is so, it chooses the weighted average and its (internal or external) uncertainty. Otherwise, the program chooses the unweighted average. In either case, it may expand the uncertainty to cover the most precise input value.

  17. Simple Example X= 500±1 1000±100 n=N - 1 X(avg)= 500 ± 5 2 2 c c (critical) =6.6 Data are discrepant =25, n n We change to 500±100 (Same statistical weights). Then X(avg)= 750 ± 250

  18. 44Ti Half-life T1/2$REF HALF-LIFE 99Wi01 60.7 1.2 98Ah03 59.0 0.6 98Go05 60.3 1.3 98No06 62.0 2.0 90Al11 66.6 1.6 83Fr27 54.2 2.1

  19. 44Ti Half-life (LWEIGHT) • 44Ti Half-life Measurements • INP. VALUE INP. UNC. R. WGHT chi**2/N-1 REFERENCE • .607000E+02 .120E+01 .141E+00 .826E-01 99Wi01 • .590000E+02 .600E+00 MIN *.563E+00* .479E+00 98Ah03 • .603000E+02 .130E+01 .120E+00 .163E-01 98Go05 • .620000E+02 .200E+01 .507E-01 .214E+00 98No06 • .666000E+02 .160E+01 .792E-01 .348E+01 90Al11 • .542000E+02 .210E+01 .460E-01 .149E+01 83Fr27 • No. of Input Values N= 6 CHI**2/N-1= 5.76 CHI**2/N-1(critical)= 3.00 • UWM :.604667E+02 .164796E+01 unweighted average • WM :.599288E+02 .450317E+00(INT.) .108057E+01(EXT.) weighted average • INP. VALUE INP. UNC. R. WGHT chi**2/N-1 REFERENCE • .607000E+02 .120E+01 .161E+00 .563E-01 99Wi01 • .590000E+02 .681E+00 *.500E+00* .487E+00 98Ah03 • * Input uncertainty increased .114E+01 times * • .603000E+02 .130E+01 .137E+00 .663E-02 98Go05 • .620000E+02 .200E+01 .580E-01 .188E+00 98No06 • .666000E+02 .160E+01 .907E-01 .334E+01 90Al11 • .542000E+02 .210E+01 .526E-01 .156E+01 83Fr27   • No. of Input Values N= 6 CHI**2/N-1= 5.63 CHI**2/N-1(critical)= 3.00 • UWM :.604667E+02 .164796E+01 unweighted average • WM :.600634E+02 .481846E+00(INT.) .114378E+01(EXT.) weighted average • LWM :.600634E+02 .114378E+01 Min. Inp. Unc.=.600000E+00 LWEIGHT value • LWM has used weighted average and external uncertainty • Recommended value: 60.0 (11) y

  20. I strongly suggest reading the following paper M.U.Rajput, T.D.Mac Mahon, Techniques for Evaluating Discrepant Data,Nucl.Instrum.Methods Phys.Res. A312, 289 (1992).

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