Discrepant Data. Program LWEIGHT Edgardo Browne

Decay Data Evaluation Project Workshop

May 12 – 14, 2008Bucharest, Romania

Statistical Analysis of Decay Data

• Relative -ray intensities• -particle intensities• Electron capture and intensities• Recommended standards for energies and

intensities• Statistical procedures for data analysis• Discrepant data

1. Relative -ray Intensities

I = A × (E)

A ± A … Spectral peak area± … Detector efficiency

Several measurements with Ge detectors:

I1= A1 × 1, I2 = A2 × 2, …

1, 2, … are determined with standard calibration sources, thus they are not independent quantities.

Best value of Iis a weighted average of IA realistic uncertainty ishould not be lower than the lowest uncertainty in the input values.

Same criterion applies to -ray energies.

Precise half-life values are important for -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%.

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.

2. -particle Intensities

I = A × A ± A … Spectral peak area………..Geometry (semiconductor detectors)is the same for all -particle energies.

Best value of Iis a weighted average of IUncertainty is the external (multiplied by ) uncertainty of the average value.

Same criterion applies to -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.

3. Electron Capture and Intensities

Most electron capture and intensities are from-ray transition intensity balances.

I or I = OUT - IN

IN

OUT

I

4. Recommended Standards for Energies and Intensities

Recommended standards for -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)

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).

5. Statistical Procedures for Data Analysis

Averages

Unweighted x(avg) = 1 / n xi

x(avg) = [ 1 / n (n – 1) (x(avg) – xi)2]1/2 Std. dev.

Weighted x(avg) = W xi / xi

2 ; W = 1 / xi-2

2 = (x(avg) – xi)2 / xi

2 Chi sqr.

2 = 1 / (n – 1) (x(avg) – xi)

2 / xi2 Red. Chi sqr

x(avg) = larger of W1/2 and W1/2 Std. dev.

Discrepant Data

•Simple definition: A set of data for which 2 > 1.

•But, 2 has a Gaussian distribution, i.e. it varies

with the number of degrees of freedom (n – 1).

•Better definition: A set of data is discrepant if 2 is

greater than 2 (critical). Where

2 (critical) is such

that there is a 99% probability that the set of data is discrepant.

2critical) [N-1]

2

critical)2

critical)

Limitation of Relative Statistical Weight Method (Program LWEIGHT)

For discrepant data (2>2

(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 recalculates 2and

produces a new average and a best value as

follows:

• If 22

(critical), the program chooses the

weighted average and its uncertainty (the larger of the internal and external values).

• If 22

(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.

Simple Example

500±1 1000±100X=

X(avg)= 500 ± 5=N - 1

2 =25, 2

(critical) =6.6 Data are discrepant

We change to 500±100 (Same statistical weights). Then

X(avg)= 750 ± 250

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

44Ti Half-life

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

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).