5.17. View of the result from calculating a mean with Gamspk1

For the weighted mean of the single line activities one obtains the following interim report for the case of a measurement of Co-60 on a HPGe detector:

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(1 + b/2L) equivalent factor for Compton BG rate: 1.120

Individual peak data:
      (pgamm*fcoin is a measure for the importance of the line!)

i E       PNRate    epsPeak pgamm   fatt   fcoin  (pgamm*fcoin)
  keV     cps          %
----------------------------------------------------------------------------------
1 1173.20 5.699E-03 0.7790  0.99850 1.0000 1.0615 1.0599  values
           2.71     1.6789  0.03000 1.0000 1.3810         u_rels in %
2 1332.50 5.360E-03 0.7030  0.99986 1.0000 1.0641 1.0640  values
           2.76     1.6245  0.00060 1.0000 1.3890         u_rels in %

Results from individual peak activities:

     A(i) = PeakNetRate(i) * (fatt(i) * fcoin(i)) / (epsPeak(i) * pgamm(i))

i  E(keV)  Activity (Bq) rel.StdDev (%)
--------------------------------------------------
1  1173.20 7.7771E-01    3.61
2  1332.50 8.1147E-01    3.63

Evaluation of the weighted mean:
weighted mean                 = 0.79379
int. std. dev. of the mean    = 2.23372E-02 ( 2.81 %) (Bayes compliant)
ext. std. dev. of the mean    = 1.85227E-02 ( 2.33 %) (not Bayes compliant)
Chi-square = test value T     = 0.68763
reduced Chi-square            = 0.68763
significance (Chi-square > T) = 4.06973 %

Note: only the internal standard deviation will be used hereafter!
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In the first table of this report the input data are shown in shortened form without reproducing their uncertainties. In this example the corrections for coincidence summing (fcoinsu) are quite significant because of the well-type counting geometry. The product (pgamm * fcoinsu) given in the last column of that table is a measure for the weighting of the individual gamma lines.

The second table of this report shows the activity values (in Bq) and their relative standard uncertainties (in %) calculated for the individual lines. What follows are the data obtained from calculating the weighted mean.

For the mean obtained by applying weighted least-squares (LSQ Mean) calculated from several peaks one obtains for the lower result-part shown above for the weighted mean:

(Note: in this case the activity variances of the two gamma lines are practically identical; thus, there are nearly no deviations between this method and that of the weighted mean.)

Evaluation of the weighted mean by least-squares:

weighted mean = 0.79358

std. dev. of the mean = 2.23360E-02 ( 2.81 %)

reduced Chi-square = 0.86694