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### Offset uncertainty

Let be the results of independent measurements and the (diagonal) covariance matrix. Let us assume that they are all affected by the same calibration constant , having a standard uncertainty . The corrected results are then . We can assume, for simplicity, that the most probable value of is 0, i.e. the detector is well calibrated. One has to consider the calibration constant as the physical quantity , whose best estimate is . A term must be added to the covariance matrix.

The covariance matrix of the corrected results is given by the transformation

 (6.25)

where . The elements of are given by

 (6.26)

In this case we get
 (6.27) Cov (6.28) (6.29) (6.30)

reobtaining the results of Section . The total uncertainty on the single measurement is given by the combination in quadrature of the individual and the common standard uncertainties, and all the covariances are equal to . To verify, in a simple case, that the result is reasonable, let us consider only two independent quantities and , and a calibration constant , having an expected value equal to zero. From these we can calculate the correlated quantities and and finally their sum ( ) and difference ( ). The results are
 (6.31) (6.32)

It follows that
 (6.33) (6.34)

as intuitively expected.

Next: Normalization uncertainty Up: Building the covariance matrix Previous: Building the covariance matrix   Contents
Giulio D'Agostini 2003-05-15