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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

 (14.14)

where . The elements of are given by

 (14.15)

In this case we get
 (14.16) Cov (14.17) (14.18) (14.19)

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
 (14.20) (14.21)

It follows that
 (14.22) (14.23)

as intuitively expected.

Next: Normalization uncertainty Up: Matrice di covarianza di Previous: Matrice di covarianza di   Indice
Giulio D'Agostini 2001-04-02