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Let us consider now the case where the calibration constant
is the scale factor , known with a standard uncertainty .
Also in this case, for simplicity and without losing generality,
let us suppose that the most probable value of is 1.
Then
, i.e.
, and
.
Then
|
|
|
(6.35) |
Cov |
|
|
(6.36) |
|
|
|
(6.37) |
|
|
|
(6.38) |
To verify the results let us consider two independent measurements
and ; let us
calculate the correlated quantities and , and finally their
product (
) and their ratio (
):
It follows that
Just as an unknown common offset error cancels in differences
and is enhanced in sums, an unknown normalization error has
a similar effect
on the ratio and the product. It is also interesting to calculate
the standard uncertainty of a difference in the case of a normalization error:
The contribution from an unknown
normalization error vanishes if the two
values are equal.
Next: General case
Up: Building the covariance matrix
Previous: Offset uncertainty
Contents
Giulio D'Agostini
2003-05-15