Examples of type B uncertainties

- Previous measurements of
__other__particular quantities, performed in similar conditions, have provided a repeatability standard deviation^{6.4}of : - A manufacturer's calibration certificate states that the uncertainty,
defined as
__standard deviations__, is ``'': - A result
is reported
in a publication
as
,
stating that the average has been performed on four measurements
and the uncertainty is a confidence interval.
One has to conclude that the confidence interval has been calculated
using the
:*Student* - A manufacturer's specification states that the
error on a quantity should not exceed . With this
limited information one has to assume a
__uniform distribution__: - A physical parameter of a Monte Carlo is believed to lie in the
interval of
around its best value,
but not with uniform distribution:
the degree of belief that the parameter is
at centre is higher than the degree of belief that it is at the edges of the
interval. With this information a
__triangular distribution__can be reasonably assumed:**Note**that the coefficient in front of changes from the of the previous example to the of this. If the interval were a interval then the coefficient would have been equal to . These variations -- to be considered extreme -- are smaller than the statistical fluctuations of empirical standard deviations estimated from measurements. This shows that one should not be worried that the type B uncertainties are less accurate than type A, especially if one tries to model the distribution of the physical quantity*honestly*. - The absolute energy calibration of an electromagnetic
calorimeter module is not
known exactly and is estimated to be between the nominal one
and . The ``statistical'' error is known by test beam
measurements to be
. What is the uncertainty
on the energy measurement of an electron which has apparently released
30 GeV?
- There is no type A uncertainty, since only one measurement has been performed.
- The energy has to be
__corrected__for the best estimate of the calibration constant: , with an uncertainty of due to sampling (the ``statistical'' error):GeV - Then one has to take into account the uncertainty due to absolute energy
scale calibration:
- assuming a
__uniform__distribution of the true calibration constant, GeV:GeV - assuming, more reasonably, a
__triangular__distribution, GeV,GeV

- assuming a
- Interpreting the maximum deviation from the nominal calibration
as uncertainty
(see comment at the end of Section ),
GeV GeVAs already mentioned earlier in these notes, while reasonable assumptions (in this case the first two) give consistent results, this is not true if one makes inconsistent use of the information just for the sake of giving ``safe'' uncertainties.

**Note added**: the original version of the*primer*contained at this point a ``more realistic and slightly more complicated example'', which requires, instead, a next-to-linear treatment [45], which was not included in the notes, neither is it in this new version. Therefore, I prefer to skip this example in order to avoid confusion.