8.10 Determination of the Measurement Uncertainty
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assessment of the spread. When information on the resolution is given, the rectangular
distribution is used, assuming that any value from the range is equally probable. This
is also assumed in the case of correction factors and errors. Knowing the value and
sign (plus or minus), those influences can be treated as random, with assigned the
rectangular distribution with the full width at half maximum (FWHM), equal to the
limiting values of those influences.
Expanded uncertainty is calculated by multiplication of the coverage factor and
the combined standard uncertainty:
U k · u c (y)
(8.8)
Expanded uncertainty is given for an arbitrarily defined confident level. In most
cases, it is accepted as 95%. The value of the coverage factor for a given confidence
level is determined by the probability distribution of the output value.
The combined standard uncertainty can be composed from a number of constituent uncertainties. Some of them can be determined on the basis of the results of
a measurement series, characterized by a spread. Other constituents of the combined
standard uncertainty, which cannot be assessed on the basis of the obtained spread of
results—for example, uncertainties stemming from imperfections of the measuring
equipment—are also evaluated by standard deviations, calculated on the basis of
predicted probability distributions.
Those two groups of uncertainty, different in the way they are obtained, are a
criterion according to which the uncertainties are divided into type A, which are
determined with the help of statistical methods, and type B, which are determined
through the use of other methods.
A commonly used way of evaluating uncertainty is to describe the measuring procedure in the form of a mathematical equation (model equation), which includes the
input quantities (factors that influence the result) and output quantity (the measured
quantity). A mathematical model of measurement is expressed with the functional
dependency
Y f (X )
(8.9)
where Y is a single output quantity, and X represents N input quantities. Each input
quantity Xi (from X 1 to X N ) is a random variable with the expected value of x i .
Usually, the symbols of quantities are designated with capital letters, X and Y,
respectively, and their estimates with small letters, x and y, respectively.
133
assessment of the spread. When information on the resolution is given, the rectangular
distribution is used, assuming that any value from the range is equally probable. This
is also assumed in the case of correction factors and errors. Knowing the value and
sign (plus or minus), those influences can be treated as random, with assigned the
rectangular distribution with the full width at half maximum (FWHM), equal to the
limiting values of those influences.
Expanded uncertainty is calculated by multiplication of the coverage factor and
the combined standard uncertainty:
U k · u c (y)
(8.8)
Expanded uncertainty is given for an arbitrarily defined confident level. In most
cases, it is accepted as 95%. The value of the coverage factor for a given confidence
level is determined by the probability distribution of the output value.
The combined standard uncertainty can be composed from a number of constituent uncertainties. Some of them can be determined on the basis of the results of
a measurement series, characterized by a spread. Other constituents of the combined
standard uncertainty, which cannot be assessed on the basis of the obtained spread of
results—for example, uncertainties stemming from imperfections of the measuring
equipment—are also evaluated by standard deviations, calculated on the basis of
predicted probability distributions.
Those two groups of uncertainty, different in the way they are obtained, are a
criterion according to which the uncertainties are divided into type A, which are
determined with the help of statistical methods, and type B, which are determined
through the use of other methods.
A commonly used way of evaluating uncertainty is to describe the measuring procedure in the form of a mathematical equation (model equation), which includes the
input quantities (factors that influence the result) and output quantity (the measured
quantity). A mathematical model of measurement is expressed with the functional
dependency
Y f (X )
(8.9)
where Y is a single output quantity, and X represents N input quantities. Each input
quantity Xi (from X 1 to X N ) is a random variable with the expected value of x i .
Usually, the symbols of quantities are designated with capital letters, X and Y,
respectively, and their estimates with small letters, x and y, respectively.