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8 Measurement Uncertainty
In the case where we can assume that the probability of the occurrence of a value
close to the mean value is bigger, it is recommended to use the triangular distribution. For example, the volume of the measuring flask provided by the manufacturer
is 100 mL ± 0.1 mL. The nominal value is the most probable; hence, taking the
triangular distribution, the standard uncertainty is 0.1/
√
6 0.04 mL. After the normalization of the statistical parameters and calculating the absolute values into the
relative values, we can sum the variances—that is, the square roots of relevant values
of the standard uncertainties. After evaluating the square root of the value, we obtain
a combined standard uncertainty for a given measuring procedure (in specified conditions). Knowing the value of the combined standard uncertainty, we can, through
correlation with strictly statistical parameters, asses that it is a range in which the
true result lies with the probability of approximately 68%.
That value is very useful when there is a need to compare results on the basis of
their standard uncertainty. However, when submitting the results to its end-users, it is
recommended to submit the expanded uncertainty—that is, the value of the standard
uncertainty multiplied by the coverage factor k, determined for the specific level of
trust. In the laboratory practice, the most often used is k 2; k 3 is used less often.
The determined value of the expanded uncertainty does not have a direct bearing
on the degrees of freedom, as in that case we do not have the information about
the number of repetitions. Nonetheless, it is assumed that, in approximation, the use
of the factor k 2 or k 3 allows the range of uncertainty that corresponds to the
probability of finding the result in specified range of 95% or 99% respectively, to be
determined.
Uncertainty is a parameter that includes, apart from the precision of the measurement, many other factors that influence the changeability of the measurement result.
Most of the measured quantities in the field of chemistry are determined indirectly
through the measurements of other quantities measured directly. The measured quantity, designated with the symbol y, is called the output quantity, and the quantities
x i (for i 1, 2, …, N) are the input quantities. The output quantity and the input
quantities are treated as random variables, for which the probability spread needs
to be evaluated. Moreover, two other parameters should be determined, namely the
expected value and the standard deviation. For each input quantity x i , the following
values are of importance: the mean value x mean and quantities influencing the input
quantity. The influencing quantities are characterized with zero expected values and
always non-zero standard deviation. Among influencing quantities, we can include
the result spread or the bias of the measuring procedure, caused, for example, by a
systematic error of the measuring device.
The measure of the spread is usually the standard deviation of the experimentally
obtained results; therefore, normal distribution is assigned to it. In each case, when
a sufficiently large set of results is available, it is possible to use the statistical
8 Measurement Uncertainty
In the case where we can assume that the probability of the occurrence of a value
close to the mean value is bigger, it is recommended to use the triangular distribution. For example, the volume of the measuring flask provided by the manufacturer
is 100 mL ± 0.1 mL. The nominal value is the most probable; hence, taking the
triangular distribution, the standard uncertainty is 0.1/
√
6 0.04 mL. After the normalization of the statistical parameters and calculating the absolute values into the
relative values, we can sum the variances—that is, the square roots of relevant values
of the standard uncertainties. After evaluating the square root of the value, we obtain
a combined standard uncertainty for a given measuring procedure (in specified conditions). Knowing the value of the combined standard uncertainty, we can, through
correlation with strictly statistical parameters, asses that it is a range in which the
true result lies with the probability of approximately 68%.
That value is very useful when there is a need to compare results on the basis of
their standard uncertainty. However, when submitting the results to its end-users, it is
recommended to submit the expanded uncertainty—that is, the value of the standard
uncertainty multiplied by the coverage factor k, determined for the specific level of
trust. In the laboratory practice, the most often used is k 2; k 3 is used less often.
The determined value of the expanded uncertainty does not have a direct bearing
on the degrees of freedom, as in that case we do not have the information about
the number of repetitions. Nonetheless, it is assumed that, in approximation, the use
of the factor k 2 or k 3 allows the range of uncertainty that corresponds to the
probability of finding the result in specified range of 95% or 99% respectively, to be
determined.
Uncertainty is a parameter that includes, apart from the precision of the measurement, many other factors that influence the changeability of the measurement result.
Most of the measured quantities in the field of chemistry are determined indirectly
through the measurements of other quantities measured directly. The measured quantity, designated with the symbol y, is called the output quantity, and the quantities
x i (for i 1, 2, …, N) are the input quantities. The output quantity and the input
quantities are treated as random variables, for which the probability spread needs
to be evaluated. Moreover, two other parameters should be determined, namely the
expected value and the standard deviation. For each input quantity x i , the following
values are of importance: the mean value x mean and quantities influencing the input
quantity. The influencing quantities are characterized with zero expected values and
always non-zero standard deviation. Among influencing quantities, we can include
the result spread or the bias of the measuring procedure, caused, for example, by a
systematic error of the measuring device.
The measure of the spread is usually the standard deviation of the experimentally
obtained results; therefore, normal distribution is assigned to it. In each case, when
a sufficiently large set of results is available, it is possible to use the statistical