8.16 Calculating Combined Standard Uncertainty
139
Statistical distributions used for the evaluation of standard uncertainty
Normal distribution (known as Gaussian distribution): is used when a large set of data are
available, which depends only on the randomly distributed parameters. In normal distribution,
most of results are clustered symmetrically on both sides of a central value, as fewer occurred
unbounded on both sides. Values near the mean are more likely than values far from the mean.
Standard uncertainty is equal to the standard deviation, with a confidence level of 68.3%
Rectangular distribution: all values lie with equal probability in the given range between −a
and +a. Thus, estimated standard uncertainty can be calculated as u(x) a/
√
3.
Rectangular distribution should be applied for the information given in certificates or other
documents, where the information that the value lies between the range is given. e.g., the purity
of cupper standards is quoted as (99.99 ± 0.01)%
Triangle distribution: this describes the situation where it is expected that values near the mean
are more likely than those far from the mean, close to the extremes of the range. Thus, the
estimated standard uncertainty can be calculated as u(x) a/
√
6.
Triangle distribution approach should be applied for the data given in the specification of e.g.
volumetric glassware volume is quoted as (100 ± 0.1) mL (in 20 °C).
8.16 Calculating Combined Standard Uncertainty
In cases where the measurement result is calculated from values of other quantities, the law of propagation of standard uncertainties of all components is applied,
which results in combined standard uncertainty. Assuming that the input quantities
are independent of one other (uncorrelated), the combined standard uncertainty is
calculated by the use of the following equations:
u c (y)
K
k1
∂ f
∂ x k
2
· u 2 (x k )
(8.17)
u
2
c (y)
∂f
∂x i
2
·
u
2 (x i )
2
(8.18)
If both methods of evaluation of uncertainty—type A and B—were used, then the
following equation should be used to determine the combined standard uncertainty:
u(x)
u
2
A (x) + u
2
B (x)
1
n(n − 1)
n
i1
(x i − ¯
x) 2 +
( e x) 2
6
+
( e x) 2
3
(8.19)
139
Statistical distributions used for the evaluation of standard uncertainty
Normal distribution (known as Gaussian distribution): is used when a large set of data are
available, which depends only on the randomly distributed parameters. In normal distribution,
most of results are clustered symmetrically on both sides of a central value, as fewer occurred
unbounded on both sides. Values near the mean are more likely than values far from the mean.
Standard uncertainty is equal to the standard deviation, with a confidence level of 68.3%
Rectangular distribution: all values lie with equal probability in the given range between −a
and +a. Thus, estimated standard uncertainty can be calculated as u(x) a/
√
3.
Rectangular distribution should be applied for the information given in certificates or other
documents, where the information that the value lies between the range is given. e.g., the purity
of cupper standards is quoted as (99.99 ± 0.01)%
Triangle distribution: this describes the situation where it is expected that values near the mean
are more likely than those far from the mean, close to the extremes of the range. Thus, the
estimated standard uncertainty can be calculated as u(x) a/
√
6.
Triangle distribution approach should be applied for the data given in the specification of e.g.
volumetric glassware volume is quoted as (100 ± 0.1) mL (in 20 °C).
8.16 Calculating Combined Standard Uncertainty
In cases where the measurement result is calculated from values of other quantities, the law of propagation of standard uncertainties of all components is applied,
which results in combined standard uncertainty. Assuming that the input quantities
are independent of one other (uncorrelated), the combined standard uncertainty is
calculated by the use of the following equations:
u c (y)
K
k1
∂ f
∂ x k
2
· u 2 (x k )
(8.17)
u
2
c (y)
∂f
∂x i
2
·
u
2 (x i )
2
(8.18)
If both methods of evaluation of uncertainty—type A and B—were used, then the
following equation should be used to determine the combined standard uncertainty:
u(x)
u
2
A (x) + u
2
B (x)
1
n(n − 1)
n
i1
(x i − ¯
x) 2 +
( e x) 2
6
+
( e x) 2
3
(8.19)