E1C05 09/14/2010
14:36:27 Page 184
The propagation of elemental random uncertainties due to the K random errors in measuring x
is given by the measurement random standard uncertainty, s x , as estimated by the RSS method of
Equation 5.2:
s x ¼ s x
ð Þ
2
1 þ s x
ð Þ
2
2 þ Á Á Á þ s x
ð Þ
2
K
h
i 1=2
ð5:24Þ
where each
s x
ð Þ k ¼ s x k =
ffiffiffiffiffiffi
N k
p
ð5:25Þ
The measurement random standard uncertainty represents a basic measure of the uncertainty due
to the known elemental errors affecting the variability of variable x at a one standard deviation
confidence level. The degrees of freedom, n, in the standard random uncertainty, is estimated using
the Welch–Satterthwaite formula (2):
v ¼
P K
k¼1
s
2
x
À Á
k
2
P K
k¼1
s 4
x
Á
k
=v k
ð5:26Þ
where k refers to each elemental error with its own v k ¼ N k À 1.
The propagation of elemental systematic uncertainties due to K systematic errors in measuring
x is treated in a similar manner. The measurement systematic standard uncertainty, b x , is given by
b x ¼ b x
ð Þ
2
1 þ b x
ð Þ
2
2 þ Á Á Á þ b x
ð Þ
2
K
h
i 1=2
ð5:27Þ
The measurement systematic standard uncertainty, b x , represents a basic measure of the uncertainty due to the elemental systematic errors that affect the measurement of variable x at a one
standard deviation confidence level.
The combined standard uncertainty in x is reported as a combination of the systematic standard
uncertainty and random standard uncertainty in x at a one standard deviation confidence level,
u x ¼ b
2
x þ s
2
x
À
Á 1=2
ð5:28Þ
For a normal distribution, this confidence interval is equivalent to a 68% probability level. A more
general form of this equation extends it to include the total uncertainty at other confidence levels
through the use of an appropriate t value as
u x ¼ t n;P b
2
x þ s
2
x
À
Á 1=2 P%
ð Þ
ð5:29Þ
In this form, the equation is called the expanded uncertainty in x at the reported confidence level
(1, 2). The degrees of freedom in Equation 5.29 is found by (2)
v ¼
P K
k¼1
s
2
x
À Á
k
þ b
2
x
À Á
k
2
P K
k¼1
s 4
x
Á
k
=v k
þ
P K
k¼1
b
4
x
Á
k
=v k
ð5:30Þ
The degrees of freedom in an uncertainty are calculated from measurements or prior information;
otherwise the degrees of freedom may be assumed to be large (n > 30 in Table 4.4). Reference 1
184 Chapter 5 Uncertainty Analysis
14:36:27 Page 184
The propagation of elemental random uncertainties due to the K random errors in measuring x
is given by the measurement random standard uncertainty, s x , as estimated by the RSS method of
Equation 5.2:
s x ¼ s x
ð Þ
2
1 þ s x
ð Þ
2
2 þ Á Á Á þ s x
ð Þ
2
K
h
i 1=2
ð5:24Þ
where each
s x
ð Þ k ¼ s x k =
ffiffiffiffiffiffi
N k
p
ð5:25Þ
The measurement random standard uncertainty represents a basic measure of the uncertainty due
to the known elemental errors affecting the variability of variable x at a one standard deviation
confidence level. The degrees of freedom, n, in the standard random uncertainty, is estimated using
the Welch–Satterthwaite formula (2):
v ¼
P K
k¼1
s
2
x
À Á
k
2
P K
k¼1
s 4
x
Á
k
=v k
ð5:26Þ
where k refers to each elemental error with its own v k ¼ N k À 1.
The propagation of elemental systematic uncertainties due to K systematic errors in measuring
x is treated in a similar manner. The measurement systematic standard uncertainty, b x , is given by
b x ¼ b x
ð Þ
2
1 þ b x
ð Þ
2
2 þ Á Á Á þ b x
ð Þ
2
K
h
i 1=2
ð5:27Þ
The measurement systematic standard uncertainty, b x , represents a basic measure of the uncertainty due to the elemental systematic errors that affect the measurement of variable x at a one
standard deviation confidence level.
The combined standard uncertainty in x is reported as a combination of the systematic standard
uncertainty and random standard uncertainty in x at a one standard deviation confidence level,
u x ¼ b
2
x þ s
2
x
À
Á 1=2
ð5:28Þ
For a normal distribution, this confidence interval is equivalent to a 68% probability level. A more
general form of this equation extends it to include the total uncertainty at other confidence levels
through the use of an appropriate t value as
u x ¼ t n;P b
2
x þ s
2
x
À
Á 1=2 P%
ð Þ
ð5:29Þ
In this form, the equation is called the expanded uncertainty in x at the reported confidence level
(1, 2). The degrees of freedom in Equation 5.29 is found by (2)
v ¼
P K
k¼1
s
2
x
À Á
k
þ b
2
x
À Á
k
2
P K
k¼1
s 4
x
Á
k
=v k
þ
P K
k¼1
b
4
x
Á
k
=v k
ð5:30Þ
The degrees of freedom in an uncertainty are calculated from measurements or prior information;
otherwise the degrees of freedom may be assumed to be large (n > 30 in Table 4.4). Reference 1
184 Chapter 5 Uncertainty Analysis