E1C05 09/14/2010
14:36:30 Page 197
which is stated at the one-standard deviation confidence level and where
b X Y ¼
X 1
h¼1
b X
À Á
1
b Y
À Á
1
¼ b X b Y
COMMENT In this case, the correlated systematic errors have a large impact on the systematic
uncertainty. This is not always the case as it depends on the functional relationship itself. We leave it
to the reader to show that if R ¼ X/Y, then the covariance term in this problem would have little
impact on the systematic standard uncertainty in the result.
There are situations where random errors can be correlated (10). In those cases, the random
standard uncertainty in a result is estimated by
s R ¼
X L
i¼1
u i s x t
À
Á 2 þ 2
X LÀ1
i¼1
X L
j¼iþ1
u i u j r x i x j s x i s x j
"
# 1=2
ð5:42Þ
where r x i x j is the correlation coefficient between x i and x j as given by equation 4.41.
5.10 NONSYMMETRICAL SYSTEMATIC UNCERTAINTY INTERVAL
There are situations where the error must be bounded on one side or skewed about the
measured mean value such that the systematic uncertainty is better described by a nonsymmetrical interval. To develop this case, we assume that the limits of possible systematic
error are known, as is the mean of the measured data set x. Let x þ B
þ
x and x À B
À
x be the upper
and lower limits of the systematic uncertainty relative to the measured mean value (Fig. 5.8),
with B
À
x ¼ 2b
À
x and B
þ
x ¼ 2b
þ
x where b
À
x and b
þ
x are the lower and upper systematic standard
uncertainty values. The modeling approach assumes that the errors are symmetric about some
mean value of the error distribution but asymmetric about the measured mean value.
If we model the error distribution as a normal distribution, then we assume that the limits
defined using B
À
x to B
þ
x cover 95% of the error distribution. Define the offset between the mean of the
x–B -
–
x –
B -
x –
B +
x –
x+B -
–
x –
x+q –
x –
B
Mean of error distribution
Sample mean value
lower
error
limit
upper
error
limit
B
q
Figure 5.8 Relation between the measured
mean value, the mean value of the distribution of errors, and the systematic
uncertainties for treating nonsymmetrical
uncertainties.
5.10 Nonsymmetrical Systematic Uncertainty Interval 197
14:36:30 Page 197
which is stated at the one-standard deviation confidence level and where
b X Y ¼
X 1
h¼1
b X
À Á
1
b Y
À Á
1
¼ b X b Y
COMMENT In this case, the correlated systematic errors have a large impact on the systematic
uncertainty. This is not always the case as it depends on the functional relationship itself. We leave it
to the reader to show that if R ¼ X/Y, then the covariance term in this problem would have little
impact on the systematic standard uncertainty in the result.
There are situations where random errors can be correlated (10). In those cases, the random
standard uncertainty in a result is estimated by
s R ¼
X L
i¼1
u i s x t
À
Á 2 þ 2
X LÀ1
i¼1
X L
j¼iþ1
u i u j r x i x j s x i s x j
"
# 1=2
ð5:42Þ
where r x i x j is the correlation coefficient between x i and x j as given by equation 4.41.
5.10 NONSYMMETRICAL SYSTEMATIC UNCERTAINTY INTERVAL
There are situations where the error must be bounded on one side or skewed about the
measured mean value such that the systematic uncertainty is better described by a nonsymmetrical interval. To develop this case, we assume that the limits of possible systematic
error are known, as is the mean of the measured data set x. Let x þ B
þ
x and x À B
À
x be the upper
and lower limits of the systematic uncertainty relative to the measured mean value (Fig. 5.8),
with B
À
x ¼ 2b
À
x and B
þ
x ¼ 2b
þ
x where b
À
x and b
þ
x are the lower and upper systematic standard
uncertainty values. The modeling approach assumes that the errors are symmetric about some
mean value of the error distribution but asymmetric about the measured mean value.
If we model the error distribution as a normal distribution, then we assume that the limits
defined using B
À
x to B
þ
x cover 95% of the error distribution. Define the offset between the mean of the
x–B -
–
x –
B -
x –
B +
x –
x+B -
–
x –
x+q –
x –
B
Mean of error distribution
Sample mean value
lower
error
limit
upper
error
limit
B
q
Figure 5.8 Relation between the measured
mean value, the mean value of the distribution of errors, and the systematic
uncertainties for treating nonsymmetrical
uncertainties.
5.10 Nonsymmetrical Systematic Uncertainty Interval 197