E1C05 09/14/2010
14:36:30 Page 196
where H is the number of elemental errors that are correlated between variables x i and x j and h is a
counter for each correlated error. Note that Equation 5.40 reduces to Equation 5.36 when no errors
are correlated (i.e., when H ¼ 0, b x i x j ¼ 0). References 2 and 9 discuss treatment of correlated
systematic errors in extended detail.
Example 5.16
Suppose a result is a function of three variables, x 1 , x 2 , x 3 . There are four systematic errors
associated with both x 1 and x 2 and five associated with x 3 . Only the first and second systematic
elemental errors associated with the second and third variable (x 2 and x 3 ) are determined to be
correlated because these errors arise from common sources. Express the systematic standard
uncertainty in the result and find the covariance term.
SOLUTION Equation 5.40 is expanded to the form
b R ¼ u 1 b x 1
ð
Þ
2 þ u 1 b x 2
ð
Þ
2 þ u 3 b x 3
À
Á 2 þ 2u 1 u 2 b x 1 x 2 þ 2u 1 u 3 b x 1 x 3 þ 2u 2 u 3 b x 2 x 3
h
i 1=2
For the two correlated errors (H ¼ 2) associated with variables 2 and 3 (i ¼ 2, j ¼ 3), the systematic
uncertainty in the result reduces to
b R ¼ u 1 b x 1
ð
Þ
2 þ u 2 b x 2
ð
Þ
2 þ u 3 b x 3
À
Á 2 þ 2u 2 u 3 b x 2 x 3
h
i 1=2
where the covariance term is
b x 2 x 3 ¼
X 2
h¼1
b x 2
ð Þ h b x 3
À Á
h
¼ b x 2
ð Þ 1 b x 3
À Á
1
þ b x 2
ð Þ 2 b x 3
À Á
2
Example 5.17
Suppose a result R is a function of two variables, X and Y, such that R ¼ X þ Y, and each variable
having one elemental error. If X ¼ 10:1 V with b X ¼ 1.1 Vand Y ¼ 12:2 V with b Y ¼ 0.8 V, estimate
the systematic standard uncertainty in the result if the systematic errors are (1) uncorrelated and (2)
correlated.
SOLUTION From the stated information, R ¼ 10:1 þ 12:2 ¼ 22:3 V. For the uncorrelated
case, the systematic standard uncertainty is estimated as
b R unc ¼ u X b X
À
Á 2 þ u Y b Y
À
Á 2
h
i 1=2 ¼ 1 Â 1:1
ð
Þ
2 þ 1 Â 0:8
ð
Þ
2
2
! 1=2
¼ 1:36 V
For the correlated case, with H ¼ 1 and L ¼ 2, the uncertainty is estimated as
b R cor ¼ u X b X
À
Á 2 þ u Y b Y
À
Á 2 þ 2u X u Y b X Y
h
i 1=2
¼ 1 Â 1:1
ð
Þ
2 þ 1 Â 0:8
ð
Þ
2 þ 2 1
ð Þ 1
ð Þ 1:1
ð Þ 0:8
ð Þ
h
i 1=2 ¼ 3:12 V
196 Chapter 5 Uncertainty Analysis
14:36:30 Page 196
where H is the number of elemental errors that are correlated between variables x i and x j and h is a
counter for each correlated error. Note that Equation 5.40 reduces to Equation 5.36 when no errors
are correlated (i.e., when H ¼ 0, b x i x j ¼ 0). References 2 and 9 discuss treatment of correlated
systematic errors in extended detail.
Example 5.16
Suppose a result is a function of three variables, x 1 , x 2 , x 3 . There are four systematic errors
associated with both x 1 and x 2 and five associated with x 3 . Only the first and second systematic
elemental errors associated with the second and third variable (x 2 and x 3 ) are determined to be
correlated because these errors arise from common sources. Express the systematic standard
uncertainty in the result and find the covariance term.
SOLUTION Equation 5.40 is expanded to the form
b R ¼ u 1 b x 1
ð
Þ
2 þ u 1 b x 2
ð
Þ
2 þ u 3 b x 3
À
Á 2 þ 2u 1 u 2 b x 1 x 2 þ 2u 1 u 3 b x 1 x 3 þ 2u 2 u 3 b x 2 x 3
h
i 1=2
For the two correlated errors (H ¼ 2) associated with variables 2 and 3 (i ¼ 2, j ¼ 3), the systematic
uncertainty in the result reduces to
b R ¼ u 1 b x 1
ð
Þ
2 þ u 2 b x 2
ð
Þ
2 þ u 3 b x 3
À
Á 2 þ 2u 2 u 3 b x 2 x 3
h
i 1=2
where the covariance term is
b x 2 x 3 ¼
X 2
h¼1
b x 2
ð Þ h b x 3
À Á
h
¼ b x 2
ð Þ 1 b x 3
À Á
1
þ b x 2
ð Þ 2 b x 3
À Á
2
Example 5.17
Suppose a result R is a function of two variables, X and Y, such that R ¼ X þ Y, and each variable
having one elemental error. If X ¼ 10:1 V with b X ¼ 1.1 Vand Y ¼ 12:2 V with b Y ¼ 0.8 V, estimate
the systematic standard uncertainty in the result if the systematic errors are (1) uncorrelated and (2)
correlated.
SOLUTION From the stated information, R ¼ 10:1 þ 12:2 ¼ 22:3 V. For the uncorrelated
case, the systematic standard uncertainty is estimated as
b R unc ¼ u X b X
À
Á 2 þ u Y b Y
À
Á 2
h
i 1=2 ¼ 1 Â 1:1
ð
Þ
2 þ 1 Â 0:8
ð
Þ
2
2
! 1=2
¼ 1:36 V
For the correlated case, with H ¼ 1 and L ¼ 2, the uncertainty is estimated as
b R cor ¼ u X b X
À
Á 2 þ u Y b Y
À
Á 2 þ 2u X u Y b X Y
h
i 1=2
¼ 1 Â 1:1
ð
Þ
2 þ 1 Â 0:8
ð
Þ
2 þ 2 1
ð Þ 1
ð Þ 1:1
ð Þ 0:8
ð Þ
h
i 1=2 ¼ 3:12 V
196 Chapter 5 Uncertainty Analysis