E1C05 09/14/2010
14:36:29 Page 195
and the measurement random standard uncertainty is
s T ¼ s T
À Á 2
1
þ s T
À Á 2
2
þ s T
À Á 2
3
h
i 1=2 ¼ 0:46
C
The degrees of freedom are found using Equation 5.30 giving n ¼ 17, which assumes a large value
for n b T .
The combined standard uncertainty in the mean oven temperature is
u T ¼ b
2
T
þ s
2
T
h
i 1=2 ¼ 0:55 % 0:6
C
with a confidence level of one standard deviation. Assigning t 17,95 ¼ 2.11, the best estimate of the
mean oven temperature is
T
0
¼ T Æ t n;P b
2
T
þ s
2
T
1=2 ¼ 344:4 Æ 1:2
C ð95%Þ
5.9 CORRECTION FOR CORRELATED ERRORS
So far, we have assumed that all of the different elements of error in a test are independent from the
others. If two errors are not independent, they are ‘‘correlated.’’
For example, when the same instrument is used to measure different variables, the instrument
systematic errors between those variables are correlated. If multiple instruments are calibrated
against the same standard, then the systematic uncertainty in the standard is passed to each
instrument. Hence, these errors are correlated. The numerical effect of correlated errors on the
uncertainty depends on the functional relationship between the variables and the magnitudes of the
elemental systematic errors that are correlated. We now introduce a correction for treating the
correlated systematic errors.
Consider the result R, which is determined through the functional relationship between the
measured independent variables x i , i ¼ 1, 2, . . . , L where L is the number of independent variables
involved. Each x i is subject to elemental systematic errors with standard uncertainties, b k , where k ¼
1, 2, . . . , K, refer to each of up to any of K elements of error. Now allow that H of these K elemental
errors are correlated between variables while the rest (K – H) are uncorrelated. When correlated
errors are included, the systematic standard uncertainty in a result is estimated by
b R ¼
X L
i¼1
u i b x i
À
Á 2 þ 2
X LÀ1
i¼1
X L
j¼iþ1
u i u j b x i x j
"
# 1=2
ð5:40Þ
where index j is a counter equal to i þ 1 and with
u i ¼
qR
qx i x¼x
ð5:14Þ
Equation 5.40 introduces the covariance, b x i x j , to account for correlated errors and this is found from
b x i x j ¼
X H
h¼1
b x i
À Á
h
b x j
À Á
h
ð5:41Þ
5.9 Correction for Correlated Errors 195
14:36:29 Page 195
and the measurement random standard uncertainty is
s T ¼ s T
À Á 2
1
þ s T
À Á 2
2
þ s T
À Á 2
3
h
i 1=2 ¼ 0:46
C
The degrees of freedom are found using Equation 5.30 giving n ¼ 17, which assumes a large value
for n b T .
The combined standard uncertainty in the mean oven temperature is
u T ¼ b
2
T
þ s
2
T
h
i 1=2 ¼ 0:55 % 0:6
C
with a confidence level of one standard deviation. Assigning t 17,95 ¼ 2.11, the best estimate of the
mean oven temperature is
T
0
¼ T Æ t n;P b
2
T
þ s
2
T
1=2 ¼ 344:4 Æ 1:2
C ð95%Þ
5.9 CORRECTION FOR CORRELATED ERRORS
So far, we have assumed that all of the different elements of error in a test are independent from the
others. If two errors are not independent, they are ‘‘correlated.’’
For example, when the same instrument is used to measure different variables, the instrument
systematic errors between those variables are correlated. If multiple instruments are calibrated
against the same standard, then the systematic uncertainty in the standard is passed to each
instrument. Hence, these errors are correlated. The numerical effect of correlated errors on the
uncertainty depends on the functional relationship between the variables and the magnitudes of the
elemental systematic errors that are correlated. We now introduce a correction for treating the
correlated systematic errors.
Consider the result R, which is determined through the functional relationship between the
measured independent variables x i , i ¼ 1, 2, . . . , L where L is the number of independent variables
involved. Each x i is subject to elemental systematic errors with standard uncertainties, b k , where k ¼
1, 2, . . . , K, refer to each of up to any of K elements of error. Now allow that H of these K elemental
errors are correlated between variables while the rest (K – H) are uncorrelated. When correlated
errors are included, the systematic standard uncertainty in a result is estimated by
b R ¼
X L
i¼1
u i b x i
À
Á 2 þ 2
X LÀ1
i¼1
X L
j¼iþ1
u i u j b x i x j
"
# 1=2
ð5:40Þ
where index j is a counter equal to i þ 1 and with
u i ¼
qR
qx i x¼x
ð5:14Þ
Equation 5.40 introduces the covariance, b x i x j , to account for correlated errors and this is found from
b x i x j ¼
X H
h¼1
b x i
À Á
h
b x j
À Á
h
ð5:41Þ
5.9 Correction for Correlated Errors 195