E1C05 09/14/2010
14:36:28 Page 186
KNOWN
B 1 ¼ 0:20 N
B 2 ¼ 0:30 N
95%
ð
Þ
ASSUMPTIONS Manufacturer specifications reliable at 95% probability
FIND B ð95%Þ
SOLUTION Lacking information as to the statistics used to specify the uncertainty values
associated with the two element errors, e 1 and e 2 , we consider them as systematic uncertainties stated
at 95% confidence level. The systematic standard uncertainties are found from Equation 5.4 as b ¼ B/2,
b 1 ¼ B 1 =2 ¼ 0:10 N b 2 ¼ B 2 =2 ¼ 0:15 N
The systematic standard uncertainty in the transducer is
b ¼ b
2
1 þ b
2
2
À
Á 1=2 ¼ 0:18 N
at one standard deviation confidence. The expanded systematic uncertainty with t n,95 ¼ 2 is
B ¼ 2 b
2
1 þ b
2
2
À
Á 1=2 ¼ 0:36 N
Elemental errors due to instruments are considered to be data-acquisition source errors (Table 5.2). If
these uncertainties were found during a calibrations by the end-user, then we could list them as
calibration source errors. In the end, as long as they are accounted for, the error source grouping
assigned is not important.
COMMENT Note that the estimate for expanded systematic error due to the instrument is equal
in value to the estimate of u c in a design-stage analysis. The resolution error uncertainty u 0 used in
Example 5.1 is now built into the data scatter and so included as part of the random uncertainty
estimate in Example 5.8.
Example 5.10
In Examples 4.8 and 4.9, a set of measured data were used to generate a polynomial curve fit
equation between variables x and y. If during a test, variable x were to be measured and y computed
from x through the polynomial, estimate the random standard uncertainty due to the data-reduction
error in the curve fit.
KNOWN Data set of Example 4.8
Polynomial fit of Example 4.9
ASSUMPTIONS Data fits curve y ¼ 1:04x þ 0:02.
FIND Random standard uncertainty in the curve fit
SOLUTION The curve fit was found to be given by y ¼ 1:04x þ 0:02 with a standard error of
the fit, s yx ¼ 0:16 based on the approximation of Equation 4.38. The standard random uncertainty
due to the curve fit, written as s x
ð Þ 1 to be consistent with Table 5.3, is
s x
ð Þ 1 ¼ s yx =
ffiffiffiffi
N
p ¼ 0:072 V
with n ¼ 3
The expanded random uncertainty interval with t 3;95 ¼ 3:18 is t 3;95 s x
ð Þ 1 ¼ 0:23 V.
186 Chapter 5 Uncertainty Analysis
14:36:28 Page 186
KNOWN
B 1 ¼ 0:20 N
B 2 ¼ 0:30 N
95%
ð
Þ
ASSUMPTIONS Manufacturer specifications reliable at 95% probability
FIND B ð95%Þ
SOLUTION Lacking information as to the statistics used to specify the uncertainty values
associated with the two element errors, e 1 and e 2 , we consider them as systematic uncertainties stated
at 95% confidence level. The systematic standard uncertainties are found from Equation 5.4 as b ¼ B/2,
b 1 ¼ B 1 =2 ¼ 0:10 N b 2 ¼ B 2 =2 ¼ 0:15 N
The systematic standard uncertainty in the transducer is
b ¼ b
2
1 þ b
2
2
À
Á 1=2 ¼ 0:18 N
at one standard deviation confidence. The expanded systematic uncertainty with t n,95 ¼ 2 is
B ¼ 2 b
2
1 þ b
2
2
À
Á 1=2 ¼ 0:36 N
Elemental errors due to instruments are considered to be data-acquisition source errors (Table 5.2). If
these uncertainties were found during a calibrations by the end-user, then we could list them as
calibration source errors. In the end, as long as they are accounted for, the error source grouping
assigned is not important.
COMMENT Note that the estimate for expanded systematic error due to the instrument is equal
in value to the estimate of u c in a design-stage analysis. The resolution error uncertainty u 0 used in
Example 5.1 is now built into the data scatter and so included as part of the random uncertainty
estimate in Example 5.8.
Example 5.10
In Examples 4.8 and 4.9, a set of measured data were used to generate a polynomial curve fit
equation between variables x and y. If during a test, variable x were to be measured and y computed
from x through the polynomial, estimate the random standard uncertainty due to the data-reduction
error in the curve fit.
KNOWN Data set of Example 4.8
Polynomial fit of Example 4.9
ASSUMPTIONS Data fits curve y ¼ 1:04x þ 0:02.
FIND Random standard uncertainty in the curve fit
SOLUTION The curve fit was found to be given by y ¼ 1:04x þ 0:02 with a standard error of
the fit, s yx ¼ 0:16 based on the approximation of Equation 4.38. The standard random uncertainty
due to the curve fit, written as s x
ð Þ 1 to be consistent with Table 5.3, is
s x
ð Þ 1 ¼ s yx =
ffiffiffiffi
N
p ¼ 0:072 V
with n ¼ 3
The expanded random uncertainty interval with t 3;95 ¼ 3:18 is t 3;95 s x
ð Þ 1 ¼ 0:23 V.
186 Chapter 5 Uncertainty Analysis