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Measurement Uncertainty Previous: 2.3
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2.4 The ASME measurementuncertainty
formulation
The conventions advocated here are adapted from Abernethy and Ringheiser
(1985), and are the basis for standards recently adopted by many engineering
societies. There are five central features of this methodology:

1.

Measurement uncertainties are classified into two categories, those contributing
to random uncertainty (or precision) and those contributing to systematic
uncertainty (or bias). This classification of errors depends on the measurement
process; in a particular process, random errors can be reduced by repeating
the process and averaging results while the systematic errors remain constant.

2.

The recommended uncertainty interval to be reported and used in analyses
of precision is the 95% confidence interval, which corresponds to approximately
two standard deviations in a Gaussian error distribution. For small samples
(where the number of degrees of freedom is less than 30) the Studentt
statistic should be used. That is, the deviation should be reported that
has 5% probability of occurring in a Studentt distribution with the given
number of degrees of freedom. (This will always be larger than the deviation
in a Gaussian distribution corresponding to 95% probability.) As
far as possible, estimates of bias should also represent 95% coverage of
expected variations.^{2.4}

3.

To obtain the combined precision resulting from the net effects of many
uncorrelated sources, the sample standard deviations are combined in quadrature,^{2.5}
and the number of degrees of freedom is estimated from the WelchSatterthwaite
equation, given later; cf. (2.14).

4.

Bias limits are also combined in quadrature. If some limits are asymmetrical,
the positive and negative limits are combined separately to obtain separate
upper and lower bias limits.

5.

An overall characterization of uncertainty that includes effects of random
and systematic uncertainties can be combined in either of two ways, but
the two components should also be reported separately. The two possibilities
are to add the bias and (95% confidence) random uncertainties linearly
or in quadrature. The latter leads to consistency with the convention that
all quoted limits provide a best estimate of 95% coverage, but either is
acceptable under the conventions adopted by engineering societies. A complete
uncertainty report must also include an estimate of the number of degrees
of freedom associated with the result.
It is also a useful convention that error sources are tabulated with associated
estimates of precision, degrees of freedom, and bias. Such tabulations
make it possible to isolate major sources of error, to consider the validity
of other investigators' estimates of error sources, and to repeat the analyses
for a new case when only one of the contributions has changed. Also, tabulated
uncertainty reports should separate the effects introduced by calibration,
data collection, and data analysis.
An important aspect of this methodology is that the degrees of freedom
associated with cited estimates of precision should be calculated and quoted.
This becomes important when the number of degrees of freedom in the result
is small, so that error limits and propagated errors have nonGaussian
character. Even if it is assumed that the individual measurements are distributed
according to a Gaussian error distribution, the true standard deviation
for an average of n samples, ,
is not known and must be estimated from the observations. The test statistic
(where
is the average of n measurements,
is the true value of x, and S_{n} is the estimated
standard deviation of the average
about ,
determined from )
will not be Gaussian distributed. The appropriate distribution for such
averages is the Studentt distribution. The difference between the Gaussian
and Studentt distributions is generally insignificant when the number
of degrees of freedom^{2.6}
exceeds about thirty, but for small sample sizes the differences can be
quite important. For this reason, when n<30, the confidence limits
used should be taken from the Studentt distribution rather than from the
normal distribution. Figure 2.2 shows the relationship between the 95%
confidence limits and the t statistic for the Studentt distribution.

Figure 2.2: Confidence limits for the Student's t distribution.
If the final number of degrees of freedom is larger than thirty, the
range to select for the precision error limit in the result is easily determined
by use of twice the estimated standard deviation in the result, the approximate
95% confidence limit. Otherwise, it is necessary to know the effective
number of degrees of freedom in the final result, as shown in Fig. . The
WelchSatterthwaite formula provides an estimate:

(2.14) 
where n_{r} is the number of degrees of freedom in the
final result, S_{Y}_{,i} is the standard
deviation in Y that would result from error source i alone,
and n_{i} is the number of degrees of freedom in that error
source.
An uncertainty analysis presented in this format should include these
components:

1.

A description of the measurement system and a discussion of the limits
within which the analysis is valid. For example, the uncertainty in
measurements of wind for a research aircraft might be specified for straightandlevel
flight within three hours of takeoff, perhaps within some altitude range.

2.

A tabulation and classification of the elemental error sources. An
example is shown in Table 2.1. Each elemental error source should be listed
with its associated bias limit (B_{i}), precision ( t_{(95)i}S_{i}),
and number of degrees of freedom (n_{i}), where t_{(95)i}
is the 95% confidence limit in the t statistic for n_{i}
degrees of freedom (and is about 2 for ).
It is convenient to tabulate the effect of the error source on the measurement,
so that tables contain
and ,
where S_{i} and B_{i} are the estimated standard
deviation and bias limit in x_{i}. This simplifies error
propagation to the final result, although special treatment is still needed
in cases where the error contributions are correlated. The error sources
should be separated into groups contributing to calibration, data acquisition,
and data processing.

3.

A discussion of each elemental error source and a description of the
basis for the error estimates. These discussions should reflect the
evidence for the tabulated values.

4.

A calculation of the resulting net estimates of overall precision and
bias, and calculation of a comprehensive limit that combines systematic
and random components of the uncertainty. Although not part of the
standard, it should be a goal to use estimates that provide 95% coverage
wherever possible.

5.

A summary of the results and the uncertainty limitations of the measurement.
It is usually helpful here to highlight the main sources of error and
possible actions that could improve the measurements.
Tables 2.1 and 2.2 show examples of this procedure applied to pressure
and temperature measured by a research aircraft (King Air N312D) operated
by the National Center for Atmospheric Research. Further explanations of
the origins of these and other elemental uncertainty estimates can be found
in NCAR Technical Notes such as Brown (19XX), Brown (19XX), and Cooper
(19XX).
Table 2.1: Elemental Uncertainties Affecting Static Pressure:
Table 2.1a. Calibration Uncertainties, wingmounted 1201 sensor
item 
elemental uncertainty 
B [mb] 
t_{95}S [mb] 
n 
1 
operation of deadweight standard 
0.10 
 
 
2 
calibration of deadweight standard 
 
 

3 
repeatability of 1501 transfer standard 
0.10 
 

4 
height uncertainty in 1501 calibration 
 
 

5 
50% of dynamic inaccuracy of 1501 transfer standard 
0.05 
0.05 
>30 
6 
stability of 1501 transfer standard 
0.10 
 

7 
resolution of 1501 transfer standard 
0.02 
 

8 
leaks in the lines during calibration 
 
 

9 
height uncertainty during calibration 
0.04 
 

10 
curve fit inaccuracy 
0.10 



1201 transducer (p_{W}): 



11 
airborne data system digitization 
0.03 
0.12 
>30 
12 
1201 static inaccuracy, hysteresis 
0.21 
 

13 
1201 static inaccuracy, repeatability 
 
0.14 
>30 
14 
1201 static inaccuracy, voltage variations 
 
0.08 
>30 

1501 transducer (p_{F}): 



15 
repeatability of the transfer standard 
0.10 
 

16 
50% of dynamic inaccuracy of 1501 sensor 
0.05 
0.05 
>30 

overall calibration uncertainty, 1201: 
0.30 
0.19 
>30 

overall calibration uncertainty, 1501: 
0.24 
0.07 
>30 





Dashes
indicate negligible contribution relative to other uncertainties.
Table 2.1b: Data Acquisition Uncertainties, 1201 Sensor (p_{W})
item 
elemental uncertainty 
B [mb] 
t_{95}S [mb] 
n 
17 
calibration uncertainty (Table 1a) 
0.36 
 

18 
dependence on temperature 
1.0 
 

19 
dependence on temperature change 
0.20 
0.20 
>30 
19 
" " , 1000 ft/min height change 
3.0 
 

20 
1201 dynamic accuracy, vibration 
 
0.2 
>30 
21 
1201 dynamic accuracy, noise 
 
0.04 
>30 
22 
line leaks 
 
 

23 
time lag 
 
 

24 
airborne data system digitization 
0.03 
0.12 
>30 
25 
aerodynamic effects 
0.20 
0.14 
>30 
26 
static defect (Appendix A) 
 
 

27 
truncation during data processing 
 
 

Not
applicable in level flight
Table 2.1c: Data Acquisition Uncertainties, 1501 Sensor (p_{F})
item 
elemental uncertainty 
B [mb] 
t_{95}S [mb] 
n 
28 
calibration uncertainty from Table 1a 
0.25 
 

29 
digitization by the digital transducer 
 
0.02 
>30 
30 
static error 
0.10 
0.10 
>30 
31 
1501 dynamic accuracy, acceleration and vibration 
 
0.20 
>30 
32 
1501 dynamic accuracy, voltage variations 
 
 

33 
longterm stability 
0.03 
 

34 
response time 
0.05 
 

34 
response time, 2000 ft/min descent 
0.10 
 

35 
line leaks 
 
 

36 
static defect (Appendix A) 
0.60 
2.0 
>30 
37 
truncation during data processing 
 
 

38 
effects of airspeed and attack angle 
 
 

Not
applicable in level flight
Table 2.1d: SUMMARY OF MEASUREMENT UNCERTAINTY
item 
instrument 
B [mb] 
t_{95}S [mb] 
U_{RSS}^{*} [mb] 

1201 Sensor (P_{W}): 
1.10 mb 
0.31 mb 
1.14 mb 

1201 Sensor, rapid climb/descent: 
3.2 
0.31 
3.2 mb 

1501 Sensor (P_{F}): 
0.67 mb 
2.0 mb 
2.12 mb 
^{*}U_{RSS} = (B^{2} + (t_{95}S) ,
where B is the bias limit and (t_{95}S) is the 95% confidence limit
for random error.
Table 2.2: Elemental Uncertainties Affecting Temperature:
Table 2.2a. Calibration Uncertainties, Rosemount 102E2A1 sensor
item 
elemental uncertainty 
B [C] 
t_{95}S [C] 
n 

Calibration of 102 sensor at NCAR (Cal. step 1): 



1 
bath uniformity 
0.02 
<0.02 
>30 
2 
selfheating of the sensor 
 
0.01 
 
3 
stability and repeatability of standard 
0.01 
0.01 
>30 
4 
accuracy of the Wheatstone bridge 
0.05 
 

5 
lead resistance during calibration 
<0.01 
 


Calibration of working standard (Cal. step 2): 



6 
repeatability of standard 
 
<0.01 
>30 
7 
Stability of the standard from cal. to use 
<0.05 
 

8 
selfheating (calibrated out) 
 
 


Calibration of factory standard (Cal. step 3): 



9 
reported resolution 
0.001 
 

10 
calibration assumed to introduce negl. error 
 
 


Calibration of resistance measurement (step 4): 



11 
resistance box accuracy 
0.05 
0.01 
>30 
12 
lead resistance differences 
0.01 
0.01 
>30 
13 
airborne data system characteristics 
 
0.01 
>30 
14 
environmental effects (temperature, etc.) 
 
 

15 
quadratic representation 
 
 


Calibration of resistance box (steps 5,6): 



16 
assumed to indroduce negl. error 
 
 


Net uncertainty, calibration: 
0.091 
0.03 
>30 





Dashes
indicate negligible contribution relative to other uncertainties.
Table 2.2b: Data Acquisition Uncertainties, 102E2A1 Sensor
item 
elemental uncertainty 
B [C] 
t_{95}S [C] 
n 

Sensor characteristics: 



17 
Calibration (from Table 2a) 
0.10 


18 
Selfheating 
 
 

19 
Longterm stability 
0.10 
 

20 
Effects of conduction from housing 
0.05 
0.05 
>30 
21 
radiative effects 
<0.001 
<0.001 
>30 
22 
airflow direction (maximum, extreme AOA) 
0.05 
0.05 
>30 
23 
stresses on sensor 
0.05 
0.05 
>30 

Data system characteristics: 



24 
random error and drift, 510BH 
0.05 
0.05 
5 
25 
temperature sensitivity, 510BH 
<0.04 
 

26 
airborne data system 
 
0.01 
>30 
27 
Roundoff, machine precision 
 
 

28 
Recovery factor used 
0.10 
0.10 
30? 
29 
Mach number used 
0.03 
0.05 
>30 
30 
Variation in C_{p} with humidity 
 
 

Table 2.2c: Summary


B [C] 
t_{95}S [C] 
U_{RSS } 

Rosemount Temperature 
0.21 
0.15 
0.26 C 
Next: 2.5
Propagation of uncertainty Up: 2.
Measurement Uncertainty Previous: 2.3
More terminology
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