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calibration errors. But if additional information is available, we can get a better idea of the
uncertainty in a measurement. So, an advanced-stage uncertainty analysis permits taking designstage analysis further by considering procedural and test control errors that affect the measurement.
We consider it as a method for a thorough uncertainty analysis when a large data set is not available.
This is often the case in the early stages of a test program or for certain tests where repeating
measurements may not be possible. Such an advanced-stage analysis, also known as singlemeasurement uncertainty analysis (3, 6), can be used: (1) in the advanced design stage of a
test to estimate the expected uncertainty, beyond the initial design stage estimate; and (2) to report
the results of a test program that involved measurements over a range of one or more parameters but
with no or relatively few repeated measurements of the pertinent variables at each test condition.
Essentially, the method assesses different aspects of the main test by quantifying potential errors
though various well focused verification tests.
In this section, the goals are either to estimate the uncertainty in some measured value x or in
some general result R through an estimation of the uncertainty in each of the factors that may affect x
or R. We present a technique that uses a step-by-step approach for identifying and estimating the
uncertainty associated with errors. We seek the combined value of the estimates at each step. We
assume that the errors follow a normal distribution, but some errors might be better described by
other distributions and these can be used. For example, calibration errors that specify only a range or
operating condition errors that are related to establishing a controlled set point are well modeled by a
rectangular distribution. The standard uncertainty for a rectangular distribution defined by the
interval (b À a) is given by
u x ¼ b À a
ð
Þ=
ffiffiffiffiffi
12
p
ð5:21Þ
which covers a 58% confidence level. Multiplying by 2 provides for two standard deviations
coverage. If we assume that the errors propagate to the result with a normal distribution, then a
coverage factor of 2 approximates the 95% confidence level for consistency.
Zero-Order Uncertainty
At zero-order uncertainty, all variables and parameters that affect the outcome of the measurement,
including time, are assumed to be fixed except for the physical act of observation itself. Under such
circumstances, any data scatter introduced upon repeated observations of the output value is the
result of instrument resolution alone. The value u 0 estimates the extent of variation expected in the
measured value when all influencing effects are controlled and is found using Equation 5.1. By
itself, a zero-order uncertainty is inadequate for the reporting of test results.
Higher-Order Uncertainty
Higher-order uncertainty estimates consider the controllability of the test operating conditions and
the variability of all measured variables. For example, at the first-order level, the effect of time as an
extraneous variable in the measurement might be considered. That is, what would happen if we
started the test, set the operating conditions, and sat back and watched? If a variation in the measured
value is observed, then time is a factor in the test, presumably due to some extraneous influence
affecting process control or simply inherent in the behavior of the variable being measured.
In practice, the uncertainty at this first level would be evaluated for each particular measured
variable by operating the test facility at some single operating condition that would be within the
5.7 Advanced-Stage Uncertainty Analysis 177
14:36:27 Page 177
calibration errors. But if additional information is available, we can get a better idea of the
uncertainty in a measurement. So, an advanced-stage uncertainty analysis permits taking designstage analysis further by considering procedural and test control errors that affect the measurement.
We consider it as a method for a thorough uncertainty analysis when a large data set is not available.
This is often the case in the early stages of a test program or for certain tests where repeating
measurements may not be possible. Such an advanced-stage analysis, also known as singlemeasurement uncertainty analysis (3, 6), can be used: (1) in the advanced design stage of a
test to estimate the expected uncertainty, beyond the initial design stage estimate; and (2) to report
the results of a test program that involved measurements over a range of one or more parameters but
with no or relatively few repeated measurements of the pertinent variables at each test condition.
Essentially, the method assesses different aspects of the main test by quantifying potential errors
though various well focused verification tests.
In this section, the goals are either to estimate the uncertainty in some measured value x or in
some general result R through an estimation of the uncertainty in each of the factors that may affect x
or R. We present a technique that uses a step-by-step approach for identifying and estimating the
uncertainty associated with errors. We seek the combined value of the estimates at each step. We
assume that the errors follow a normal distribution, but some errors might be better described by
other distributions and these can be used. For example, calibration errors that specify only a range or
operating condition errors that are related to establishing a controlled set point are well modeled by a
rectangular distribution. The standard uncertainty for a rectangular distribution defined by the
interval (b À a) is given by
u x ¼ b À a
ð
Þ=
ffiffiffiffiffi
12
p
ð5:21Þ
which covers a 58% confidence level. Multiplying by 2 provides for two standard deviations
coverage. If we assume that the errors propagate to the result with a normal distribution, then a
coverage factor of 2 approximates the 95% confidence level for consistency.
Zero-Order Uncertainty
At zero-order uncertainty, all variables and parameters that affect the outcome of the measurement,
including time, are assumed to be fixed except for the physical act of observation itself. Under such
circumstances, any data scatter introduced upon repeated observations of the output value is the
result of instrument resolution alone. The value u 0 estimates the extent of variation expected in the
measured value when all influencing effects are controlled and is found using Equation 5.1. By
itself, a zero-order uncertainty is inadequate for the reporting of test results.
Higher-Order Uncertainty
Higher-order uncertainty estimates consider the controllability of the test operating conditions and
the variability of all measured variables. For example, at the first-order level, the effect of time as an
extraneous variable in the measurement might be considered. That is, what would happen if we
started the test, set the operating conditions, and sat back and watched? If a variation in the measured
value is observed, then time is a factor in the test, presumably due to some extraneous influence
affecting process control or simply inherent in the behavior of the variable being measured.
In practice, the uncertainty at this first level would be evaluated for each particular measured
variable by operating the test facility at some single operating condition that would be within the
5.7 Advanced-Stage Uncertainty Analysis 177