E1C05 09/14/2010
14:36:25 Page 164
5.3 DESIGN-STAGE UNCERTAINTY ANALYSIS
Design-stage uncertainty analysis refers to an analysis performed in the formulation stage prior to a
test. It provides only an estimate of the minimum uncertainty based on the instruments and method
chosen. If this uncertainty value is too large, then alternate approaches will need to be found. So, it is
useful for selecting instruments and selecting measurement techniques. At the test design stage, the
measurement system and associated procedures may be but a concept. Often little may be known
about the instruments, which in many cases might still be just pictures in a catalog. Major facilities
may need to be built and equipment ordered with a considerable lead time. Uncertainty analysis at
this time is used to assist in selecting equipment and test procedures based on their relative
performance. In the design stage, distinguishing between systematic and random errors might be too
difficult to be of concern. So for this initial discussion, consider only sources of error and their
assigned uncertainty in general. A measurement system usually consists of sensors and instruments,
each with their respective contributions to system uncertainty. We first discuss individual contributions to uncertainty.
Even when all errors are otherwise zero, a measured value must be affected by our ability to
resolve the information provided by the instrument. This zero-order uncertainty of the instrument,
u 0 , assumes that the variation expected in the measured values will be only that amount due to
instrument resolution and that all other aspects of the measurement are perfectly controlled.
Essentially, u 0 is an estimate of the expected random uncertainty caused by the data scatter due to
instrument resolution.
In lieu of any other information, assign a numerical value to u 0 of one-half of the analog
instrument resolution
1 or to equal to its digital least count. This value will reasonably represent the
uncertainty interval on either side of the reading with a probability of 95%. Then,
u 0 ¼
1
2
resolution ¼ 1 LSD
ð5:1Þ
where LSD refers to the least significant digit of the readout.
Note that because we assume that the error has a normal distribution with its uncertainty applied
equally to either side of the reading, we could write this as
u 0 ¼ Æ
1
2
resolution ð95%Þ
But unless specifically stated otherwise, the Æ sign for the uncertainty will be assumed for any
computed uncertainty value and applied only when writing the final uncertainty interval of a result.
The second piece of information that is usually available is the manufacturer’s statement
concerning instrument error. We can assign this stated value as the instrument uncertainty, u c .
Essentially, u c is an estimate of the expected systematic uncertainty due to the instrument. If no
probability level is provided with such information, a 95% level can be assumed.
Sometimes the instrument errors are delineated into parts, each part due to some contributing
factor (Table 1.1). A probable estimate in u c can be made by combining the uncertainties of known
errors in some reasonable manner. An accepted approach of combining uncertainties is termed the
root-sum-squares (RSS) method.
1 It is possible to assign a value for u 0 that differs from one-half the scale resolution. Discretion should be used. Instrument
resolution is likely described by either a normal or a rectangular distribution, depending on the instrument.
164 Chapter 5 Uncertainty Analysis
14:36:25 Page 164
5.3 DESIGN-STAGE UNCERTAINTY ANALYSIS
Design-stage uncertainty analysis refers to an analysis performed in the formulation stage prior to a
test. It provides only an estimate of the minimum uncertainty based on the instruments and method
chosen. If this uncertainty value is too large, then alternate approaches will need to be found. So, it is
useful for selecting instruments and selecting measurement techniques. At the test design stage, the
measurement system and associated procedures may be but a concept. Often little may be known
about the instruments, which in many cases might still be just pictures in a catalog. Major facilities
may need to be built and equipment ordered with a considerable lead time. Uncertainty analysis at
this time is used to assist in selecting equipment and test procedures based on their relative
performance. In the design stage, distinguishing between systematic and random errors might be too
difficult to be of concern. So for this initial discussion, consider only sources of error and their
assigned uncertainty in general. A measurement system usually consists of sensors and instruments,
each with their respective contributions to system uncertainty. We first discuss individual contributions to uncertainty.
Even when all errors are otherwise zero, a measured value must be affected by our ability to
resolve the information provided by the instrument. This zero-order uncertainty of the instrument,
u 0 , assumes that the variation expected in the measured values will be only that amount due to
instrument resolution and that all other aspects of the measurement are perfectly controlled.
Essentially, u 0 is an estimate of the expected random uncertainty caused by the data scatter due to
instrument resolution.
In lieu of any other information, assign a numerical value to u 0 of one-half of the analog
instrument resolution
1 or to equal to its digital least count. This value will reasonably represent the
uncertainty interval on either side of the reading with a probability of 95%. Then,
u 0 ¼
1
2
resolution ¼ 1 LSD
ð5:1Þ
where LSD refers to the least significant digit of the readout.
Note that because we assume that the error has a normal distribution with its uncertainty applied
equally to either side of the reading, we could write this as
u 0 ¼ Æ
1
2
resolution ð95%Þ
But unless specifically stated otherwise, the Æ sign for the uncertainty will be assumed for any
computed uncertainty value and applied only when writing the final uncertainty interval of a result.
The second piece of information that is usually available is the manufacturer’s statement
concerning instrument error. We can assign this stated value as the instrument uncertainty, u c .
Essentially, u c is an estimate of the expected systematic uncertainty due to the instrument. If no
probability level is provided with such information, a 95% level can be assumed.
Sometimes the instrument errors are delineated into parts, each part due to some contributing
factor (Table 1.1). A probable estimate in u c can be made by combining the uncertainties of known
errors in some reasonable manner. An accepted approach of combining uncertainties is termed the
root-sum-squares (RSS) method.
1 It is possible to assign a value for u 0 that differs from one-half the scale resolution. Discretion should be used. Instrument
resolution is likely described by either a normal or a rectangular distribution, depending on the instrument.
164 Chapter 5 Uncertainty Analysis