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14:36:25 Page 162
This chapter approaches uncertainty analysis as an evolution of information from test design
through final data analysis. While the structure of the analysis remains the same at each step, the number
of errors identified and their uncertainty values may change as more information becomes available. In
fact, the uncertainty in the result may increase. There is no exact answer to an analysis, just the result
from a reasonable approach using honest numbers. This is the nature of an uncertainty analysis.
There are two accepted professional documents on uncertainty analysis. The American
National Standards Institute/American Society of Mechanical Engineers (ANSI/ASME) Power
Test Codes (PTC) 19.1 Test Uncertainty (2) is the United States engineering test standard, and our
approach favors that method. The International Organization on Standardization’s ‘‘Guide to the
Expression of Uncertainty in Measurement’’ (ISO GUM) (1) is an international metrology standard.
The two differ in some terminology and how errors are cataloged. For example, PTC 19.1 refers to
random and systematic errors, terms that classify errors by how they manifest themselves in the
measurement. ISO GUM refers to type A and type B errors, terms that classify errors by how their
uncertainties are estimated. These differences are real but they are not significant to the outcome.
Once past the classifications, the two methods are quite similar. The important point is that the end
outcome of an uncertainty analysis by either method will yield a similar result!
Upon completion of this chapter, the reader will be able to
explain the relation between an error and an uncertainty,
execute an appropriate uncertainty analysis regardless of the level and quantity of information
available,
explain the differences between systematic and random errors and treat their assigned
uncertainties,
analyze a test system and test approach from test design through data presentation to assign
and propagate uncertainties, and
propagate uncertainties to understand their impact on the final statement of a result.
5.2 MEASUREMENT ERRORS
In the discussion that follows, errors are grouped into two categories: systematic error and random
error. We do not consider measurement blunders that result in obviously fallacious data—such data
should be discarded.
Consider the repeated measurement of a variable under conditions that are expected to produce
the same value of the measured variable. The relationship between the true value of the population
and the measured data set, containing both systematic and random errors, can be illustrated as in
Figure 5.1. The total error in a set of measurements obtained under seemingly fixed conditions can
be described by the systematic errors and the random errors in those measurements. The systematic
errors shift the sample mean away from the true mean by a fixed amount, and within a sample of
many measurements, the random errors bring about a distribution of measured values about the
sample mean. Even a so-called accurate measurement contains small amounts of systematic and
random errors.
Measurement errors enter during all aspects of a test and obscure our ability to ascertain the
information that we desire: the true value of the variable measured. If the result depends on more
than one measured variable, these errors further propagate to the result. In Chapter 4, we stated that
the best estimate of the true value sought in a measurement is provided by its sample mean value and
162 Chapter 5 Uncertainty Analysis
14:36:25 Page 162
This chapter approaches uncertainty analysis as an evolution of information from test design
through final data analysis. While the structure of the analysis remains the same at each step, the number
of errors identified and their uncertainty values may change as more information becomes available. In
fact, the uncertainty in the result may increase. There is no exact answer to an analysis, just the result
from a reasonable approach using honest numbers. This is the nature of an uncertainty analysis.
There are two accepted professional documents on uncertainty analysis. The American
National Standards Institute/American Society of Mechanical Engineers (ANSI/ASME) Power
Test Codes (PTC) 19.1 Test Uncertainty (2) is the United States engineering test standard, and our
approach favors that method. The International Organization on Standardization’s ‘‘Guide to the
Expression of Uncertainty in Measurement’’ (ISO GUM) (1) is an international metrology standard.
The two differ in some terminology and how errors are cataloged. For example, PTC 19.1 refers to
random and systematic errors, terms that classify errors by how they manifest themselves in the
measurement. ISO GUM refers to type A and type B errors, terms that classify errors by how their
uncertainties are estimated. These differences are real but they are not significant to the outcome.
Once past the classifications, the two methods are quite similar. The important point is that the end
outcome of an uncertainty analysis by either method will yield a similar result!
Upon completion of this chapter, the reader will be able to
explain the relation between an error and an uncertainty,
execute an appropriate uncertainty analysis regardless of the level and quantity of information
available,
explain the differences between systematic and random errors and treat their assigned
uncertainties,
analyze a test system and test approach from test design through data presentation to assign
and propagate uncertainties, and
propagate uncertainties to understand their impact on the final statement of a result.
5.2 MEASUREMENT ERRORS
In the discussion that follows, errors are grouped into two categories: systematic error and random
error. We do not consider measurement blunders that result in obviously fallacious data—such data
should be discarded.
Consider the repeated measurement of a variable under conditions that are expected to produce
the same value of the measured variable. The relationship between the true value of the population
and the measured data set, containing both systematic and random errors, can be illustrated as in
Figure 5.1. The total error in a set of measurements obtained under seemingly fixed conditions can
be described by the systematic errors and the random errors in those measurements. The systematic
errors shift the sample mean away from the true mean by a fixed amount, and within a sample of
many measurements, the random errors bring about a distribution of measured values about the
sample mean. Even a so-called accurate measurement contains small amounts of systematic and
random errors.
Measurement errors enter during all aspects of a test and obscure our ability to ascertain the
information that we desire: the true value of the variable measured. If the result depends on more
than one measured variable, these errors further propagate to the result. In Chapter 4, we stated that
the best estimate of the true value sought in a measurement is provided by its sample mean value and
162 Chapter 5 Uncertainty Analysis