Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition

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226 Measurement and Data Analysis for Engineering and Science

done for the histogram above). This essentially amounts to overlaying

the predicted values on the histogram constructed for the ﬁrst problem.

Use Scott’s formula. Assume that the mean and the standard deviation

of the normal distribution are the same as for the velocity data. (d) How

well do the velocities compare with those predicted, assuming a normal

distribution? What does it mean physically if the velocities are normally

distributed for this situation?

Bibliography

[1] D. Salsburg. 2001. The Lady Tasting Tea: How Statistics Revolutionized

Science in the Twentieth Century. New York: W.H. Freeman and Com-

pany.

[2] Student. 1908. The Probable Error of a Mean. Biometrika 4: 1-25.

[3] Figliola, R. and D. Beasley. 2002. Theory and Design for Mechanical

Measurements. 3rd ed. New York: John Wiley and Sons.

[4] H. Young. 1962. Statistical Treatment of Experimental Data. New York:

McGraw-Hill.

[5] J. Mandel. 1964. The Statistical Analysis of Experimental Data. New

York: Dover.

[6] A.J. Hayter. 2002. Probability and Statistics for Engineers and Scientists.

2nd ed. Paciﬁc Grove: Duxbury/Thomson Learning.

[7] J.L. Devore. 2000. Probability and Statistics for Engineering and the Sci-

ences. Paciﬁc Grove: Duxbury/Thomson Learning.

[8] R.A. Fisher. 1935. The Design of Experiments. Edinburgh: Oliver and

Boyd.

[9] A. Hald. 1952. Statistical Theory with Engineering Applications. New

York: John Wiley and Sons.

[10] L.B. Barrentine. 1999. Introduction to Design of Experiments: A Simpli-

ﬁed Approach. Milwaukee: ASQ Quality Press.

[11] Gunst, R.F., and R.L. Mason. 1991. How to Construct Fractional Facto-

rial Experiments. Milwaukee: ASQ Quality Press.

[12] D.C. Montgomery. 2000. Design and Analysis of Experiments. 5th ed.

New York: John Wiley and Sons.

[13] Brach, R.M. and Dunn, P.F. 2009. Uncertainty Analysis for Forensic

Science. 2nd ed. Tucson: Lawyers & Judges Publishing Company, Inc.

[14] Guttman, I., Wilks, S. and J. Hunter. 1982. Introductory Engineering

Statistics. 3rd ed. New York: John Wiley and Sons.

[15] Montgomery, D.C. and G.C. Runger. 1994. Applied Statistics and Prob-

ability for Engineers. New York: John Wiley and Sons.

227

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Uncertainty Analysis

CONTENTS

7.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

7.3 Comparing Theory and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

7.4 Uncertainty as an Estimated Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

7.5 Systematic and Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

7.6 Measurement Process Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

7.7 Quantifying Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7.8 Measurement Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

7.9 General Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

7.9.1 Single-Measurement Measurand Experiment . . . . . . . . . . . . . . . . . . . . . 244

7.9.2 Single-Measurement Result Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 250

7.10 Detailed Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

7.10.1 Multiple-Measurement Measurand Experiment . . . . . . . . . . . . . . . . . . . 261

7.10.2 Multiple-Measurement Result Experiment . . . . . . . . . . . . . . . . . . . . . . . 262

7.11 Uncertainty Analysis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

7.12 *Finite-Diﬀerence Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7.12.1 *Derivative Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

7.12.2 *Integral Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

7.12.3 *Uncertainty Estimate Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

7.13 *Uncertainty Based upon Interval Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

7.14 Problem Topic Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.15 Review Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.16 Homework Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Whenever we choose to describe some device or process with mathematical

equations based on physical principles, we always leave the real world behind,

to a greater or lesser degree. ... These approximations may, in individual

cases, be good, fair, or poor, but some discrepancy between modeled and real

behavior always exists.

E.O. Doebelin. 1995. Engineering Experimentation. New York: McGraw-Hill.

A measurement result is complete only if it is accompanied by a quantitative

expression of its uncertainty. The uncertainty is needed to judge whether the

result is adequate for its intended purpose and whether it is consistent with

other similar results.

Ferson, S., Kreinovich, V., Hajagos, J., Oberkampf, W. and Ginzburg, L. 2007.

Experimental Uncertainty Estimation and Statistics for Data Having Interval

Uncertainty. SAND2007-0939. Albuquerque: Sandia National Laboratories.

229

230 Measurement and Data Analysis for Engineering and Science

7.1 Chapter Overview

Uncertainty is one part of life that cannot be avoided. Its presence is a con-

stant reminder of our limited knowledge and inability to control each and

every factor that inﬂuences us. This especially holds true in the physical

sciences. Whenever a process is quantiﬁed, by either modeling or experi-

ments, uncertainty is present. In the beginning of this chapter the uncertain-

ties present in modeling and experiments are identiﬁed. Agreement between

modeling and experiment is described in the context of uncertainty. Mea-

surement uncertainties are studied in detail. Conventional methods on how

to characterize, quantify, and propagate them are presented. The generic

cases of single or multiple measurements of a measurand or a result are con-

sidered. Then, numerical uncertainties associated with measurements are

discussed. Finally, uncertainty that results from a lack of knowledge about

a variable’s speciﬁc value is considered.

7.2 Uncertainty

Aristotle ﬁrst addressed uncertainty over 2300 years ago when he pondered

the certainty of an outcome. However, it was not until the late 18th cen-

tury that scientists considered the quantiﬁable eﬀects of errors in measure-

ments [1]. Continual progress on characterizing uncertainty has been made

since then. Within the last 55 years, various methodologies for quantifying

measurement uncertainty have been proposed [2]. In 1993, an international

experimental uncertainty standard was developed by the International Orga-

nization for Standards (ISO) [3]. Its methodology now has been adopted by

most of the international scientiﬁc community. In 1997 the National Con-

ference of Standards Laboratories (NCSL) produced a U.S. guide almost

identical to the ISO guide [4], “to promote consistent international methods

in the expression of measurement uncertainty within U.S. standardization,

calibration, laboratory accreditation, and metrology services.” The Ameri-

can National Standards Institute with the American Society of Mechanical

Engineers (ANSI/ASME) [5] and the American Institute of Aeronautics and

Astronautics (AIAA) [6] also have new standards that follow the ISO guide.

These new standards diﬀer from the ISO guide only in that they use diﬀer-

ent names for the two categories of errors, random and systematic instead

of Type A and Type B, respectively.

How is uncertainty categorized in the physical sciences? Whenever a

physical process is quantiﬁed, uncertainties associated with modeling and

computer simulation and/or with measurements can arise. Modeling and

simulation uncertainties occur during the phases of “conceptual modeling

Uncertainty Analysis 231

of the physical system, mathematical modeling of the conceptual model,

discretization and algorithm selection for the mathematical model, com-

puter programming of the discrete model, numerical solution of the com-

puter program model, and representation of the numerical solution” [7].

Such predictive uncertainties can be subdivided into modeling and numer-

ical uncertainties [10]. Modeling uncertainties result from the assumptions

and approximations made in mathematically describing the physical pro-

cess. For example, modeling uncertainties occur when empirically based or

simpliﬁed sub-models are used as part of the overall model. Modeling un-

certainties perhaps are the most diﬃcult to quantify, particularly those that

arise during the conceptual modeling phase. Numerical uncertainties occur

as a result of numerical solutions to mathematical equations. These include

discretization, round-oﬀ, non-convergence, artiﬁcial dissipation, and related

uncertainties. No standard for modeling and simulation uncertainty has been

established internationally. Experimental or measurement uncertainties are

inherent in the measurement stages of calibration and data acquisition. Nu-

merical uncertainties also can occur in the analysis stage of the acquired

data.

The terms uncertainty and error each have diﬀerent meanings in mod-

eling and experimental uncertainty analysis. Modeling uncertainty is de-

ﬁned as a potential deﬁciency due to a lack of knowledge and modeling

error as a recognizable deﬁciency not due to a lack of knowledge [7]. Ac-

cording to Kline [8], measurement error is the diﬀerence between the

true value and the measured value. It is a speciﬁc value. Measurement

uncertainty is an estimate of the error in a measurement. It represents a

range of possible values that the error might assume for a speciﬁc measure-

ment. Additional uncertainty can arise because of a lack of knowledge of a

speciﬁc measurand value within an interval of possible values, as described

in Section 7.13.

By convention, the reported value of x is expressed with the same pre-

cision as its uncertainty, U

, such as 1.25 ± 0.05. The magnitude of U

depends upon the assumed conﬁdence, the uncertainties that contribute to

, and how the contributing uncertainties are combined. The approach

taken to determine U

involves adopting an uncertainty standard, such as

that presented in the ISO guide, identifying and categorizing all of the con-

tributory uncertainties, assuming a conﬁdence for the estimate, and then,

ﬁnally, combining the contributory uncertainties to determine U

. The types

of error that contribute to measurement uncertainty must be identiﬁed ﬁrst.

The remainder of this chapter focuses primarily on measurement uncertainty

analysis. Its associated numerical uncertainties are considered near the end

of this chapter.

232 Measurement and Data Analysis for Engineering and Science

FIGURE 7.1

Graphical presentation of a comparison between predictive and experimental

results.

7.3 Comparing Theory and Measurement

Given that both modeling and experimental uncertainties exist, what does

agreement between the two mean? How should such a comparison be illus-

trated? Conventionally, the experimental uncertainty typically is denoted

in a graphical presentation by error bars centered on measurement values

and the modeling uncertainty by dashed or dotted curves on both sides of

the theoretical curve. Both uncertainties should be estimated with the same

statistical conﬁdence. An example is shown in Figure 7.1. When all data

points and their error bars are within the model’s uncertainty curves, then

the experiment and theory are said to agree completely within the assumed

conﬁdence. When some data points and either part or all of their error bars

lay within the predictive uncertainty curves, then the experiment and the-

ory are said to agree partially within the assumed conﬁdence. There is no

agreement when all of the data points and their error bars are outside of

the predictive uncertainty curves.

Does agreement between experiment and theory imply that both cor-

rectly represent the process under investigation? Not all of the time. Cau-

tion must be exercised whenever such comparisons are made. Agreement

between theory and experiment necessarily does not imply correctness. This

Uncertainty Analysis 233

issue has been addressed by Aharoni [11], who discusses the interplay be-

tween good and bad theory and experiments. There are several possibilities.

A bad theory can agree with bad data. The wrong theory can agree with a

good experiment by mere coincidence. A correct theory may disagree with

the experiment simply because an unforeseen variable was not considered or

controlled during the experiment. Therefore, agreement does not necessarily

assure correctness.

Caution also must be exercised when arguments are made to support

agreement between theory and experiment. Scientiﬁc data can be misused

[12]. The data may have been presented selectively to ﬁt a particular hypoth-

esis while ignoring other hypotheses. Indicators may have been chosen with

units to support an argument, such as the number of automobile deaths per

journey instead of the number per kilometer travelled. Inappropriate scales

may have been used to exaggerate an eﬀect. Taking the logarithm of a vari-

able appears graphically as having less scatter. This can be deceptive. Other

illogical mistakes may have been made, such as confusing cause and eﬀect,

and implicitly using unproven assumptions. Important factors and details

may have been neglected that could lead to a diﬀerent conclusion. Caveat

emptor!

Example Problem 7.1

Statement: Lemkowitz et al. [12] illustrate how diﬀerent conclusions can be drawn

from the same data. Consider the number of fatalities per 100 million passengers for

two modes of transport given in the 1992 British fatality rates. For the automobile,

there were 4.5 fatalities per journey and 0.4 fatalities per km. For the airplane, there

were 55 fatalities per journey and 0.03 fatalities per km. Therefore, on a per km basis

airplane travel had approximately 10 times fewer fatalities than the automobile. Yet,

on a per journey basis, the auto had approximately 10 times fewer fatalities. Which of

the two modes of travel is safer and why?

Solution: There is no unique answer to this question. In fact, on a per hour basis,

the automobile and airplane have the the same fatality rate, which is 15 fatalities

per 100 million passengers. Perhaps driving for shorter distances and ﬂying for longer

distances is safer than the converse.

7.4 Uncertainty as an Estimated Variance

When measurements are made under ﬁxed conditions, the recorded values

of a variable still will vary to an extent. This implicitly is caused by small

variations in uncontrolled variables. The extent of these variations can be

characterized by the variance of the values. Further, as the number of ac-

quired values becomes large (N ≥ 30), the sample variance approaches its

true variance, σ

, and the distribution evolves to a normal distribution.

234 Measurement and Data Analysis for Engineering and Science

Thus, if uncertainty is considered to represent the range within which an

acquired value will occur, then uncertainty can be viewed as an estimate of

the true variance of a normal distribution.

Consider the most general situation of a result, r, where r = r(x

, x

, ...)

and x

, x

, ..., represent measured variables whose distributions are normal.

In uncertainty analysis, a result is deﬁned as a variable that is not measured

but is related functionally to variables that are measured, which are called

measurands. The result’s variance is deﬁned as

= lim

N→∞

i=1

− r

)

, (7.1)

where N is the number of determinations of r based upon the set of x

, x

, ...

measurands. The diﬀerence between a particular r

value and its mean value,

, can be expressed in terms of a Taylor series expansion and the measur-

ands’ diﬀerences by

− r

) ' (x

− x

)



∂r

∂x



+ (x

− x

)



∂r

∂x



+ ... . (7.2)

In this equation the higher order terms involving second derivatives and

beyond are assumed to be negligible. Equation 7.2 can be substituted into

Equation 7.1 to yield

' lim

N→∞

i=1



− x

)



∂r

∂x



+ (x

− x

)



∂r

∂x



+ ...



' lim

N→∞

i=1

[(x

− x

)



∂r

∂x



+ (x

− x

)



∂r

∂x



+2(x

− x

)(x

− x

)



∂r

∂x



∂r

∂x



+ ...]. (7.3)

The ﬁrst two terms on the right side of Equation 7.3 are related to the

variances of x

and x

, where

= lim

N→∞

i=1

− x

)

(7.4)

and

= lim

N→∞

i=1

− x

)

. (7.5)

The third term is related to the covariance of x

and x

, σ

, where

= lim

N→∞

i=1

− x

)(x

− x

)

. (7.6)

Uncertainty Analysis 235

When x

and x

are statistically independent, σ

= 0. Substituting Equa-

tions 7.4, 7.5, and 7.6 into Equation 7.3 gives

' σ



∂r

∂x



+ σ



∂r

∂x



+ 2σ



∂r

∂x



∂r

∂x



+ ... . (7.7)

Equation 7.7 can be extended to relate the uncertainty in a result as a

function of measurand uncertainties. Deﬁning the squared uncertainty u

an estimate of the variance σ

, Equation 7.7 becomes

' u



∂r

∂x



+ u



∂r

∂x



+ 2u



∂r

∂x



∂r

∂x



+ ... . (7.8)

This equation shows that the uncertainty in the result is a function of the

estimated variances (uncertainties) u

and u

and their estimated covari-

ance u

. It forms the basis for more detailed uncertainty expressions that

are developed in the remainder of this chapter and used to estimate the

overall uncertainty in a variable. u

is called the combined estimated

variance. The combined standard uncertainty is u

. This is denoted

by u

for a result and by u

for a measurand. In order to determine the

combined standard uncertainty, the types of errors that contribute to the

uncertainty must be examined ﬁrst.

7.5 Systematic and Random Errors

When a single measurement is performed, a number is assigned that repre-

sents the magnitude of the sensed physical variable. Because the measure-

ment system used is not perfect, an error is associated with that number. If

the system’s components have been calibrated against more accurate stan-

dards, these standards have their own inaccuracies. The act of calibration

itself introduces further uncertainty. All these factors contribute to the mea-

surement uncertainty of a single measurement. Further, when the measure-

ment is repeated, its value most likely will not be the same as it was the

ﬁrst time. This is because small, imperceptible changes in variables that af-

fect the measurement have occurred in the interim, despite any attempts to

perform a controlled experiment. Fortunately, almost all experimental un-

certainties can be estimated, provided there is a consistent framework that

identiﬁes the types of uncertainties and establishes how to quantify them.

The ﬁrst step in this process is to identify the types of errors that give rise

to measurement uncertainty.

Following the convention of the 1998 ANSI/ASME guidelines [5], the

errors that arise in the measurement process can be categorized into either