Table Of ContentManfredDrosg
DealingwithUncertainties
Manfred Drosg
Dealing with
Uncertainties
A Guide to Error Analysis
With24Figures
123
Univ.Prof.Dr.ManfredDrosg
UniversitätWien
Strudlhofgasse4
A-1090Wien,Austria
Manfred.Drosg@univie.ac.at
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I dedicate this book to my American friends,
in particular to those I met in New Mexico,
and to Peter Weinzierl,
who patronized me by arranging for me,
with the magnificent support
of R.F. Taschek, former P-Division leader of LANL,
to become the first foreign postdoctoral fellow at LANL.
Preface
For many people uncertainty as it occurs in the scientific context is still a
matter of speculation and frustration. One of the reasons is that there are
several ways of approaching this subject, depending upon the starting point.
The theoretical part has been well established over centuries. However, the
application of this knowledge on empirical data, freshly produced (e.g., by
an experiment) or when evaluating data, can often present a problem. In
some cases this is triggered by the word “error” that is an alternative term
for uncertainty. For many the word error means something that is wrong.
However,aswillbeshown,anuncertaintyisjustonecharacteristicofscientific
dataanddoesnotindicatethatthesedataarewrong.Toavoidanyassociation
with something being wrong,the term erroris avoidedin this book whenever
possible, and the term uncertainty is used instead. This appears to be in
agreement with the general tendency in modern science.
The philosopher Sir Karl Popper made it clear that any scientific truth
is uncertain. Usually, uncertainty is mentally associated only with measured
data for which an “error analysis” is mandatory, as many know. This makes
peoplethinkthatuncertaintyhasonlytodowithmeasurements.However,all
scientific truths, even predictions of theories and of computer models, should
beassigneduncertainties.Whereastheuncertaintyofmeasureddataisrather
easy to determine, it is too difficult, if not impossible, to establish reliable
uncertainties for theoretical data of either origin.
Thus this book deals mainly with uncertainties of empirical data, even
if much of it is applicable in a more general way. In particular, I want to
promote a deeper understanding of the phenomenon of uncertainty and to
removeatleasttwomajorhurdlesenroute.Oneistoemphasizetheexistence
of internal uncertainties. Usually only external uncertainties are considered
because they are the direct result of the theoretical approach. The former
arethe resultofa deductive approachto uncertainties,whereasthe latter are
obtained inductively. The other hurdle is the so-called systematic error. This
term is not used unambiguously, giving cause to many misunderstandings. It
is used both for correlated (or systematic) uncertainties and for systematic
VIII Preface
deviations of data. The latter just means that these data are wrong, that is,
thattheyshouldhavebeencorrectedforthatdeviation.Thereareevenbooks
in which both meanings are intermingled!
Not using the term error will make such misconceptions less likely. So I
speak of uncorrelated uncertainty instead of random error, and of correlated
uncertainty (and of systematic deviation, respectively) instead of systematic
error. In addition, it will be shown that these two types of uncertainties are
ofthe same nature.Thus a remarktakenfroma morerecentbooklike “there
is no evidence that you cannot treat random and systematic errors the same
way” is self-evident.
Myfirstinterestinthesubjectofthisbookgoesbackto1969,whenNelson
(Bill)JarmieatLosAlamosNationalLaboratory,USA,who wasapioneerin
accurate measurements of cross sections, introduced me to various subtleties
inthis field.Iamindebtedto himfor manyinsights.Notsurprisingly,quite a
few examples dealwith nuclear physics.In this field (and in electronics)I am
most experienced and, even more important, uncertainties of data based on
radioactive decay can easily be determined both deductively and inductively.
The essence of this book is found already in work sheets that I prepared
for undergraduate students in an advanced practical physics course when it
became clear that nothing like it was available in either German or English
books. This lack is the reason for not including a reference list.
Students and colleagues have contributed by asking the right questions,
my colleague Prof. Gerhard Winkler by way of enlightening discussions and
very valuable suggestions and M.M. Steurer, MS, by reporting a couple of
mistakes. My sincere thanks to all of them.
I sincerely urge my readers to contact me at Manfred.Drosg@univie.ac.at
whenever they can reporta mistake or wantto suggestsome additionaltopic
to be included in this book. Any such corrections or additions I will post at
http://homepage.univie.ac.at/Manfred.Drosg/uncertaintybook.htm.
Vienna, September 2006 Manfred Drosg
Foreword by the Translator
Myfirstcontactwiththetopic“uncertainties”datesbacktomyfirstpractical
physics courseatthe university.The theory andpracticalprocedurewerenot
explained very well. I was quite confused, so I asked my dad (M. Drosg) to
explain it to me, and that helped! Now, several years later, he asked me to
translate the German version of this book into English to make the answers
to those questions that bugged me (early on in my studies) available to a
greater number of people. This was a great idea and quite a challenge for
me! Although I am US-born, I spent only a little time in American schools,
but, several months at Los Alamos National Laboratory, where I worked as
a summer student. My mentor during this time was Robert C. Haight, who
taught me science in English—I am very thankful for this great support!
Nevertheless, the translation work was not always easy, so it was a great
help thatIcouldrely onmy dadfordouble-checkingthe text, andfor finding
the correct technical term when I was not sure. Although the aim was a full
and correct translation of the German original, it is not unlikely that a few
mistakes escaped the multiple proofreadings. I apologize for that.
Inparticular,IwanttothankAliceC.Wynne,Albuquerque,NewMexico,
a long-time family friend, for her thorough proofreading of the manuscript.
Vienna, September 2006 Roswitha Drosg
Prolog. Seven Myths in Error Analysis
Myth 1. Randomerrorscanalwaysbedeterminedbyrepeatingmeasurements
under identical conditions.
Althoughwehaveshowninonecase(Problem6.3.)thattheinductiveandthe
deductive methodprovidepracticallythe samerandomerrors,this statement
is true only for time-related random errors (Sect.6.2.5).
Myth 2. Systematic errors can be determined inductively.
It should be quite obvious that it is not possible to determine the scale error
from the pattern of data values (Sect.7.2.4).
Myth 3. Measuring is the cause of all errors.
Thestandardexampleofrandomerrors,measuringthecountrateofradiation
from a radioactivesource repeatedly,is notbased onmeasurementerrorsbut
on the intrinsic properties of radioactive sources (Sect.6.2.1). Usually, the
measurement contribution to this error is negligible.
Justasradiationhazardismostfearedofallhazardsbecauseitisbestun-
derstood,measurementsarethoughttobetheintrinsiccauseoferrorsbecause
their errors are best understood.
Myth 4. Counting can be done without error.
Usually, the counted number is an integer and therefore without (rounding)
error.However,the bestestimateofascientificallyrelevantvalueobtainedby
counting will always have an error. These errors can be very small in cases
of consecutive counting, in particular of regular events, e.g., when measuring
frequencies (Sect.2.1.4).
Myth 5. Accuracy is more important than precision.
For single best estimates, be it a mean value or a single data value, this
question does not arise because in that case there is no difference between
accuracy and precision. (Think of a single shot aimed at a target, Sect.7.6.)
Generally, it is good practice to balance precision and accuracy. The actual
requirements will differ from case to case.
XII Prolog. Seven Myths in Error Analysis
Myth 6. It is possible to determine the sign of an error.
It is possible to find the signed deviation of an individual data value but the
sign of the error of a best estimate, be it systematic or random, cannot be
determined because the true value cannot be known (Sect.7.2.1). The use of
the term systematic error for a systematic deviation is misleading because a
deviation is not an uncertainty at all.
Myth 7. It is all right to “guess” an error.
Theuncertainty(theerror)isoneofthecharacteristicsofabestestimate,just
likeitsvalue,andnearlyasimportant.Correcterroranalysissavesmeasuring
time and total cost. A factual example for that is given (Sect.10.1.1) where
correct error analysis could have saved 90% of the cost.
Contents
1 Introduction............................................... 1
1.1 The Exactness of Science ................................. 2
1.2 Data Without Uncertainty................................ 4
2 Basics on Data ............................................ 7
2.1 What Is a Measurement?................................. 8
2.2 Analog vs. Digital ....................................... 11
2.3 Dealing With Data (Numerals)............................ 12
3 Basics on Uncertainties (Errors) ........................... 15
3.1 Typical Sources of Internal Uncertainties ................... 16
3.2 Definitions.............................................. 17
3.3 Uncertainty of Data Depending on One Variable ............ 26
3.4 Multiple Uncertainty Components (Quadratic Sum) ......... 27
4 Radioactive Decay, a Model for Random Events ........... 33
4.1 Time Interval Distribution of Radioactive Events ............ 33
4.2 Inductive Approach to Uncertainty (Example) .............. 39
5 Frequency and Probability Distributions................... 51
5.1 Frequency Distribution (Spectrum) ........................ 51
5.2 Probability Distributions ................................. 58
5.3 Statistical Confidence .................................... 65
5.4 Dealing with Probabilities ................................ 66
6 Deductive Approach to Uncertainty ....................... 71
6.1 Theoretical Situation .................................... 71
6.2 Practical Situation....................................... 71
6.3 Regression Analysis (Least-Squares Method) ................ 81
6.4 Data Consistency Within Data Sets........................ 87
