Measurements and their Uncertainties: A practical guide to modern error analysis

153 Pages·2010·2.862 MB·English

by  Ifan Hughes, Thomas Hase

Uncertainties in single-variable
functions
The functional approach to obtain the uncertainty in a function Z = f (A), Z=f(A)
when Ahasbeenmeasuredto(cid:2)be(cid:3)A¯±αA:(cid:4) (cid:3) (cid:4)(cid:2) f(A¯+αA)
αZ =(cid:2)f A¯+αA − f A¯ (cid:2). f(A¯)
Thisisshownschematicallyinthefigure.
Assuming small uncertainties, such that the function can be approximated
asastraightlineinthevicinityof A¯,thenthecalculus-basedapproximationto A¯ A¯+αA
thisresultis:
(cid:2) (cid:2)
αZ =(cid:2)(cid:2)(cid:2)dZ(cid:2)(cid:2)(cid:2) αA. AdinrecutnlyceinrttaoinatnyuαncAeritnainthtyeαvZariinabthleefAunmctaiopns
dA Z = f(A).
Table1 Results for the propagation of uncertainties in single-variable func-
tions. The results for the trigonometric functions assume that the angles and
theiruncertaintiesareinradians.
dZ
Function, Z(A) Uncertainty
dA
(cid:2) (cid:2) (cid:2) (cid:2)
1 1 α (cid:2)α (cid:2) (cid:2)α (cid:2)
− α = A = Z2α OR(cid:2) Z(cid:2)=(cid:2) A(cid:2)
A A2 z A2 A Z A
expA expA α =expAα = Z α
z A A
1 α
lnA α = A
z
A A
1 α
logA α = A
An nlnA(1n−0)1 A αz =(cid:2)(cid:2)lnnA(1n0−)1(cid:2)(cid:2)Aα OR(cid:2)(cid:2)(cid:2)αZ(cid:2)(cid:2)(cid:2)=(cid:2)(cid:2)(cid:2)nαA(cid:2)(cid:2)(cid:2)
z A
Z A
10A 10Aln(10) α =10A ln(10)α
z A
sinA cosA α =|cosA| α
z A
cosA −sinA α =|sinA| α
z A
(cid:3) (cid:4)
tanA 1+tan2 A α = 1+Z2 α
z A
MEASUREMENTSANDTHEIRUNCERTAINTIES
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Measurements and their
Uncertainties
A practical guide to modern error analysis
IFAN G. HUGHES
PhysicsDepartment,DurhamUniversity
THOMAS P. A. HASE
PhysicsDepartment,WarwickUniversity
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Preface
Thisbookgrewoutoftheprocessofrevampingthefirst-yearpracticalcourse
at the physics department at Durham University. During the restructuring of
the laboratory course, we noted the significant changes in the skill set of the
studentcohortoverthepreviousdecade.Wesearchedforabookthatcouldbe
arecommendedtextforthetreatmentofuncertainties.Thereisnoshortageof
bookswhichdealwithuncertaintiesinmeasurementsanderroranalysis.Most
of these books treat error analysis in the traditional, old-fashioned approach
whichdoesnottakeintoaccountmoderndevelopments—indeed,errorpropa-
gationisoftentreatedasanexerciseincalculusofmanyvariables.Inmodern
laboratories computers are used extensively for data taking and analysis, and
studentsnowhaveaccessto,andfamiliaritywith,spreadsheetsformanipula-
tionofdata.Thisbookiswrittenassumingthatmostofthenumbercrunching
will be done by computer. Traditional textbooks have appendixes which list,
e.g.Gaussianintegrals,cumulativedistributionfunctions,tablesofchi-squared
distributionlikelihoods,andsoon.Ouremphasisisoncalculatingtherelevant
numberswithinroutinelyavailablespreadsheetsoftwarepackages.Incontrast
totraditionalbooks,wehavedecidedtoproduceahands-onguidebook:key
points illustrated with worked examples, concise and in a handy format—
sufficiently user-friendly that students actually bring the book along and use
itintheteachinglaboratory.
The scope of this book is to cover all the necessary groundwork for lab-
oratory sessions in a first- and second-year undergraduate physics laboratory
and to contain enough material to be useful for final-year projects, graduate
studentsandpractisingprofessionalscientistsandengineers.
In contrast to the mathematically rigorous treatment of other textbooks
we have adopted a ‘rule of thumb’ approach, and encourage students to use
computerstoassistwithasmanyofthecalculationsaspossible.Lastcentury
if a student had knowledge of an angle and its uncertainty, and was testing
thevalidityofRutherford’sscatteringlawwithitssin−4(θ/2)dependence,the
suggested approach to error propagation was to turn this into an exercise in
differentiation. Nowadays, a student can create a spreadsheet, calculate the
valueofthefunctionatthedesiredangle,attheangle+errorbar,anddeducethe
uncertaintyinpropagationthroughRutherford’sformulamuchmorequickly.
Throughout the book we encourage a functional approach to calculations, in
preference to the calculus-based approximation. A large number of end-of-
chapterexercisesareincluded,aserroranalysisisaparticipation,ratherthana
spectator,sport.
The only prerequisite is suitably advanced mathematics (an A-level, or
equivalent) which is a compulsory qualification for studying physics and
viii Preface
engineering at most universities. Although written originally for use in a
physicslaboratorywebelievethebookwouldbeausefulhands-onguidefor
undergraduatesstudyingthephysicalsciencesandengineering.
A preliminary version of this book (‘the little red book’) was used by
more than 1500 students following the Discovery Skills module at Durham
University; we are grateful to all of those who helped identify and eradicate
typographicalerrors,inconsistenciesandsourcesofconfusion.
Several colleagues involved in the teaching laboratories at Durham have
contributedtothisbookviaanumberofdiscussions.Wewouldliketorecord
ourgratitudetoDelAtkinson(werealisethebookis‘fulloferrors’),Richard
Bower, David Carty, Paula Chadwick, Simon Cornish, Graham Cross, Nigel
Dipper,KenDurose,DavidFlower,DouglasHalliday,MichaelHunt,Gordon
Love, John Lucey, Lowry McComb, Simon Morris, Robert Potvliege, Steve
Rayner, Marek Szablewski, and Kevin Weatherill. Many colleagues kindly
donatedtheirtimetoproofreadvariouschapters,andweareindebtedtothem
forthisservice,includingCharlesAdams,MatthewBrewer,PaulaChadwick,
Stewart Clark, Malcolm Cooper, David Flower, Patrick Hase, Jony Hudson,
Martin Lees, Phillip Petcher, Jon Pritchard, and Marek Szablewski. Simon
Gardiner gave invaluable TEX advice. For any remaining flaws or lack of
clarity the authors alone are responsible. While performing error analysis in
research, the authors have benefited from discussions with Ed Hinds, Jony
Hudson, Clemens Kaminski, Ben Sauer, Peter Smith, Derek Stacey, and
DuncanWalker.DamianHampshirepointedoutthat‘truth’shouldbesought
in the department of theology, not an error-analysis book. Simon Cornish
and Lara Bogart kindly provided us with data for some of the figures. We
are grateful to Bethan Mair and Tony Shaw for advice on writing a book,
Mike Curry for providing food and lodging, and Stewart Clark and Marek
Szablewskiforeradicatingthelonelinessoflong-distanceruns.
We would like to thank our families for support and encouragement, and
SönkeAdlungandAprilWarmanatOUPfortheirenthusiasmandpatience.
December2009 DurhamandWarwick