studio measurement and uncertainty analysis · measurements & uncertainty analysis 4 university...
TRANSCRIPT
Measurements & Uncertainty Analysis
1 Department of Physics and Astronomy
MeasurementandUncertaintyAnalysisGuide
“Itisbettertoberoughlyrightthanpreciselywrong.”–AlanGreenspan
TableofContentsTHEUNCERTAINTYOFMEASUREMENTS..............................................................................................................2RELATIVE(FRACTIONAL)UNCERTAINTY.............................................................................................................4RELATIVEERROR.......................................................................................................................................................5TYPESOFUNCERTAINTY..........................................................................................................................................6ESTIMATINGEXPERIMENTALUNCERTAINTYFORASINGLEMEASUREMENT................................................9ESTIMATINGUNCERTAINTYINREPEATEDMEASUREMENTS...........................................................................9STANDARDDEVIATION..........................................................................................................................................12STANDARDDEVIATIONOFTHEMEAN(STANDARDERROR).........................................................................14WHENTOUSESTANDARDDEVIATIONVSSTANDARDERROR......................................................................14ANOMALOUSDATA.................................................................................................................................................15BIASESANDTHEFACTOROFN–1.......................................................................................................................15SIGNIFICANTFIGURES............................................................................................................................................16UNCERTAINTY,SIGNIFICANTFIGURES,ANDROUNDING.................................................................................17PROPAGATIONOFUNCERTAINTY........................................................................................................................20THEUPPER-LOWERBOUNDMETHODOFUNCERTAINTYPROPAGATION...................................................20QUADRATURE..........................................................................................................................................................21COMBININGANDREPORTINGUNCERTAINTIES................................................................................................26CONCLUSION:“WHENDOMEASUREMENTSAGREEWITHEACHOTHER?”...................................................27MAKINGGRAPHS....................................................................................................................................................29USINGEXCELFORDATAANALYSISINPHYSICSLABS......................................................................................31GETTINGSTARTED.................................................................................................................................................31CREATINGANDEDITINGAGRAPH......................................................................................................................32ADDINGERRORBARS............................................................................................................................................33ADDINGATRENDLINE...........................................................................................................................................33DETERMININGTHEUNCERTAINTYINSLOPEANDY-INTERCEPT..................................................................34INTERPRETINGTHERESULTS...............................................................................................................................34FINALSTEP–COPYINGDATAANDGRAPHSINTOAWORDDOCUMENT.....................................................34USINGLINESTINEXCEL......................................................................................................................................35APPENDIXI:PROPAGATIONOFUNCERTAINTYBYQUADRATURE.................................................................39APPENDIXII.PRACTICALTIPSFORMEASURINGANDCITINGUNCERTAINTY............................................41
Measurements & Uncertainty Analysis
2 University of North Carolina
TheUncertaintyofMeasurementsSomenumericalstatementsareexact:Maryhas3brothers,and2+2=4.However,allmeasurementshavesomedegreeofuncertaintythatmaycomefromavarietyofsources.Theprocessofevaluatingtheuncertaintyassociatedwithameasurementresultisoftencalleduncertaintyanalysisorsometimeserroranalysis.Thecompletestatementofameasuredvalueshouldincludeanestimateofthelevelofconfidenceassociatedwiththevalue.Properlyreportinganexperimentalresultalongwithitsuncertaintyallowsotherpeopletomakejudgmentsaboutthequalityof the experiment, and it facilitates meaningful comparisons with other similarvaluesoratheoreticalprediction.Withoutanuncertaintyestimate,itisimpossibleto answer the basic scientific question: “Does my result agree with a theoreticalprediction or results from other experiments?” This question is fundamental fordecidingifascientifichypothesisisconfirmedorrefuted.Whenmakingameasurement,wegenerallyassume thatsomeexactor truevalueexistsbasedonhowwedefinewhatisbeingmeasured.Whilewemayneverknowthis true value exactly, we attempt to find this ideal quantity to the best of ourability with the time and resources available. As we make measurements bydifferent methods, or even whenmakingmultiple measurements using the samemethod,wemayobtainslightlydifferentresults.Sohowdowereportourfindingsforourbestestimateofthiselusivetruevalue?Themostcommonwaytoshowtherangeofvaluesthatwebelieveincludesthetruevalueis:
measurement=(bestestimate±uncertainty)unitsAsanexample,supposeyouwanttofindthemassofagoldringthatyouwouldliketoselltoafriend.Youdonotwanttojeopardizeyourfriendship,soyouwanttogetanaccuratemassoftheringinordertochargeafairmarketprice.Youestimatethemasstobebetween10and20gramsfromhowheavyitfeelsinyourhand,butthisisnotaverypreciseestimate.Aftersomesearching,youfindanelectronicbalancethat gives amass reading of 17.43 grams.While thismeasurement ismuchmoreprecise than the original estimate, how do you know that it isaccurate, and howconfident are you that this measurement represents the true value of the ring’smass? Since the digital display of the balance is limited to 2 decimal places, youcouldreportthemassasm=17.43±0.01g.Supposeyouusethesameelectronicbalance and obtain several more readings: 17.46 g, 17.42 g, 17.44 g, so that theaveragemass appears to be in the range of 17.44± 0.02 g. By now youmay feelconfidentthatyouknowthemassof thisringtothenearesthundredthofagram,buthowdoyouknowthatthetruevaluedefinitelyliesbetween17.43gand17.45g? Since you want to be honest, you decide to use another balance that gives areadingof17.22g.Thisvalueisclearlybelowtherangeofvaluesfoundonthefirstbalance,andundernormalcircumstances,youmightnotcare,butyouwant tobe
Measurements & Uncertainty Analysis
3 Department of Physics and Astronomy
fairtoyourfriend.Sowhatdoyoudonow?Theanswerliesinknowingsomethingabouttheaccuracyofeachinstrument.Tohelpanswerthesequestions,wefirstdefinethetermsaccuracyandprecision:
Accuracyistheclosenessofagreementbetweenameasuredvalueandatrueoracceptedvalue.Measurementerroristheamountofinaccuracy.Precisionisameasureofhowwellaresultcanbedetermined(withoutreferencetoatheoreticalortruevalue).Itisthedegreeofconsistencyandagreementamongindependentmeasurementsofthesamequantity;alsothereliabilityorreproducibilityoftheresult.
Theaccuracyandprecisioncanbepicturedasfollows:
highprecision,lowaccuracy lowprecision,highaccuracy
Figure1.AccuracyvsPrecision
Measurements & Uncertainty Analysis
4 University of North Carolina
Theuncertainty estimate associatedwith ameasurement should account for boththeaccuracyandprecisionof themeasurement.Precision indicates thequalityofthe measurement, without any guarantee that the measurement is “correct.”Accuracy, on the other hand, assumes that there is an ideal “true” value, andexpresseshow far your answer is from that “correct” answer.These concepts aredirectlyrelatedtorandomandsystematicmeasurementuncertainties(nextsection).Note:Unfortunatelythetermserroranduncertaintyareoftenusedinterchangeablyto describe both imprecision and inaccuracy. This usage is so common that it isimpossibletoavoidentirely.Wheneveryouencountertheseterms,makesureyouunderstandwhethertheyrefertoaccuracyorprecision,orboth.Inthisdocument,wewillemphasizetheterm“uncertainty”butwillusetheterm“error,”asnecessary,toavoidconfusionwithcommonlyfoundexamplesandstandardusageoftheterm.Inordertodeterminetheaccuracyofaparticularmeasurement,wehavetoknowthe ideal, truevalue, sometimes referred to as the “gold standard.” Sometimeswehavea“textbook”measuredvalue,whichiswellknown,andweassumethatthisisour“ideal”value,anduseit toestimatetheaccuracyofourresult.Othertimesweknowatheoreticalvalue,whichiscalculatedfrombasicprinciples,andthisalsomaybetakenasan“ideal”value.Butphysics isanempiricalscience,whichmeansthatthetheorymustbevalidatedbyexperiment,andnottheotherwayaround.Wecanescapethesedifficultiesandretainausefuldefinitionofaccuracybyassumingthat,evenwhenwedonotknowthetruevalue,wecanrelyonthebestavailableacceptedvaluewithwhichtocompareourexperimentalvalue.For thegoldringexample, there isnoacceptedvaluewithwhich tocompare,andbothmeasuredvalueshavethesameprecision,sothereisnoreasontobelieveonemorethantheother.Wecouldlookuptheaccuracyspecificationsforeachbalanceas provided by the manufacturer, but the best way to assess the accuracy of ameasurementistocompareitwithaknownstandard.Forthissituation,itmaybepossible to calibrate the balanceswith a standardmass that is accuratewithin anarrow tolerance and is traceable to a primary mass standard at the NationalInstitute of Standards and Technology (NIST). Calibrating the balances shouldeliminatethediscrepancybetweenthereadingsandprovideamoreaccuratemassmeasurement.
Relative(Fractional)UncertaintyPrecisionisoftenreportedquantitativelybyusingrelativeorfractionaluncertainty,givenbytheratiooftheuncertaintydividedbytheaveragevalue:
Relative Uncertainty = uncertaintymeasured quantity
(1)
Measurements & Uncertainty Analysis
5 Department of Physics and Astronomy
Therelativeuncertainty isdimensionlessbut isoftenreportedasapercentageor inparts permillion (ppm) to emphasize the fractional nature of the value. A scientistmightalsomakethestatementthatthismeasurement“isgoodtoabout1partin500”or“precisetoabout0.2%”.Therelativeuncertaintyisimportantbecauseitisusedinpropagating uncertainty in calculations using the result of a measurement, asdiscussedinalatersection.Forexample,m=75.5±0.5ghasarelativeuncertaintyof
%7.0600.05.755.0 ==gg
RelativeErrorAccuracyisoftenreportedquantitativelybyusingrelativeerror:
Relative Error = measured value - expected valueexpected value
(2)
Iftheexpectedvalueformis80.0g,thentherelativeerroris:CriticalNotes:
• Theminus sign indicates that themeasuredvalue is less than theexpectedvalue–unlessexplicitlystated,theterm“relativeerror”issignedanddoesnotinandofitselfrefertoamagnitude.
• The denominator is neither the measured value nor the average of themeasuredandexpectedvalue–therelativeerrorcanonlybecitedwhenthereisaknownexpectedvalueorgoldstandard.
%6.5056.00.800.805.75 −=−=−
Measurements & Uncertainty Analysis
6 University of North Carolina
TypesofUncertaintyMeasurement uncertainties may be classified as either random or systematic,depending on how themeasurement was obtained (an instrument could cause arandomuncertaintyinonesituationandasystematicuncertaintyinanother).Random uncertainties are statistical fluctuations (in either direction) in themeasureddata.Theseuncertaintiesmayhavetheirorigininthemeasuringdevice,orinthefundamentalphysicsunderlyingtheexperiment.Therandomuncertaintiesmaybemaskedbytheprecisionoraccuracyofthemeasurementdevice. Randomuncertainties can be evaluated through statistical analysis and can be reduced byaveraging over a large number of observations (see “standard error” later in thisdocument).Systematicuncertaintiesarereproducibleinaccuraciesthatareconsistentlyinthe“samedirection,”andcouldbecausedbyanartifactinthemeasuringinstrument,oraflawintheexperimentaldesign(becauseofthesepossibilities,itisnotuncommonto see the term “systematic error”). Theseuncertaintiesmaybedifficult to detectandcannotbeanalyzedstatistically.Ifasystematicuncertaintyorerrorisidentifiedwhen calibrating against a standard, applying a correction or correction factor tocompensate for the effect can reduce the bias. Unlike random uncertainties,systematicuncertaintiescannotbedetectedorreducedbyincreasingthenumberofobservations.When making careful measurements, the goal is to reduce as many sources ofuncertainty aspossible and tokeep trackof those that cannotbe eliminated. It isusefultoknowthetypesofuncertaintiesthatmayoccur,sothatwemayrecognizethem when they arise. Common sources of uncertainty in physics laboratoryexperimentsinclude:Incomplete definition (may be systematic or random) - One reason that it isimpossible to make exact measurements is that the measurement is not alwaysclearlydefined.Forexample,iftwodifferentpeoplemeasurethelengthofthesamestring, theywouldprobablygetdifferent resultsbecauseeachpersonmaystretchthestringwithadifferenttension.Thebestwaytominimizedefinitionuncertaintyis to carefully consider and specify the conditions that could affect themeasurement.Failuretoaccountforafactor(usuallysystematic)–Themostchallengingpartofdesigning an experiment is trying to control or account for all possible factorsexcept theone independentvariable that isbeinganalyzed.For instance,youmayinadvertently ignore air resistance when measuring free-fall acceleration, or youmayfailtoaccountfortheeffectoftheEarth’smagneticfieldwhenmeasuringthefieldnearasmallmagnet.Thebestwaytoaccountforthesesourcesofuncertaintyistobrainstormwithyourpeersaboutallthefactorsthatcouldpossiblyaffectyour
Measurements & Uncertainty Analysis
7 Department of Physics and Astronomy
result.Thisbrainstormshouldbedonebeforebeginningtheexperimentinordertoplan and account for the confounding factors before taking data. Sometimes acorrectioncanbeappliedtoaresultaftertakingdatatoaccountforanuncertaintythatwasnotdetectedearlier.Environmental factors (systematic or random) - Be aware of uncertaintyintroducedbytheimmediateworkingenvironment.Youmayneedtotakeaccountoforprotectyourexperimentfromvibrations,drafts,changesintemperature,andelectronicnoiseorothereffectsfromnearbyapparatus.Instrumentresolution(random)-Allinstrumentshavefiniteprecisionthatlimitsthe ability to resolve small measurement differences. For instance, a meter stickcannotbeusedtodistinguishdistancestoaprecisionmuchbetterthanabouthalfofitssmallestscaledivision(typically0.5mm).Oneof thebestwaystoobtainmoreprecise measurements is to use a null differencemethod instead of measuring aquantitydirectly.Nullorbalancemethodsinvolveusinginstrumentationtomeasurethe difference between two similar quantities, one of which is known veryaccurately and is adjustable. The adjustable reference quantity is varied until thedifference is reduced to zero. The two quantities are then balanced, and themagnitude of the unknown quantity can be found by comparison with ameasurement standard. With this method, problems of source instability areeliminated,andthemeasuringinstrumentcanbeverysensitiveanddoesnotevenneedascale.Thistypeofmeasurementismoresophisticatedandwilltypicallynotbeusedintheintroductoryphysicscourses.Calibration (systematic) –Whenever possible, the calibration of an instrumentshouldbecheckedbeforetakingdata.Ifacalibrationstandardisnotavailable,theaccuracy of the instrument should be checked by comparing with anotherinstrumentthatisatleastasprecise,orbyconsultingthetechnicaldataprovidedbythemanufacturer. Calibrationerrorsareusually linear (measuredasa fractionofthefullscalereading),sothatlargervaluesresultingreaterabsoluteerrors.Zerooffset(systematic)-Whenmakingameasurementwithamicrometercaliper,electronicbalance,orelectricalmeter,alwayscheckthezeroreadingfirst.Re-zerothe instrument if possible, or at leastmeasure and record the zero offset so thatreadings can be corrected later. It is also a good idea to check the zero readingthroughouttheexperiment. Failuretozeroadevicewillresultinaconstantoffsetthatismoresignificantforsmallermeasuredvaluesthanforlargerones.Physicalvariations(random)-Itisalwayswisetoobtainmultiplemeasurementsover the widest range possible. Doing so often reveals variations that mightotherwisegoundetected.Thesevariationsmaycallforcloserexamination,ortheymaybecombinedtofindanaveragevalue.
Measurements & Uncertainty Analysis
8 University of North Carolina
Parallax (systematic or random) - This error can occur whenever there is somedistance between the measuring scale and the indicator used to obtain ameasurement. If the observer’s eye is not squarely alignedwith the pointer andscale,thereadingmaybetoohighorlow(someanalogmetershavemirrorstohelpwiththisalignment).Instrument drift (systematic) - Most electronic instruments have readings thatdriftovertime.Theamountofdriftisgenerallynotaconcern,butoccasionallythissourceofuncertaintycanbesignificant.Lag time andhysteresis (systematic) - Somemeasuring devices require time toreach equilibrium, and taking ameasurement before the instrument is stablewillresult in a measurement that is too high or low. A common example is takingtemperaturereadingswithathermometerthathasnotreachedthermalequilibriumwithitsenvironment.Asimilareffectishysteresis,whereintheinstrumentreadingslag behind and appear to have a “memory” effect, as data are taken sequentiallymoving up or down through a range of values. Hysteresis is most commonlyassociatedwithmaterialsthatbecomemagnetizedwhenachangingmagneticfieldisapplied.Last but not least, some uncertainties are the result of carelessness, poortechnique, or bias on the part of the experimenter. The experimentermay use ameasuringdeviceincorrectly,ormayusepoortechniqueintakingameasurement,ormayintroduceabiasintomeasurementsbyexpecting(andinadvertentlyforcing)the results toagreewith theexpectedoutcome.Grossuncertaintiesof thisnaturecanbereferredtoasmistakesorblunders,andshouldbeavoidedandcorrected ifdiscovered.Asarule,theseuncertaintiesareexcludedfromanyuncertaintyanalysisdiscussion because it is generally assumed that the experimental result wasobtainedbyfollowingcorrectandwell-intentionedprocedures–thereisnopointtoperforminganexperimentandthenreportingthatitwasknowntobeincorrect.Thetermhumanerrorshouldbeavoidedinuncertaintyanalysisdiscussionsbecauseitistoogeneraltobeuseful.
Measurements & Uncertainty Analysis
9 Department of Physics and Astronomy
EstimatingExperimentalUncertaintyforaSingleMeasurementAny measurement will have some uncertainty associated with it, no matter theprecisionofthemeasuringtool.Howisthisuncertaintydeterminedandreported?Theuncertaintyofasinglemeasurementislimitedbytheprecisionandaccuracyofthemeasuringinstrument,alongwithanyotherfactorsthatmightaffecttheabilityoftheexperimentertomakethemeasurement.For example, if you are trying to use ameter stick tomeasure the diameter of atennis ball, the uncertaintymight be±5mm, but if you use a Vernier caliper, theuncertaintycouldbereducedtomaybe±2mm.The limitingfactorwiththemeterstickisparallax,whilethesecondcaseislimitedbyambiguityinthedefinitionofthetennisball’sdiameter(it’sfuzzy!).Inbothofthesecases,theuncertaintyisgreaterthanthesmallestdivisionsmarkedonthemeasuringtool(likely1mmand0.05mmrespectively). Unfortunately, there is no general rule for determining theuncertaintyinallmeasurements.Theexperimenteristheonewhocanbestevaluateandquantifytheuncertaintyofameasurementbasedonallthepossiblefactorsthataffecttheresult.Therefore,thepersonmakingthemeasurementhastheobligationtomakethebestjudgmentpossibleandreporttheuncertaintyinawaythatclearlyexplainswhattheuncertaintyrepresents:Measurement=(measuredvalue±standarduncertainty)(unitofmeasurement)
where “± standard uncertainty” indicates approximately a 68% confidenceinterval(seesectionsonStandardDeviationandReportingUncertainties).
Example:Diameteroftennisball=6.7±0.2cm
EstimatingUncertaintyinRepeatedMeasurementsSupposeyoutimetheperiodofoscillationofapendulumusingadigitalinstrument(that you assume is measuring accurately) and find that T = 0.44 seconds. Thissingle measurement of the period suggests a precision of ±0.005 s, but thisinstrument precision may not give a complete sense of the uncertainty, and youshouldavoidreportingtheuncertainty inthis fashionifpossible. Ifyourepeatthemeasurementseveraltimesandexaminethevariationamongthemeasuredvalues,youcangetabetterideaoftheuncertaintyintheperiod.Forexample,herearetheresultsof5measurements,inseconds:0.46,0.44,0.45,0.44,0.41.Forthissituation,thebestestimateoftheperiodistheaverage,ormean:
N
xxx N+++= ... (mean) Average 21
Measurements & Uncertainty Analysis
10 University of North Carolina
Whenever possible, repeat ameasurement several times and average the results.Thisaverageisgenerallythebestestimateofthe“true”value(unlessthedatasetisskewedbyoneormoreoutlierswhichshouldbeexaminedtodetermineiftheyarebaddatapointsthatshouldbeomittedfromtheaverageorvalidmeasurementsthatrequire further investigation). Generally, the more repetitions you make of ameasurement,thebetterthisestimatewillbe,butbecarefultoavoidwastingtimetakingmoremeasurementsthanisnecessaryfortheprecisionrequired.Consider, as another example, themeasurement of thewidth of a piece of paperusingameterstick.Beingcarefultokeepthemeterstickparalleltotheedgeofthepaper (to avoid a systematic error which would cause themeasured value to beconsistentlyhigherthanthecorrectvalue),thewidthofthepaperismeasuredatanumberofpointsonthesheet,andthevaluesobtainedareenteredinadatatable.Notethatthelastdigit isonlyaroughestimate,sinceit isdifficulttoreadametersticktothenearesttenthofamillimeter(0.01cm)–weretainthelastdigitfornowtomakeapointlater.
Observation Width(cm)#1 31.33#2 31.15#3 31.26#4 31.02#5 31.20
Table1.FiveMeasurementsoftheWidthofaPieceofPaper
Average = sum of observed widthsnumber of observations
= 155.96 cm5
= 31.19 cm
Thisaverageisthebestavailableestimateofthewidthofthepieceofpaper,butitisnot exact. We would have to average an infinite number of measurements toapproachthetruemeanvalue,andeventhen,wearenotguaranteedthatthemeanvalueisaccuratebecausethereisstill likelysomesystematicuncertaintyfromthemeasuringtool,whichisdifficulttocalibrateperfectlyunlessitisthegoldstandard.Sohowdoweexpresstheuncertaintyinouraveragevalue?Oneway toexpress thevariationamong themeasurements is touse theaveragedeviation. Thisstatistictellsusonaverage(with50%confidence)howmuchtheindividualmeasurementsvaryfromthemean.
The average deviation is a sufficient measure of uncertainty; however, it isimportant to understand the distribution of measurements. The Central Limit
Nxxxxxxd N ||...|||| Deviation, Average 21 −++−+−=
Measurements & Uncertainty Analysis
11 Department of Physics and Astronomy
Theoremproves thatas thenumberof independentmeasurements increases, andassumingthatthevariations inthesemeasurementsarerandom(i.e., therearenosystematic uncertainties), the distribution of measurements will approach thenormaldistribution,morecommonlyknownasabellcurve. Inthiscourse,wewillassumethatourmeasurements,performedinsufficientnumber,willproduceabellcurve(normal)distribution.Inthiscase,thestandarddeviationisanalternatewayto characterize the spread of the data. The standard deviation is always slightlygreater than the average deviation, and is used because of its mathematicalassociationwiththenormaldistribution.
Measurements & Uncertainty Analysis
12 University of North Carolina
StandardDeviationTocalculatethestandarddeviationforasampleofNmeasurements: 1.SumallthemeasurementsanddividebyNtogettheaverage,ormean. 2.SubtractthisaveragefromeachoftheNmeasurementstoobtainN“deviations.” 3.SquareeachoftheNdeviationsandaddthemtogether. 4.Dividethisresultby(N–1)andtakethesquareroot.Toconvertthisintoaformula,lettheNmeasurementsbecalledx1,x2,…,xN.LettheaverageoftheNvaluesbecalled x .Theneachdeviationisgivenby
xxx ii −=δ ,fori=1,2,...,NThestandarddeviationisthen:
( )( ) ( )11
... 2222
21
−=
−+++
= ∑Nx
Nxxx
s iN δδδδ
Inthemeterstickandpaperexample,theaveragepaperwidth x is31.19cm.Thedeviationsare:
Observation Width(cm) Deviation(cm)#1 31.33 +0.14 =31.33-31.19#2 31.15 -0.04 =31.15-31.19#3 31.26 +0.07 =31.26-31.19#4 31.02 -0.17 =31.02-31.19#5 31.20 +0.01 =31.20-31.19
Table1(completed).FiveMeasurementsoftheWidthofaSheetofPaper
Theaveragedeviationis: d =0.09cmThestandarddeviationis:The significance of the standard deviation is this: if you now make one moremeasurement using the samemeter stick, you can reasonably expect (with about68%confidence)thatthenewmeasurementwillbewithin0.12cmoftheestimatedaverage of 31.19 cm. In fact, it is reasonable to use the standarddeviation as theuncertaintyassociatedwiththissinglenewmeasurement.However,theuncertainty
cm 12.015
)01.0()17.0()07.0()04.0()14.0( 22222
=−
++++=s
Measurements & Uncertainty Analysis
13 Department of Physics and Astronomy
oftheaveragevalueisthestandarddeviationofthemean,whichisalwayslessthanthestandarddeviation(seenextsection).Consideranexampleof100measurementsofaquantity, forwhichtheaverageormean value is 10.50 and the standard deviation is s = 1.83. Figure 2 below is ahistogram of the 100 measurements, which shows how often a certain range ofvalueswasmeasured.Forexample,in20ofthemeasurements,thevaluewasintherange9.50to10.50,andmostofthereadingswereclosetothemeanvalueof10.50.Thestandarddeviations forthissetofmeasurementsisroughlyhowfarfromtheaveragevaluemost of the readings fell. Fora largeenoughsample, approximately68%ofthereadingswillbewithinonestandarddeviation(“1-sigma”)ofthemeanvalue,95%ofthereadingswillbeintheinterval x ±2s(“2-sigma”),andnearlyall(99.7%) of the readings will lie within 3 standard deviations (“3-sigma”) of themean.Thesmoothcurvesuperimposedonthehistogramisthenormaldistributionpredictedbytheoryformeasurementsinvolvingrandomerrors.Asmoreandmoremeasurementsaremade,thehistogramwillbetterapproximateabell-shapedcurve,butthestandarddeviationofthedistributionwillremainapproximatelythesame.
sx
→±← 1 sx
→±← 2 sx →±← 3 sx
Figure2.ANormalDistribution(BellCurve)Basedon100Measurements
Measurements & Uncertainty Analysis
14 University of North Carolina
StandardDeviationoftheMean(StandardError)When reporting the average valueofNmeasurements, theuncertainty associatedwith this average value is the standard deviation of the mean, often called thestandarderror(SE).
Standard Deviation of the Mean, or Standard Error (SE), σ x = sN (3)
Thestandarderrorissmallerthanthestandarddeviationbyafactorof N1 .Thisreflects the fact thatweexpect theuncertaintyof theaveragevaluetogetsmallerwhen we use a larger number of measurements. In the previous example, wedividedthestandarddeviationof0.12by√5toget thestandarderrorof0.05cm.Thefinalresultshouldthenbereportedas“averagepaperwidth=31.19±0.05cm.”Forthisexample,therelativeuncertaintywouldbe(0.05/31.19),or≈0.2%.
WhentoUseStandardDeviationvsStandardErrorForrepeatedmeasurements,thesignificanceofthestandarddeviationsisthatyoucanreasonablyexpect(withabout68%confidence)thatthenextmeasurementwillbe within s of the estimated average. It may be reasonable to use the standarddeviationas theuncertaintyassociatedwith thismeasurement;however, asmoremeasurementsaremade, thevalueof thestandarddeviationwillberefinedbut itwill not significantly decrease as the number of measurements is increased;therefore, it will not be the best estimate of the uncertainty of a set ofmeasurements. In contrast, if youare confident that the systematicuncertainty inyourmeasurement is very small, then it is reasonable to assume that your finitesample of all possiblemeasurements is not biased away from the “true” value. Inthis case, the uncertainty of the average value can be expressed as the standarddeviationofthemean,whichisalwayslessthanthestandarddeviationbyafactorof√N.
Figure3.StandardDeviationvsStandardError
Measurements & Uncertainty Analysis
15 Department of Physics and Astronomy
Ifyouarenotconfidentthatthesystematicuncertaintyinyourmeasurementisverysmall,theuncertaintythatshouldbereportedisthestandardcombineduncertainty(Uc)thatincludesallknownuncertaintyestimates(seesectiononCombiningandReportingUncertaintieslaterinthisdocument).
AnomalousDataThefirststepyoushouldtakeinanalyzingdata(andevenwhiletakingdata)istoexaminethedatasetasawholetolookforpatternsandoutliers.Anomalousdatapointsthatlieoutsidethegeneraltrendofthedatamaysuggestaninterestingphenomenonthatcouldleadtoanewdiscovery,ortheymaysimplybetheresultofamistakeorrandomfluctuations.Inanycase,anoutlierrequirescloserexaminationtodeterminethecauseoftheunexpectedresult.Extremedatashouldneverbe“thrownout”withoutclearjustificationandexplanation,becauseyoumaybediscardingthemostsignificantpartoftheinvestigation!However,ifyoucanclearlyjustifyomittinganinconsistentdatum,thenyoumayexcludetheoutlierfromyouranalysissothattheaveragevalueisnotskewedfromthe“true”mean.Thereareanumberofstatisticalmeasuresthathelpquantifythedecisiontodiscardoutliers,buttheyarebeyondthescopeofthisdocument.Beawareofthepossibilityofanomalousdata,andaddressthetopicasneededinthediscussionincludedwithalabreportorlabnotebook.
BiasesandtheFactorofN–1Youmayfinditsurprisingthatthebestvalue(average)iscalculatedbynormalizing(dividing)byN,whereasthestandarddeviationiscalculatedbynormalizingtoN–1.ThereasonisbecausenormalizingtoNisknowntounderestimatethecorrectvalueofthewidthofanormaldistribution,unlessNislarge.Thisunderestimateisreferredtoasabiasandistheresultofincompletesampling(thatis,thepopulationofmeasurements,orsample,fallsshortoftheentirepopulationofmeasurementsthatcouldbetaken),alsoknownas“samplingerror.”Ifthenumberofsamplesislessthanorabout10,eventheN–1term(knownasBessel’scorrection)isknowntoinduceabias.Determiningtheexactcorrectiontominimizeoreliminatebiasdependsonthedistributionofthedata,andthereisnosimpleexactequationthatcanbeapplied;however,forsmallsamplesizesthatarequitecommoninintroductoryphysicsclasses,acorrectionofN–1.5maybemoreappropriate.Ifyouuseacorrectionfactorof1.5inyourlabreports,youmustmakethisclearinyouranalysisandcitethisGuideasareference.
Measurements & Uncertainty Analysis
16 University of North Carolina
SignificantFigures Thenumberofsignificantfigures(sigfigs)inavaluecanbedefinedasallthedigitsbetweenand including the firstnon-zerodigit fromthe left, throughthe lastdigit.Forinstance,0.44hastwosigfigs,andthenumber66.770has5sigfigs.Zeroesaresignificantexceptwhenusedtolocatethedecimalpoint,asinthenumber0.00030,whichhas2 sig figs.Zeroesmayormaynotbe significant fornumbers like1200,whereitisnotclearwhethertwo,three,orfoursigfigsareindicated.Toavoidthisambiguity, suchnumbersshouldbeexpressed inscientificnotation(e.g.,1.20×103indicates3sigfigs).Whenusingacalculator,thedisplaywilloftenshowmanydigits,onlysomeofwhichare meaningful (significant in a different sense). For example, if you want toestimate theareaof a circularplaying field, youmightpaceoff the radius tobe9metersandusetheformulaA=πr2.Whenyoucomputethisarea,thecalculatorwillreportavalueof254.4690049m2.Itwouldbeextremelymisleadingtoreportthisnumberastheareaofthefield,becauseitwouldsuggestthatyouknowtheareatoanabsurddegreeofprecision–towithinafractionofasquaremillimeter!Sincetheradiusisonlyknowntoonesignificantfigure, it isconsideredbestpracticetoalsoexpressthefinalanswertoonlyonesignificantfigure:Area=3×102m2.Fromthisexample,wecanseethatthenumberofsignificantfiguresreportedforavalue impliesacertaindegreeofprecisionandcansuggestaroughestimateoftherelativeuncertainty: 1significantfiguresuggestsarelativeuncertaintyofabout10%to100% 2significantfiguressuggestarelativeuncertaintyofabout1%to10% 3significantfiguressuggestarelativeuncertaintyofabout0.1%to1%To understand this connection more clearly, consider a value with 2 significantfigures, like99,whichsuggestsanuncertaintyof±1,ora fractionaluncertaintyof±1/99=±1%(onemightarguethattheimplieduncertaintyin99is±0.5sincetherange of values that would round to 99 are 98.5 to 99.4; however, since theuncertainty here is only a rough estimate, there is not much point debating thefactor of two.) The smallest 2-significant-figure number, 10, also suggests anuncertainty of±1,which in this case is a fractional uncertainty of±1/10=±10%.The ranges for other numbers of significant figures can be reasoned in a similarmanner.Warning: thisprocedure isopen toawiderangeof interpretation; therefore,oneshould use caution when using significant figures to imply uncertainty, and themethod should only be used if there is no other better way to determineuncertainty. An explicit warning to this effect should accompany the use of thismethod.
Measurements & Uncertainty Analysis
17 Department of Physics and Astronomy
Subject to the above warning, significant figures can be used to find a possiblyappropriateprecisionforacalculatedresultforthefourmostbasicmathfunctions.
• Formultiplicationanddivision,thenumberofsignificantfiguresthatarereliablyknowninaproductorquotientisthesameasthesmallestnumberofsignificantfiguresinanyoftheoriginalnumbers.
Example: 6.6 (2significantfigures) ×7328.7 (5significantfigures) 48369.42=48x103 (2significantfigures)
• Foradditionandsubtraction,theresultshouldberoundedofftothelastdecimalplacereportedfortheleastprecisenumber.
Examples: 223.64 5560.5 +54 +0.008 278 5560.5Critical Note: if a calculated number is to be used in further calculations, it ismandatorytokeepguarddigitstoreduceroundingerrorsthatmayaccumulate.Thefinalanswercanthenberoundedaccordingtotheaboveguidelines.Thenumberofguarddigitsrequiredtomaintaintheintegrityofacalculationdependsonthetypeof calculation. For example, the number of guard digits must be larger whenperformingpowerlawcalculationsthanwhenadding.
Uncertainty,SignificantFigures,andRoundingFor the same reason that it is dishonest to report a result with more significantfiguresthanarereliablyknown, theuncertaintyvalueshouldalsonotbereportedwithexcessiveprecision.Forexample,itwouldbeunreasonabletoreportaresultinthefollowingway:
measureddensity=8.93±0.475328g/cm3 WRONG!Theuncertaintyinthemeasurementcannotpossiblybeknownsoprecisely!Inmostexperimentalwork, the confidence in theuncertainty estimate is notmuchbetterthanabout±50%becauseofallthevarioussourcesoferror,noneofwhichcanbeknownexactly.
Therefore,unlessexplicitlyjustified,uncertaintyvaluesshouldbestated(rounded)toonesignificantfigure.
Measurements & Uncertainty Analysis
18 University of North Carolina
Tohelpgiveasenseoftheamountofconfidencethatcanbeplacedinthestandarddeviation as a measure of uncertainty, the following table indicates the relativeuncertainty associatedwith the standard deviation for various sample sizes.Notethat in order for a standard deviation to be reported to a precision of 3 significantfigures,morethan10,000readingswouldberequired!
N RelativeUncertainty*
SignificantFiguresValid
ImpliedUncertainty
2 71% 1 ±10%to100%3 50% 1 ±10%to100%4 41% 1 ±10%to100%5 35% 1 ±10%to100%10 24% 1 ±10%to100%20 16% 1 ±10%to100%30 13% 1 ±10%to100%50 10% 2 ±1%to10%100 7% 2 ±1%to10%10000 0.7% 3 ±0.1%to1%
Table2.ValidSignificantFiguresinUncertainties
*Therelativeuncertaintyinthestandarddeviationisgivenbytheapproximateformula:When an explicit uncertainty estimate is made using the standard deviation, theuncertainty termindicateshowmanysignificant figuresshouldbereported in themeasured value (not the otherway around!). For example, the uncertainty in thedensitymeasurementaboveisabout0.5g/cm3,whichsuggeststhatthedigitinthetenthsplaceisuncertain,andshouldbethelastonereported.Theotherdigitsinthehundredthsplaceandbeyondareinsignificant,andshouldnotbereported:
measureddensity=8.9±0.5g/cm3 RIGHT!Anexperimentalvalueshouldberoundedtobeconsistentwiththemagnitudeofitsuncertainty. This generally means that the last significant figure in any reportedvalue should be in the same decimal place as the uncertainty, and unless a largenumberofmeasurementshavebeen taken, theuncertainty shouldbe reported toonlyonesigfig.Inmostinstances,thispracticeofroundinganexperimentalresulttobeconsistentwith the uncertainty estimate gives the samenumber of significant figures as therules discussed earlier for simple propagation of uncertainties for adding,subtracting,multiplying,anddividing.
)1(21−
=Nσ
σσ
Measurements & Uncertainty Analysis
19 Department of Physics and Astronomy
When conducting an experiment, it is important to keep inmind thatprecision isexpensive (both in termsof timeandmaterial resources).Donotwaste your timetrying toobtainaprecise resultwhenonlya roughestimate is required.The costincreases exponentially with the amount of precision required, so the potentialbenefitofthisprecisionmustbeweighedagainsttheextracost.Important Notes about sig figs when interpreting both this guide and coursematerials:
a) Onexams,homeworks,orsolutions,youmayseeaquantitycitedimprecisely(forexample,“1200m”insteadof“1200.0m”).Thisistypicallydoneforreadability,anddoesnotimplyanythingaboutuncertaintyunlessthequantityistiedexplicitlytoanexperiment.Youmayconsidersuchvaluestobeexact.
b) Incontrast,youmayseeexamplesorsolutionsthatappeartoretainanexcessivenumberofsigfigs(includinginthisdocument).Suchexamplesmayhaveskippedthefinalroundingstepforeitherclarityorreadability.Often,theseexamplesretainhigherprecisiontohelpyouascertainthatyouhaveduplicatedacalculationcorrectly,ortodistinguishthemfromincorrectsolutionsthathappentobeclosetothecorrectanswer.Again,unlesssuchexamplesaretiedexplicitlytoanexperimentaluncertainty,youmayignoresuchinconsistencies;however,whenyoupresentanansweronanassignmentorexam,youshouldminimizeyourprecisionunlessotherwisedirected.
c) Normally,whenreviewingexperimentalquestions(suchasarefoundonthepracticumorlabreports),graderscarefullyscrutinizetheuseofmorethan1sigfigwhencitinguncertainty.Oneexceptionhasalreadybeendiscussed,andisembodiedinTable2:ifenoughsampleshavebeencollectedtojustify2sigfigs,thenthismustbeexplicitlydiscussedinyouranswers.Asecondexceptioncouldbewhentheleadingsignificantfigureisa“1”–itispossibletoinduceabiaswhenroundingtoonesigfiginthiscase.Considertheexampleofrounding±1.4asopposedto±9.4,toonesigfig.Thevalueof±1.0representsa40%discrepancyfrom±1.4(verylarge),whereasthevalueof±9.0representsa4%discrepancyfrom±9.4(muchsmaller,albeitnonzero).Thebestapproachhere,asforvirtuallyallexamandassignmentexamples,istocitethehighprecisionanswer,whichindicatesthatyouknowthedetailsofthecalculation,andthenyourfinalanswerwithsuitableroundingandjustification.
Practicaltipsformeasuringandcitinguncertainty,anduncertaintycalculationexamples,canbefoundinAppendixII.
Measurements & Uncertainty Analysis
20 University of North Carolina
PropagationofUncertaintySupposewewanttodetermineaquantityf,whichdependsonxandmaybeseveralothervariablesy,z,etc.Wewanttoknowtheuncertaintyinfifwemeasurex,y,etc,withuncertaintiesσX,σY, etc.That is,wewant to findouthow theuncertainty inone set of variables (usually the independent variables) propagates to theuncertaintyinanothersetofvariables(usuallythedependentvariables).Therearetwoprimarymethodsofperformingthispropagationprocedure:
• upper-lowerbound• quadrature
Theupper-lowerboundmethodissimplerinconcept,buttendstooverestimatetheuncertainty,whilethequadraturemethodismoresophisticated(andcomplicated)butprovidesabetterstatisticalestimateoftheuncertainty.
TheUpper-LowerBoundMethodofUncertaintyPropagationThismethodusestheuncertaintyrangesofeachvariabletocalculatethemaximumand minimum values of the function. You can also think of this procedure asexaminingthebestandworstcasescenarios.Asanexample,considerthedivisionoftwovariables–acommonexampleisthecalculationofaveragespeed:
vavg =ΔxΔt
Let’ssayanexperimentisdonerepeatedlyandmeasuresadistancetravelledof30±0.5mduringatimeof2±0.1sec.Tofindtheupperandlowerboundofvavg,theuncertaintiesmustbesettocreatethe“worstcasescenario”fortheuncertaintyinvavg:
vavg−max =30 + 0.5 m2 – 0.1 sec
= 16.05 m/s vavg−min =30 – 0.5 m2 + 0.1 sec
= 14.05 m/s
Thebest(expected)valuefortheaveragespeedis30/2=15.00m/s.Theupperboundis1.05m/shigherbutthelowerboundis0.95m/slower(differentfrom1.05).Thisisatypicaloutcomewhenusingtheupper-lowerboundmethod.Theuncertaintyshouldbeexpressedasthemostconservativevalue.Thus: vavg=15.00±1.05m/s PREFERRED!Notethatitisnotcorrecttotakethedifferenceanddividebytwo: vavg=15.05±1.00m/s NOTPREFERRED!
Measurements & Uncertainty Analysis
21 Department of Physics and Astronomy
Althoughthelastresultsatisfiessymmetrybetweenthebounds,itexplicitlycalculatesanincorrectvalueofthebest-knownexpectedvalueoftheaveragespeed.Manytimes,thedifferencebetweentheso-called“preferred“and“notpreferred”approachesisnotsignificantenoughtobeanissue.Citingtheuncertaintyinthisexampleto3sigfigswouldrequireastrongjustification.Theacceptedapproach(andtheonethatshouldbefollowedintheabsenceofsuchjustification)istoroundto15.0±1.1m/s,andthenroundagainto15±1m/s.Withthisapproach,argumentsaboutwhetherthe“actualanswer”is15.00or15.05areirrelevant,becausetheexperimentdoesnotjustifysuchprecision.This asymmetry arises in the case of non-linear functions as well. For example,supposeyoumeasureanangletobeθ=25°±1°andyouneedtofindf=cosθ:fmin=cos(26°)=0.8988 f=cos(25°)=0.9063 fmax=cos(24°)=0.9135
Thedifferencesaref–fmin=0.0075andfmax–f=0.0072.Whenroundingtheuncertaintyto1sigfig,themostconservativevalueshouldbechosen(0.008).Thustheanswerwouldbecitedasf=0.906±0.008.Notethefollowingaboutthisexample:
• themaximumθisassociatedwiththeminimumf,andviceversa• guarddigitsareretainedtomakeinformedchoicesaboutthefinalanswer• althoughθwasonlymeasuredto2sigfigs,fisknownto3sigfigs
The upper-lower bound method is especially useful when the functionalrelationshipisnotclearorisincomplete.Onepracticalapplicationisforecastingtheexpected range in an expense budget. In this case, some expenses may be fixed,whileothersmaybeuncertain,andtherangeoftheseuncertaintermscouldbeusedtopredicttheupperandlowerboundsonthetotalexpense.
QuadratureThe quadrature method yields a standard uncertainty estimate (with a 68%confidence interval) and is especially useful and effective in the case of severalvariables that weight the uncertainty non-uniformly. Themethod is derivedwithseveralexamplesshownbelow.Forasingle-variablefunctionf(x),thedeviationinfcanberelatedtothedeviationinxusingcalculus:
xdxdff δδ ⎟
⎠⎞⎜
⎝⎛=
Measurements & Uncertainty Analysis
22 University of North Carolina
Takingthesquareandtheaverageyields:
22
2 xdxdff δδ ⎟
⎠⎞⎜
⎝⎛=
Usingthedefinitionofσyields:
xf dxdf σσ =
Examplesforpowerlaws: f=√x f=x2
dfdx
= 12 x
dfdx
= 2x
σ f =σ x
2 xor
σ f
f= 12σ x
x
σ f
f= 2σ x
x
Notethatbyjudiciouslynormalizing,itiseasytoexpresstherelative(fractional)uncertaintyinonevariablewithrespecttotherelative(fractional)uncertaintyinanother.Notealsothattheweightingisdirectlyrelatedtothepowerexponentofthefunction.Nowreconsiderthetrigexamplefromtheupper-lowerboundsection: f=cosθ
θ
θsin−=
ddf
θσθσ sin=f Notethatinthissituation,σθmustbeinradians.Forθ=25°±1°(0.727±0.017)
σf=|sinθ|σθ=(0.423)(π/180)=0.0074Thisisessentiallythesameresultasupper-lowerboundmethod.Thefractionaluncertaintyfollowsimmediatelyas:
θσθσ
tan=ff
Measurements & Uncertainty Analysis
23 Department of Physics and Astronomy
Thedeeperpowerofthequadraturemethodisevidentinthecasewherefdependsontwoormorevariables;thederivationabovecanberepeatedwithminormodification.Fortwovariables,f(x,y):
yyfx
xff δδδ ⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞⎜
⎝⎛∂∂
=
Thepartial derivativexf∂∂
meansdifferentiating fwith respect tox,whileholding
theothervariablesfixed.Takingthesquareandtheaverageyieldsthegeneralizedlawofpropagationofuncertaintybyquadrature:
(δ f )2 = ∂ f∂x
⎛⎝⎜
⎞⎠⎟2
(δ x)2 + ∂ f∂y
⎛⎝⎜
⎞⎠⎟
2
(δ y)2 + 2 ∂ f∂x
⎛⎝⎜
⎞⎠⎟
∂ f∂y
⎛⎝⎜
⎞⎠⎟δ xδ y (4)
Ifthemeasurementsofxandyareuncorrelated,then 0=yxδδ ,andthisreducestoitsmostcommonform:
σ f =∂ f∂x
⎛⎝⎜
⎞⎠⎟2
σ x2 + ∂ f
∂y⎛⎝⎜
⎞⎠⎟
2
σ y2
AdditionandSubtractionExample: f=x±y
∂ f∂x
= 1, ∂ f∂y
= ±1 → σ f = σ x2 +σ y
2
Whenadding(orsubtracting)independentmeasurements,theabsoluteuncertaintyofthesum(ordifference)istherootsumofsquares(RSS)oftheindividualabsoluteuncertainties.Whenaddingcorrelatedmeasurements,theuncertaintyintheresultissimplythesumoftheabsoluteuncertainties,whichisalwaysalargeruncertaintyestimatethanaddinginquadrature(RSS).Addingorsubtractingaconstantdoesnotchangetheabsoluteuncertaintyofthecalculatedvalueaslongastheconstantisanexactvalue.
Measurements & Uncertainty Analysis
24 University of North Carolina
Multiplicationexample: f=xy
∂ f∂x
= y, ∂ f∂y
= x → σ f = y2σ x2 + x2σ y
2
Dividingtheaboveequationbyf=xyyields:
22
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞⎜
⎝⎛=
yxfyxf σσσ
Divisionexample: f=x/y
∂ f∂x
= 1y
, ∂ f∂y
= − xy2 → σ f =
1y
⎛⎝⎜
⎞⎠⎟
2
σ x2 + x
y2
⎛⎝⎜
⎞⎠⎟
2
σ y2
Dividingthepreviousequationbyf=x/yyields:
22
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞⎜
⎝⎛=
yxfyxf σσσ
Note that the relative (fractional) uncertainty in f has the same form formultiplication and division: the relative uncertainty in a product or quotientdependsontherelativeuncertaintyofeachindividualterm.Asanotherexample,considerpropagatingtheuncertaintyinthespeedv=at,wheretheaccelerationisa=9.8±0.1m/s2andthetimeist=1.2±0.1s.
( ) ( ) 3.1%or 031.0029.0010.02.11.0
8.91.0 22
2222
=+=⎟⎠⎞⎜
⎝⎛+⎟
⎠⎞⎜
⎝⎛=⎟
⎠⎞⎜
⎝⎛+⎟
⎠⎞⎜
⎝⎛=
tavtav σσσ
Whenmultiplying(ordividing)independentmeasurements,therelativeuncertaintyoftheproduct(quotient)istheRSSoftheindividualrelativeuncertainties.Whenmultiplyingcorrelatedmeasurements,theuncertaintyintheresultisjustthesumoftherelativeuncertainties,whichisalwaysalargeruncertaintyestimatethanaddinginquadrature(RSS).Multiplyingordividingbyaconstantdoesnotchangetherelativeuncertaintyofthecalculatedvalue.
Measurements & Uncertainty Analysis
25 Department of Physics and Astronomy
Notice that the relative uncertainty in t (2.9%) is significantly greater than therelative uncertainty for a (1.0%), and therefore the relative uncertainty in v isessentiallythesameasfort(about3%).Graphically,theRSSislikethePythagoreantheorem:Thetotaluncertaintyisthelengthofthehypotenuseofarighttrianglewithlegsthelengthofeachuncertaintycomponent.Timesavingapproximation:“Achainisonlyasstrongasitsweakestlink.”Ifoneof theuncertainty terms ismorethan3 timesgreater thantheother terms,the root-squares formula canbe skipped, and the combineduncertainty is simplythe largest uncertainty. This shortcut can save a lot of time without losing anyaccuracyintheestimateoftheoveralluncertainty.Thequadraturemethodcanbegeneralizedtoallpowerlawsinthefollowingway:
f=xnym
σ f
f= n2 σ x
x⎛⎝⎜
⎞⎠⎟2
+m2 σ y
y⎛⎝⎜
⎞⎠⎟
2
(5)
TheproofofthisisshowninAppendixI.
Theuncertaintyestimatefromtheupper-lowerboundmethodisgenerallylargerthanthestandarduncertaintyestimatefoundfromthequadraturemethod,butbothmethodswillgiveareasonableestimateoftheuncertaintyinacalculatedvalue.Note:Onceyouhaveanunderstandingofthequadraturemethod,itisnotrequiredtoperformthepartialderivativeeverytimeyouarepresentedwithapropagationofuncertainty problem in any of the above forms! Instead, simply apply the correctformulafortherelativeuncertainties.
1.0% 3.1%
2.9%
Measurements & Uncertainty Analysis
26 University of North Carolina
CombiningandReportingUncertaintiesIn1993, the InternationalStandardsOrganization (ISO)published the firstofficialworldwideGuidetotheExpressionofUncertaintyinMeasurement.Beforethistime,uncertainty estimates were evaluated and reported according to differentconventions depending on the context of the measurement or the scientificdiscipline.Hereareafewkeypointsfromthis100-pageguide,whichcanbefoundinmodifiedformontheNISTwebsite(seeReferences).Whenreportingameasurement,themeasuredvalueshouldbereportedalongwithanestimateofthetotalcombinedstandarduncertaintyUcofthevalue.Thetotaluncertainty is found by combining the uncertainty components based on the twotypesofuncertaintyanalysis:TypeAevaluationofstandarduncertainty–methodofevaluationofuncertaintybythestatisticalanalysisofaseriesofobservations.Thismethodprimarilyincludesrandomuncertainties.TypeBevaluationofstandarduncertainty–methodofevaluationofuncertaintybymeansother thanthestatisticalanalysisofseriesofobservations.Thismethodincludes systematic uncertainties and errors and any other factors that theexperimenterbelievesareimportant.The individual uncertainty components ui should be combined using the law ofpropagation of uncertainties, commonly called the “root-sum-of-squares” or “RSS”method.Whenthisisdone,thecombinedstandarduncertaintyshouldbeequivalentto the standard deviation of the result,making this uncertainty value correspondwith a 68% confidence interval. If a wider confidence interval is desired, theuncertaintycanbemultipliedbyacoveragefactor(usuallyk=2or3)toprovideanuncertaintyrangethatisbelievedtoincludethetruevaluewithaconfidenceof95%(fork=2)or99.7%(fork=3).Ifacoveragefactorisused,thereshouldbeaclearexplanationofitsmeaningsothereisnoconfusionforreadersinterpretingthesignificanceoftheuncertaintyvalue.You should be aware that the ± uncertainty notation might be used to indicatedifferentconfidenceintervals,dependingonthescientificdisciplineorcontext.Forexample,apublicopinionpollmayreportthattheresultshaveamarginoferrorof±3%,whichmeansthatreaderscanbe95%confident(not68%confident)thatthereportedresultsareaccuratewithin3percentagepoints.Similarly,amanufacturer’stoleranceratinggenerallyassumesa95%or99%levelofconfidence.
Measurements & Uncertainty Analysis
27 Department of Physics and Astronomy
Conclusion:“Whendomeasurementsagreewitheachother?”Wenowhavetheresourcestoanswerthefundamentalscientificquestionthatwasaskedatthebeginningofthiserroranalysisdiscussion:“Doesmyresultagreewithatheoreticalpredictionorresultsfromotherexperiments?”Generally speaking, a measured result agrees with a theoretical prediction if theprediction lies within the range of experimental uncertainty. Similarly, if twomeasured values have standard uncertainty ranges that overlap, then themeasurements are said to be consistent (they agree). If the uncertainty ranges donotoverlap, then themeasurementsaresaid tobediscrepant (theydonotagree).However, you should recognize that these overlap criteria can give two oppositeanswers depending on the evaluation and confidence level of the uncertainty. Itwould be unethical to arbitrarily inflate the uncertainty range just to make ameasurementagreewithanexpectedvalue.Abetterprocedurewouldbetodiscussthe size of the difference between the measured and expected values within thecontextof theuncertainty,andtry todiscover thesourceof thediscrepancy if thedifferenceistrulysignificant.Example:A=1.2±0.4B=1.8±0.4These measurements agree within their uncertainties, despite the fact that thepercentdifferencebetweentheircentralvaluesis40%.Incontrast,iftheuncertaintyishalved(±0.2), thesesamemeasurementsdonotagree sincetheiruncertaintiesdonotoverlap:Furtherinvestigationwouldbeneededtodeterminethecauseforthediscrepancy.Perhapstheuncertaintieswereunderestimated,theremayhavebeenasystematicerror that was not considered, or there may be a true difference between thesevalues.
0 0.5 1 1.5 2 2.5
Measurements and their uncertainties
A
B
0 0.5 1 1.5 2 2.5
Measurements and their uncertainties
A
B
Measurements & Uncertainty Analysis
28 University of North Carolina
Analternativemethodfordeterminingagreementbetweenvaluesistocalculatethedifferencebetweenthevaluesdividedbytheircombinedstandarduncertainty.Thisratiogivesthenumberofstandarddeviationsseparatingthetwovalues.Ifthisratioislessthan1.0,thenitisreasonabletoconcludethatthevaluesagree.Iftheratioismore than2.0, then it is highlyunlikely (less thanabout5%probability) that thevaluesarethesame.
Examplefromabovewithu=0.4: 1.157.0
|8.12.1| =− AandBlikelyagree
Examplefromabovewithu=0.2: 1.228.0
|8.12.1| =− AandBlikelydonotagree
Notethefollowingexampleofoverlap:
Thesetwomeasurementsagree,despitetheirlargedifferenceinerrorbars.Thereisnorequirementthateithermeasurementbeincludedintheothermeasurement’serrorbars–onlythattheerrorbarsofthetwomeasurementsoverlap.
Measurements & Uncertainty Analysis
29 Department of Physics and Astronomy
MakingGraphsWhengraphsarerequiredinlaboratoryexercises,youwillbeinstructedto“plotAvs.B”(whereAandBarevariables).Byconvention,A(thedependantvariable)shouldbeplottedalongtheverticalaxis(ordinate),andB(theindependentvariable)shouldbeplottedalongthehorizontalaxis(abscissa).Graphsthatareintendedtoprovidenumericalinformationshouldbedrawnonruledgraphpaper.Useasharppencil(notapen)todrawgraphs,sothatmistakescanbecorrectedeasily.Itisacceptabletouseacomputer(seetheExceltutorialbelow)toproducegraphs.Thefollowinggraphisatypicalexampleinwhichdistancevs.timeisplottedforafreelyfallingobject.Examinethisgraphandnotethefollowingimportantrulesforgraphing:
Figure3.PlotofDistancevsTime
Title.Everygraphshouldhaveatitlethatclearlystateswhichvariablesappearontheplot.Ifthegraphisnotattachedtoanotheridentifyingreport,writeyournameandthedateontheplotforconvenientreference.Axislabels.Eachcoordinateaxisofagraphshouldbelabeledwiththewordorsymbolforthevariableplottedalongthataxisandtheunits(inparentheses)inwhichthevariableisplotted.ChoiceofScale.Scalesshouldbechoseninsuchawaythatdataareeasytoplotandeasytoread.Oncoordinatepaper,every5thand/or10thlineshouldbeselectedas
Measurements & Uncertainty Analysis
30 University of North Carolina
majordivisionlinesthatrepresentadecimalmultipleof1,2,or5(e.g.,0,l,2,0.05,20,500,etc.)–otherchoices(e.g.,0.3)makeitdifficulttoplotandalsoreaddata.Scalesshouldbemadenofinerthanthesmallestincrementonthemeasuringinstrumentfromwhichdatawereobtained.Forexample,datafromameterstick(whichhaslmmgraduations)shouldbeplottedonascalenofinerthanldivision=lmm,becauseascalefinerthan1div/mmwouldprovidenoadditionalplottingaccuracy,sincethedatafromthemeterstickareonlyaccuratetoabout0.5mm.Frequentlythescalemustbeconsiderablycoarserthanthislimit,inordertofittheentireplotontoasinglesheetofgraphpaper.Intheillustrationabove,scaleshavebeenchosentogivethegrapharoughlysquareboundary;avoidchoicesofscalethatmaketheaxesverydifferentinlength.Notethatitisnotalwaysnecessarytoincludetheorigin(‘zero’)onagraphaxis;inmanycases,onlytheportionofthescalethatcoversthedataneedbeplotted.DataPoints.Enterdatapointsonagraphbyplacingasuitablesymbol(e.g.,asmalldotwithasmallcirclearoundthedot)atthecoordinatesofthepoint.Ifmorethanonesetofdataistobeshownonasinglegraph,useothersymbols(e.g.,∆)todistinguishthedatasets.Ifdrawingbyhand,adraftingtemplateisusefulforthispurpose.Curves.Drawasimplesmoothcurvethroughthedatapoints.Thecurvewillnotnecessarilypassthroughallthepoints,butshouldpassascloseaspossibletoeachpoint,withabouthalfthepointsoneachsideofthecurve;thiscurveisintendedtoguidetheeyealongthedatapointsandtoindicatethetrendofthedata.AFrenchcurveisusefulfordrawingcurvedlinesegments.Donotconnectthedatapointsbystraight-linesegmentsinadot-to-dotfashion.Thiscurvenowindicatestheaveragetrendofthedata,andanypredicted(interpolatedorextrapolated)valuesshouldbereadfromthiscurveratherthanrevertingbacktotheoriginaldatapoints.Straight-lineGraphs.Inmanyoftheexercisesinthiscourse,youwillbeaskedtolinearizeyourexperimentalresults(plotthedatainsuchawaythatthereisalinear,orstraight-linerelationshipbetweengraphedquantities).Inthesesituations,youwillbeaskedtofitastraightlinetothedatapointsandtodeterminetheslopeandy-interceptfromthegraph.Intheexamplegivenabove,itisexpectedthatthefallingobject’sdistancevarieswithtimeaccordingtod=½gt2.Itisdifficulttotellwhetherthedataplottedinthefirstgraphaboveagreeswiththisprediction;however,ifdvs.t2isplotted,astraightlineshouldbeobtainedwithslope=½gandy-intercept=0.
Measurements & Uncertainty Analysis
31 Department of Physics and Astronomy
UsingExcelforDataAnalysisinPhysicsLabsStudentshaveanumberofsoftwareoptionsforanalyzinglabdataandgeneratinggraphswiththehelpofacomputer.Itisthestudent’sresponsibilitytoensurethatthecomputationalresultsarecorrectandconsistentwiththerequirementsstatedinthislabmanual.Anysuitablesoftwarecanbeusedtoperformtheseanalysesandgeneratetablesandplotsforlabreportsandassignments;however,sinceMicrosoftExceliswidelyavailableonallCCIlaptopsandinuniversitycomputerlabs,studentsareencouragedtousethisspreadsheetprogram.Inaddition,theremaybeassignmentsduringthesemesterthatspecificallyrequireanExcel(orplatform-equivalent)spreadsheettobesubmitted.
GettingStartedThistutorialwillleadyouthroughthestepstocreateagraphandperformlinearregressionanalysisusinganExcelspreadsheet.Thetechniquespresentedherecanbeusedtoanalyzevirtuallyanysetofdatayouwillencounterinyourphysicsstudio.Tobegin,openExcel.Ablankworksheetshouldappear.EnterthesampledataandcolumnheadingsshownbelowintocellsA1throughD6.Savethefiletoadiskortoyourpersonalfilespaceonthecampusnetwork.
Time(sec) Distance(m) Time± Distance±0.64 1.15 0.1 0.31.1 2.35 0.2 0.51.95 3.35 0.3 0.42.45 4.46 0.4 0.72.85 5.65 0.3 0.5
Notethattheuncertaintiesforthetimeanddistance(denoted±)havebeenincluded.Thesearenotnecessaryforabasicplot,butthestudiolabreportsandassignmentsrequireanuncertaintyanalysis,soyoushouldgetintothehabitofincludingthem.
Measurements & Uncertainty Analysis
32 University of North Carolina
CreatingandEditingaGraphNote:theseinstructionsmaynotbepreciselycorrectfordifferentversionsofExcelorondifferentplatforms;however,theseinstructionsshouldberoughlycorrectforallmodernversionsofExcelonallplatforms.Otherspreadsheetprogramsshouldhavesimilarfeatures.Wesuggestyouuseon-lineresourcesorconsultwithyourclassmatesoryourTAforspecificquestionsorissues.Youwillbecreatingagraphofthesedatawhosefinishedformlookslikethis:
Followthesestepstoaccomplishthis:1:Useyourmousetoselectallthecellsthatcontainthedatathatyouwanttograph(inthisexample,columnsAandB).Tographthesedata,selectChartonthetoolbar.2:ClickonScatterplotsandchooseXY(Scatter)orMarkedScatterwithnolines.Adefaultplotshouldappearinthespreadsheet,anditshouldbebothmoveableandresizable.3:UsingtheChartLayouttool,experimentwithsettingthetitle,axes,axistitles,gridlinesandlegends.Ataminimum,werequirethattheplotbetitledandthatthex-andy-axesaredescriptivelylabeledwithunits.Westronglysuggestthatallgridlinesandthelegendberemovedforclarity.
y=1.8885x-0.0035R²=0.9767
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3 3.5
Distance(meters)
Time(sec)
SpeedfromDistancevsTimeData
Measurements & Uncertainty Analysis
33 Department of Physics and Astronomy
Mostgraphfeaturescanbemodifiedbydouble-clickingonthefeatureyouwanttochange.Youcanalsoright-clickonafeaturetogetamenu.Trychangingthecoloroftheplotarea,thenumbersontheaxes,andtheappearanceofthedatapoints.Itisrecommendedthatyoualwaysformatthebackgroundareatowhiteusingthe“Automatic”option.
AddingErrorBarsRight-clickonadatumpointandchooseFormatDataSeries...andselectErrorBars.ClickontheappropriateErrorBartab(XorY)andchooseBothunderDisplayandCapfortheEndstyle.Fixedvaluesorpercentagescanbeset,forexample,butifyouhaveseparatecolumnsofuncertaintyvaluesforeachdatum,asshownabove,thenselectCustomtospecifythevalues.Inthesubsequentcustomerrorbarwindow,selectthepositiveerrorvaluefieldandthenclickanddraginthecorrespondingExcelcolumnofuncertainties.RepeatforthenegativevalueandclickOK.Yourcustomerrorbarswillthenbeapplied.Repeatfortheotheraxis.Notethatifyoucreateseparatecolumnsforthepositiveandnegativeerrorbars,theycanbesetindependently.Alsonotethaterrorbarsmaynotbevisibleiftheyaresmallerthanthesizeofthedatumpointontheplot.
AddingaTrendlineTheprimaryreasonforgraphingdataistoexaminethemathematicalrelationshipbetweenthetwovariablesplottedonthex-andy-axes.Toaddatrendlineanddisplayitscorrespondingequation,right-clickonanydatumpointandAddTrendline.Choosethegraphshapethatbestfitsyourdataandisconsistentwithyourtheoreticalprediction(usuallyLinear).Clickonthe"Options"tabandchecktheboxesfor"Displayequationonchart"and“DisplayR-squaredvalueonchart.”AgoodfitisindicatedbyanR2valuecloseto1.Caution:Whensearchingforamathematicalmodelthatexplainsyourdata,itisveryeasytousethetrendlinetooltoproducenonsense.Thistoolshouldbeusedtofindthesimplestmathematicalmodelthatexplainstherelationshipbetweenthetwovariablesyouaregraphing.Lookattheequationandshapeofthetrendlinecritically:
• Doesitmakesenseintermsofthephysicalprincipleyouareinvestigating?• Isthisthebestpossibleexplanationfortherelationshipbetweenthetwo
variables?Usethesimplestequationthatpassesthroughmostoftheerrorbarsonyourgraph.Youmayneedtotryacoupleoftrendlinesbeforeyougetthemostappropriateone.Toclearatrendline,right-clickonitsregressionlineandselectClear.
Measurements & Uncertainty Analysis
34 University of North Carolina
DeterminingtheUncertaintyinSlopeandY-interceptTheR2valueindicatesthequalityoftheleast-squaresfit,butthisvaluedoesnotgivetheerrorintheslopedirectly.Giventhebestfitliney=mx+b,withndatapoints,thestandarderror(uncertainty)intheslopemcanbedeterminedfromtheR2valuebyusingthefollowingformula:
21)/1( 2
−−=
nRmmσ (6)
Likewise,theuncertaintyinthey-interceptbis:
nx
mb∑=
2
σσ (7)
ThesevaluescanbecomputeddirectlyinExcelorbyusingacalculator.Forthissamplesetofdata,σm=0.1684m/s,andσb=0.333m.NotethatavalueofR2ofexactly1leadstoslopeandinterceptuncertaintiesofzero.CarefullyexamtheExcelR2value–althoughitmaydisplayasexactly1,itlikelyisnotexactly1.Ifyourvalueisindeedexactly1,itindicatesanerrorinhowyouhaveplottedyourdata.Theuncertaintyintheslopeandy-interceptcanalsobefoundbyusingtheLINESTfunctioninExcel.UsingthisfunctionissomewhattediousandisbestunderstoodfromtheHelpfeatureinExcel.
InterpretingtheResultsOncearegressionlinehasbeenfound,theequationmustbeinterpretedintermsofthecontextofthesituationbeinganalyzed.Thissampledatasetcamefromacartmovingalongatrack.Wecanseethatthecartwasmovingatnearlyaconstantspeedsincethedatapointstendtolieinastraightlineanddonotcurveupordown.Thespeedofthecartissimplytheslopeoftheregressionline,anditsuncertaintyisfoundfromtheequationabove:v=1.9±0.2m/s.(Note:Ifwehadplottedagraphoftimeversusdistance,thenthespeedwouldbetheinverseoftheslope:v=1/m)They-interceptgivesustheinitialpositionofthecart:x0=–0.0035±0.33m,whichisessentiallyzero.
FinalStep–CopyingDataandGraphsintoaWordDocumentCopyyourplotsanddatatablefromExceltoWord.Justselectthegraph(orcells)andusetheEditmenuorkeyboardshortcutstocopyandpaste.
Measurements & Uncertainty Analysis
35 Department of Physics and Astronomy
UsingLINESTinExcelLINESTisanalternativelinearleastsquarefittingfunctioninExcel.TheresultsoftheLINESTanalysisarevirtuallyidenticaltothelineartrendlineanalysisdescribedabove;however,LINESTprovidesasingle-stepcalculationofboththeslopeandinterceptuncertainties,insteadofthemulti-stepproceduredescribedabove.
• Startwithatablefortimeandvelocity(right).
• TheLINESTfunctionreturnsseveraloutputs;toprepare,selecta2by5arraybelowthedata,asshown.
• Note,velocitywasmistakenlylabeledashavingunitsofm/secinthetabletotheright.
• UndertheInsertmenu,selectionFunction,thenStatistical,andfinallyLINESTasshownbelow(andhitOK).
Measurements & Uncertainty Analysis
36 University of North Carolina
Selectthey-valuesandx-valuesfromthetableandwrite‘TRUE’forthelasttwooptions(below).Again,pressOK.
Measurements & Uncertainty Analysis
37 Department of Physics and Astronomy
TheLINESTfunctionisanarrayfunction;therefore,youmusttellExcelyouaredonewiththearray.Highlighttheformulaintheformulabarasshownbelow.PressCtrl+ShiftsimultaneouslywithEnter(Macusers,pressCommand-EnterorShift-Control-EnterdependingonyourversionofExcel)
Measurements & Uncertainty Analysis
38 University of North Carolina
Theresultisthatthearrayofblankcellsthatwasselectedpreviouslyisnowfilledwithdata(althoughthenewdataisn’tlabeled).Seetheimagebelowforthelabelsthatcanbeaddedafterthefact(e.g.,“Slope,”“R^2Value,”etc):
References:Baird,D.C.Experimentation:AnIntroductiontoMeasurementTheoryandExperimentDesign,3rd.ed.PrenticeHall:EnglewoodCliffs,1995.Bevington,PhillipandRobinson,D.DataReductionandErrorAnalysisforthePhysicalSciences,2nd.ed.McGraw-Hill:NewYork,1991.ISO.GuidetotheExpressionofUncertaintyinMeasurement.InternationalOrganizationforStandardization(ISO)andtheInternationalCommitteeonWeightsandMeasures(CIPM):Switzerland,1993.Lichten,William.DataandErrorAnalysis.,2nd.ed.PrenticeHall:UpperSaddleRiver,NJ,1999.NIST.EssentialsofExpressingMeasurementUncertainty.http://physics.nist.gov/cuu/Uncertainty/Taylor,John.AnIntroductiontoErrorAnalysis,2nd.ed.UniversityScienceBooks:Sausalito,1997.
Measurements & Uncertainty Analysis
39 Department of Physics and Astronomy
AppendixI:PropagationofUncertaintybyQuadratureConsideraquantityftobecalculatedbymultiplyingtwomeasuredquantitiesxandywhoseuncertaintiesareσxandσy,respectively.Thequestionishowtopropagatethoseuncertaintiestothecalculatedquantityf.Fromthechainruleofcalculus,thechangeinfduetochangesinxandyis:
δ f = ∂ f∂x
δ x + ∂ f∂y
δ y
Squaringandaveragingyields:
δ f( )2 = ∂ f∂x
⎛⎝⎜
⎞⎠⎟2
δ x( )2 + ∂ f∂y
⎛⎝⎜
⎞⎠⎟
2
δ y( )2 + 2 ∂ f∂x
∂ f∂y
δ xδ y
Foruncorrelatedmeasurements,δ xδ y iszero.Considertheaveragesquarechange
inquantitiestobetheuncertaintyineachofx,y,andf;thatis, δ f( )2 =σ f2 ,etc.Then:
σ f =∂ f∂x
⎛⎝⎜
⎞⎠⎟2
σ x2 + ∂ f
∂y⎛⎝⎜
⎞⎠⎟
2
σ y2
To generalize it to arbitrary powers of x and y, consider the function f = xnym;substituting this into the last equation and dividing by f yields the relativeuncertainty:
σ f
f= ∂ f
∂x⎛⎝⎜
⎞⎠⎟2 σ x
2
xnym( )2+ ∂ f
∂y⎛⎝⎜
⎞⎠⎟
2 σ y2
xnym( )2
Thepartialderivativesare∂ f∂x
= nxn−1ym and ∂ f∂y
= mxnym−1 .Substitutingtheseyields:
σ f
f= nxn−1ym( )2 σ x
2
xnym( )2+ mxnym−1( )2 σ y
2
xnym( )2
Thisexpressionlookscomplicatedbutitsimplifiestosomethingrathersimple:
σ f
f= n2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+m2 σ y
y⎛
⎝⎜⎞
⎠⎟
2
Measurements & Uncertainty Analysis
40 University of North Carolina
Theresultisthattherelative(fractional)uncertaintyinfistherootofthesquaredsum(RSS)ofindividualuncertaintiesinxandy.Examplesinthebodyofthisdocumentinclude:
f=xy σ f
f= (1)2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+ (1)2σ y
y⎛
⎝⎜⎞
⎠⎟
2
f=x/y σ f
f= (1)2 σ x
x⎛⎝⎜
⎞⎠⎟
2
+ (−1)2σ y
y⎛
⎝⎜⎞
⎠⎟
2
f=xy2 σ f
f= σ x
x⎛⎝⎜
⎞⎠⎟
2
+ 4σ y
y⎛
⎝⎜⎞
⎠⎟
2
Note that the result formultiplication and division is the same (division is just apower lawwith anegative exponent).Alsonote that variables that appearwith ahigherpowerareweightedmoreheavilyinthepropagation.For some functions, especiallynon-linear trig functions, youmayhave toperformthe derivatives to find how the uncertainty propagates; however, for manyfunctions,performingthederivativeseachtime isnotrequired–merelyapply theequationhighlightedinyellowabove.
Measurements & Uncertainty Analysis
41 Department of Physics and Astronomy
AppendixII.PracticalTipsforMeasuringandCitingUncertaintyI.InstrumentPrecisionandAccuracyInstrumentprecisionisthesmallestreadinganinstrumentcanprovide.Forsomeinstruments,thisprecisionmaybeunambiguous.Forexample,adigitalbalancewhosereadoutisdisplayedas__._ghasaninstrumentprecisionof0.1g.Incontrast,ameterstick’sinstrumentprecisionisopentointerpretation.Ifthefinestmarkonthemeterstickis1mm,itcouldbeclaimedthattheinstrumentprecisionis0.001m,or1mm;however,adifferentobservermightclaimthatitispossibletointerpolatebetweenthefinestmarks,andthereforemightclaimthattheinstrumentprecisionis0.0005m,or0.5mm.Inallcases,thechoiceofinstrumentprecisionandthereasoningshouldbeincludedinanyreportthatincorporatesthatinstrument’suse.Incontrasttotheinstrumentprecision,theinstrumentaccuracyreferstotheabilitytocorrectlymeasurethevalueofaknownstandard.Ifadigitalbalancewithameasurementprecisionof0.1gisusedtomeasureaweightindependentlyknowntohaveamassofexactly500.0g,andtheresultis495.3g,theaccuracyisnobetterthanabout5g,eventhoughtheprecisionisagain0.1g.Severalstrategiesforaddressingthisdiscrepancyarelistedbelow:a)Citethevalueas495.3±4.7g.Unacceptable:citingtheuncertaintyto2sigfigsisnotwarrantedforlessthan50measurements(Table2).b)Citethevalueas495.3±5g.Unacceptable:citingthemeasurementtoahigherprecisionthantheuncertaintyisneverwarranted.c)Citethevalueas495±5g.Acceptablebutnotoptimal:themeasurementisclearlyincorrectbecauseitisknownthattheweightisexactly500g.Ifonlyasinglemeasurementispossible,citingasshownisallowed,butshouldbeaccompaniedbyadiscussionabouttheknowndiscrepancyandtheintroductionofasystematicerror.d)Citethevalueas495.3±0.1g.Allowedbutonlyifaknowncalibrationstandardorotherinformationabouttheaccuracyisunavailable:thecalibrationstatusoftheinstrumentshouldbeconsulted.Ifthisisnotpossible,itisnotunreasonabletoassumethattheinstrumenthasanimpliedaccuracythatmatchesitsprecision,asshowninthisexample.Itemd)isthemostcommonscenarioinPhysics118activities.Anysuchassumptionaboutsubstitutingprecisionforaccuracyshouldbeincludedinadiscussion,withanacknowledgmentthatasystematicerrormayhavebeenintroduced.Therecommendedproceduretoaddressthisissueistoacquiremultiple“identical”samplesofthe“known”weight,andcomputeanaveragevalueandstandarddeviation.Thiswouldresolvethequestionofwhethercitingtheprecisionasan
Measurements & Uncertainty Analysis
42 University of North Carolina
uncertaintyiswarranted;however,itwouldnotresolvetheaccuracyquestion.Thebestapproachtotheaccuracyquestionisacalculationofthestandarderror,butitisextremelyimportanttoremoveanysystematicuncertainties.Anymeasurementsperformedwithoutknowncalibrationstandardsavailableareopentosystematicuncertainties!II.StackingAssumeyouareaskedtomeasurethemassofatypicalpenny(accordingtotheUSMint,currentlymadepennieshaveanominalmassof2.5g)withascalewhoseaccuracyisknowntobe±0.2gbutreadstoaprecisionof0.1g.Onecanimmediatelydiscardthechoiceof0.1gastheaccuracy.Measuringonepennymightyieldameasurementof2.4±0.2g,andthiswouldbetheonlymeasurementpossibleforthatonepenny.Likewise,anotherpennymightyieldameasurementof2.5±0.2g.Isthereawaytogetamoreprecisemeasurement?Inthiscase,yes,becauseyouareaskedtofindthemassofatypicalpenny.Bystackingpenniesandmeasuringmorethanoneofthematthesametime,dividingbythenumberofpenniesmeasuredcanprovideamorepreciseanswer.Forexample,assumethatyoumeasurethemassvalueoffivepenniesseparately,withtheseresults(allwithanaccuracyof±0.2g):2.4,2.4,2.5,2.4,2.6.Therelativeuncertaintyofeachmeasurementisabout8%.Furtherassumethatwhenyoumeasureallfiveatthesametime,thevalueis12.3±0.2g,yieldingarelativeuncertaintyofabout2%forthestack.Themeanvalueforatypicalpennyistherefore(retainingguarddigits)2.460g.Butwhatdoweassignastheuncertainty?Onemightarguethattheuncertaintyisstill0.2g;however,wecanplausiblyrewritethesumas:sum=(2.4±0.04)+(2.4±0.04)+(2.5±0.04)+(2.4±0.04)+(2.6±0.04)=12.3±0.2gThisisdemonstrablythesameasthestackedvalueof12.3±0.2g,andsuggeststhat,basedontheupper-lowerboundmethod,theuncertaintycanalsobedividedbyfive.Therefore,asageneralrule,wecanreasonablydivideboththestackedvalueanditsuncertaintybyN,andassertthevalueforatypicalpennyas2.46±0.04g.NotethatasperTable2,theuncertaintyiscitedtoonlyonesigfig,becauseonlyfivemeasurementsweremade.Assumeinsteadthattheaccuracyofthescaleisunknown;aworkingpropositionisthatthescale’smechanismwillcorrectlyreportthatanobjectof0.1gissomewherebetween0.05gand0.15g.Thereadingof12.3gforall5penniestogethercouldthereforehavevaluesfallingbetween12.25and12.35g.Dividingthisby5yieldsanominalvalueof2.460gwithahighandlowof2.470gand2.450g,respectively.Thevalueof2.460±0.010gcouldthenbecitedas2.46±0.01gtoonesigfig;however,althoughtheresultsseems“better,”thereisnoconfidencethatthisvalueiscorrect,Inmeasurementcasesweretheaccuracyofthemeasuringdeviceisnotknown,theresultshouldexplicitlycitethisdiscrepancy.
Measurements & Uncertainty Analysis
43 Department of Physics and Astronomy
III.ApproachtoUncertaintyWhenAccuracyisCitedasaPercentage.Instrumentaccuracymaybedescribedintermsofapercentage,suchas“±0.1%.”Forhighqualityinstruments,thisusuallymeans0.1%ofthefull-scale(FS)reading,acrosstheentiremeasurementrangeoftheinstrument.Forexample,ifadigitalbalancecanmeasureamaximumvalueof199.9g,theaccuracywouldbe0.1%ofFS(200g),or±2g(worsethantheinstrumentprecisionis0.1g).Inthiscase,followtheguidelinesinExampleI.However,theoriginofthepercentageuncertaintymaybeambiguous,andcouldimplyapercentageofthecurrentreading.Iftheinterpretationisambiguous,itcanbeevaluatedbycomparingtotheprecisionoftheinstrument.Asanexample,assumeitisclaimedthattheaccuracyofthebalanceis0.1%foranymass,andthatthemeasurementofoneobjectreads53.4g.Thepercentaccuracyof0.1%impliesandaccuracyof±0.05g,whichisbetterthantheinstrumentprecisionof0.1g!Thisinconsistencysuggeststhata“stackofone”itemisinappropriate.Stackingshouldbeused,butitcannotbeappliedarbitrarily;astackoftwowouldnotunambiguouslyresolvethequestion,butastackofthreewould.Thegeneralapproachinthiscaseistostackenoughobjectssothatthepercentageaccuracyexceedstheprecisionofthemeasurement.TheapproachinExampleIcanthenbeused;however,underallcircumstances,anyassertionofanuncertaintybeyond1sigfigisonlyjustifiedforN=50orabove(Table2).IV.UseofContainersAssumeyouaregivenaquantityofwaterandyouareaskedtomeasureitsweightusingascale.Youneedacontainer,butthecontaineritselfhasaweight.Theprocedureistoweighthecontainerbyitself,andthenthecontainerandthewatertogether,andsubtracttogettheweightofthewater.However,inthiscase,theuncertaintiesofthetwomeasurementsmustbeadded.Thisshouldbeself-evidentfromanupper-lowerboundperspective;thequadratureanalysisisamoreformalstatementofthesameresult(seeQuadraturesection).
Measurements & Uncertainty Analysis
44 University of North Carolina
V.ExplainingDiscrepanciesAsimplependulumisknowntohaveaperiodofoscillationT =1.55s.StudentAusesadigitalstopwatchtomeasurethetotaltimefor5oscillationsandcalculatesanaverageperiodT =1.25s.StudentBusesananalogwristwatchandthesameproceduretocalculateanaverageperiodforthe5oscillationsandfindsT =1.6s.
• Whichstudentmadethemoreaccuratemeasurement?ThemeasurementmadebyStudentBisclosertotheknownvalueandisthereforemoreaccurate.
• Whichmeasurementismoreprecise?ThemeasurementmadebyStudentAisreportedwithmoredigitsandisthereforemoreprecise.
• Whatisthemostlikelysourceofthediscrepancybetweentheresults?Althoughthedifferenceinperiodmeasurementsisonly0.3s,theoriginaltimingmeasurementsmayhavedifferedby5timesthisamountsincewearetoldthattheaverageperiodwascalculatedfromthetotaltimefor5oscillations.Soeventhoughreactiontime(typically~0.2s)isalikelysourceofsystematicerror,thiswouldnotexplainthediscrepancyofapproximately1.5s,oraboutoneperiod.Therefore,themostlikelysourceofthediscrepancyisthatStudentAmistakenlymeasuredonly4oscillationsinsteadof5.Thisexampleshowsthatameasurementwithgreaterprecisionisnotalwaysmoreaccurate.
VI.PropagatingUncertaintyandReportingValuesAstudentusesaprotractortomeasureanangleofθ=85° ±1°.Whatshouldshereportforsinθ?
sin(θ)=0.996±0.002.Theuncertaintycanbedeterminedfromeithertheupper/lowerboundmethodorpropagationofuncertaintyusingpartialderivatives.Bothmethodsyieldthesameresultwhenroundedtoonesignificantfigure.Notethatitwouldbeinappropriatetoroundthisfurtherto1.00±0.00,becauseitimpliesanuncertaintyofzero.
Measurements & Uncertainty Analysis
45 Department of Physics and Astronomy
VII.OutliersAgroupofstudentsaretoldtouseametersticktofindthelengthofahallway.Theytake6independentmeasurementsasfollows:3.314m,3.225m,3.332m,3.875m,3.374m,3.285m.Howshouldtheyreporttheirfindings,andwhy?
Nodatashouldbeexcludedwhenpresentingtherawnumbers.Therefore,allvaluesshouldappearinanydatatables,andallvaluesshouldappearonplots.Anexampleisshownbelowforthesedata:
Themeasurementof3.875misanoutlier(10standarddeviationsfromtheothervalues)andismostlikelyamis-measurement(see2ndbulletpointbelow).Inthiscase,thebestestimatewouldbetheaverageandstandarderrorforthe5“good”values(3.31±0.02m).Notetwoaspectsofthisapproach:
• Ifasystematicerrorissuspected,thenusingthestandarderrormayalsobesuspect.Thestandarddeviationis0.06m;therefore,amoreconservativeanswerwouldbe3.3±0.1m.
• Becarefulbeforethrowingoutdata–NobelPrizeshavebeenwonbecauseofoutliers(althoughprobablynotin118)!Alsothinkabouttheexperimentcarefully:inthisexample,thereislikelytobeamisusedmeterstick,butinthecaseofgoldprospecting,onlytheoutliersarekept,andeverythingelseisthrownout!
VIII.LimitingFactors AVerniercaliper(aninstrumentthatcanroutinelymeasurelengthsto0.001inch,orlessthan0.05mm),isusedtomeasuretheradiusofatennisball.ThevalueispresentedasR=3.2±0.1cm(halfofD=6.4±0.2cm).Whyisthisresultnotmoreprecise?
Theuncertaintyofthismeasurementisdeterminednotbytheresolutionofthecaliper,butbytheimprecisedefinitionoftheball'sdiameter(it’sfuzzy!).
Measurements & Uncertainty Analysis
46 University of North Carolina
IX.UncertaintyPropagationIEquipmentisusedtomeasuretheaccelerationofaglideronaninclinedairtrackasaccuratelyaspossible.Thedistancebetweentwophotogatesismeasuredveryaccuratelyasd=1.500±0.005m.Theexperimentisrun5timeswiththegliderstartingfromrestatthefirstphotogate,andthephotogatetimesarerecordedast(insec)=1.12,1.06,1.19,1.15,1.08.Whatisthebestestimateoftheaccelerationoftheglider?
Theaveragetimeis1.12swithastandarderrorof0.02s.Therelationshipbetweenthedistanceandaccelerationisd=½at2;thebestestimatefortheaccelerationisthereforea=2.392m/s2(retainingguarddigits).Thequadratureresultfortheuncertaintyintheaccelerationis
σ a
a=
σ d
d
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
+4 σ t
t
⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
Therelativeuncertaintyinthetimeismorethan3xtherelativeuncertaintyintheposition,sotheuncertaintyintimedominates.Therefore:
σ a =2a
σ t
t
⎛
⎝⎜⎜
⎞
⎠⎟⎟=2(2.392m/s2)
0.021.12 =0.085m/s
2
Thefinalresultisa=2.39±0.09m/s2,or2.4±0.1m/s2.
Measurements & Uncertainty Analysis
47 Department of Physics and Astronomy
X.UncertaintyPropagationIIAstudentperformsasimpleexperimenttofindtheaverageaccelerationofafallingobject.Hedropsabaseballfromabuildingandusesastringandtapemeasuretomeasuretheheightfromwhichtheballwasdropped.Heusesastopwatchtofindanaveragetimeoffallfor3trialsfromthesameheightandreportsthefollowingdata:h=6.75±0.33m,t=1.14±0.04s.Determinetheaverageacceleration,andcommentontheaccuracyoftheresult(doyouthinkanyerrorsweremade?).Whatsuggestionswouldyoumaketoimprovetheresult?
Calculatingtheaccelerationasforthepreviousexampleyieldsa=2h/t2=10.387m/s2.Inthiscase,neitherrelativeuncertaintyintimeordistancedominates,sobothshouldbeincludedinaquadraturecalculation:
σ a = a
σ h
h⎛
⎝⎜⎞
⎠⎟
2
+4 σ t
t⎛
⎝⎜⎞
⎠⎟
2
= (10.387m/s2) 0.336.75
⎛⎝⎜
⎞⎠⎟
2
+4 0.041.14
⎛⎝⎜
⎞⎠⎟
2
=0.888m/s2
Thefinalanswerwouldbewrittenas10.4±0.9m/s2.Notethatthisanswerishigherthanthevalueforginavacuum.Althoughtheuncertaintyrangeincludesthevalue9.8m/s2,wewouldexpectthevalueforaccelerationtobesomewhatorwellbelowgbecauseoftheeffectofdrag.Therefore,thisresultshouldbeviewedwithsomesuspicion.Suggestionsincludedoingmoretrialsandimprovingtheaccuracyofeachmeasurement.Accuracyinthetimeiswherethemostgainscanbemade,becausetherelativeuncertaintyofthetimeisweightedtwicethatoftherelativeuncertaintyofthedistance.
Measurements & Uncertainty Analysis
48 University of North Carolina
XI.UncertaintyfromLinearRegressionInaninvestigationtoempiricallydeterminethevalueofπ,astudentmeasuresthecircumferenceanddiameterof5circlesofvaryingsize,andusesExceltomakealinearplotofcircumferenceversusdiameter(bothinunitsofmeters).Alinearregressionfityieldstheresultofy=3.1527x–0.0502,withR2=0.9967forthe5datapointsplotted.Howshouldthisstudentreportthefinalresult?DoestheempiricalratioofC/Dagreewiththeacceptedvalueofπ?
ThetheoreticalresultisC=πD+0.Byinspection,theexperimentalvalueofπisgivenbytheslopeoftheregression,or3.1527.Theuncertaintyinthisvalueis:
σ m =m
(1/R2)−1n−2 =(3.1527) (1/0.9967)−1
5−2 =0.1047
Thefinalexperimentalvalueforπisthen3.15±0.10,or3.2±0.1.Notethattheexperimentalinterceptisnotzero,butnotenoughinformationhasbeengiventocalculatetheintercept’suncertainty.Thevalueforπcanthereforeonlybetrustedtotheextentthat–0.0502(theintercept)isexperimentallyequivalenttozero.Therefore,theuncertaintyintheinterceptmustbelargerthan0.0502;ifnot,thereismorediscussiontobewritten!